Character tables

1x1:	C₁
2x2:	C₂
3x3:	C₃ S₃
4x4:	C₄ C₂² D₅ A₄
5x5:	C₅ D₄ Q₈ D₇ F₅ C₇⋊C₃ S₄ A₅
6x6:	C₆ Dic₃ D₆ D₉ C₃⋊S₃ C₃²⋊C₄ PSU₃(𝔽₂) GL₃(𝔽₂)
7x7:	C₇ D₈ SD₁₆ Q₁₆ D₁₁ SL₂(𝔽₃) C₁₃⋊C₃ F₇ C₁₃⋊C₄ C₁₁⋊C₅ S₅ A₆
8x8:	C₈ C₂×C₄ C₂³ Dic₅ D₁₀ C₂×A₄ D₁₃ C₄²⋊C₃ CSU₂(𝔽₃) GL₂(𝔽₃) C₂²⋊A₄ F₈ C₁₇⋊C₄ C₁₃⋊C₆ C₂⁴⋊C₅ AΓL₁(𝔽₈) C₅²⋊Q₈ C₅²⋊Dic₃
9x9:	C₉ C₃² C₃×S₃ Dic₆ D₁₂ C₃⋊D₄ D₁₅ S₃² C₁₉⋊C₃ C₃⋊F₅ C₃.S₄ F₉ S₃≀C₂ C₃⋊S₄ C₁₉⋊C₆ SL₂(𝔽₅) AΓL₁(𝔽₉) C₄²⋊A₄ C₄².A₄ PGL₂(𝔽₇)
10x10:	C₁₀ C₂²⋊C₄ C₄⋊C₄ M₄(2) C₂×D₄ C₂×Q₈ C₄○D₄ Dic₇ D₁₄ D₁₇ C₅⋊C₈ C₂×F₅ C₂×C₇⋊C₃ A₄⋊C₄ C₂×S₄ C₃²⋊C₆ C₉⋊C₆ He₃⋊C₂ C₄²⋊S₃ C₂⁴⋊C₆ C₄²⋊C₆ C₂³.A₄ C₂²⋊S₄ C₂₅⋊C₄ C₅⋊F₅ C₅²⋊C₄ C₂×A₅ C₁₇⋊C₈ C₅²⋊C₆ C₂⁴⋊D₅ ASL₂(𝔽₃) C₂³.F₈

1x1 character tables

Character table of C₁

C₁: Trivial group C1 ID 1,1

class	1
size	1
ρ₁	1	trivial faithful

2x2 character tables

Character table of C₂

C₂: Cyclic group C2 ID 2,1

class	1	2
size	1	1
ρ₁	1	1	trivial
ρ₂	1	-1	linear of order 2 faithful

3x3 character tables

Character table of C₃

C₃: Cyclic group; = A₃ = triangle rotations C3 ID 3,1

class	1	3A	3B
size	1	1	1
ρ₁	1	1	1	trivial
ρ₂	1	ζ₃	ζ₃²	linear of order 3 faithful
ρ₃	1	ζ₃²	ζ₃	linear of order 3 faithful

Character table of S₃

S₃: Symmetric group on 3 letters; = D₃ = GL₂(𝔽₂) = triangle symmetries = 1^st non-abelian group S3 ID 6,1

class	1	2	3
size	1	3	2
ρ₁	1	1	1	trivial
ρ₂	1	-1	1	linear of order 2
ρ₃	2	0	-1	orthogonal faithful

4x4 character tables

Character table of C₄

C₄: Cyclic group; = square rotations C4 ID 4,1

class	1	2	4A	4B
size	1	1	1	1
ρ₁	1	1	1	1	trivial
ρ₂	1	1	-1	-1	linear of order 2
ρ₃	1	-1	-i	i	linear of order 4 faithful
ρ₄	1	-1	i	-i	linear of order 4 faithful

Character table of C₂²

C₂²: Klein 4-group V₄ = elementary abelian group of type [2,2]; = rectangle symmetries C2^2 ID 4,2

class	1	2A	2B	2C
size	1	1	1	1
ρ₁	1	1	1	1	trivial
ρ₂	1	1	-1	-1	linear of order 2
ρ₃	1	-1	1	-1	linear of order 2
ρ₄	1	-1	-1	1	linear of order 2

Character table of D₅

D₅: Dihedral group; = pentagon symmetries D5 ID 10,1

class	1	2	5A	5B
size	1	5	2	2
ρ₁	1	1	1	1	trivial
ρ₂	1	-1	1	1	linear of order 2
ρ₃	2	0	^-1-√5/₂	^-1+√5/₂	orthogonal faithful
ρ₄	2	0	^-1+√5/₂	^-1-√5/₂	orthogonal faithful

Character table of A₄

A₄: Alternating group on 4 letters; = PSL₂(𝔽₃) = L₂(3) = tetrahedron rotations A4 ID 12,3

class	1	2	3A	3B
size	1	3	4	4
ρ₁	1	1	1	1	trivial
ρ₂	1	1	ζ₃	ζ₃²	linear of order 3
ρ₃	1	1	ζ₃²	ζ₃	linear of order 3
ρ₄	3	-1	0	0	orthogonal faithful

5x5 character tables

Character table of C₅

C₅: Cyclic group; = pentagon rotations C5 ID 5,1

class	1	5A	5B	5C	5D
size	1	1	1	1	1
ρ₁	1	1	1	1	1	trivial
ρ₂	1	ζ₅	ζ₅²	ζ₅³	ζ₅⁴	linear of order 5 faithful
ρ₃	1	ζ₅²	ζ₅⁴	ζ₅	ζ₅³	linear of order 5 faithful
ρ₄	1	ζ₅³	ζ₅	ζ₅⁴	ζ₅²	linear of order 5 faithful
ρ₅	1	ζ₅⁴	ζ₅³	ζ₅²	ζ₅	linear of order 5 faithful

Character table of D₄

D₄: Dihedral group; = He₂ = AΣL₁(𝔽₄) = 2₊¹⁺² = square symmetries D4 ID 8,3

class	1	2A	2B	2C	4
size	1	1	2	2	2
ρ₁	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	-1	linear of order 2
ρ₃	1	1	1	-1	-1	linear of order 2
ρ₄	1	1	-1	-1	1	linear of order 2
ρ₅	2	-2	0	0	0	orthogonal faithful

Character table of Q₈

Q₈: Quaternion group; = C₄ .C₂ = Dic₂ = 2_-¹⁺² Q8 ID 8,4

class	1	2	4A	4B	4C
size	1	1	2	2	2
ρ₁	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	-1	linear of order 2
ρ₃	1	1	1	-1	-1	linear of order 2
ρ₄	1	1	-1	-1	1	linear of order 2
ρ₅	2	-2	0	0	0	symplectic faithful, Schur index 2

Character table of D₇

D₇: Dihedral group D7 ID 14,1

class	1	2	7A	7B	7C
size	1	7	2	2	2
ρ₁	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	linear of order 2
ρ₃	2	0	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	orthogonal faithful
ρ₄	2	0	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	orthogonal faithful
ρ₅	2	0	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	orthogonal faithful

Character table of F₅

F₅: Frobenius group; = C₅ ⋊C₄ = AGL₁(𝔽₅) = Aut(D₅) = Hol(C₅) = Sz(2) F5 ID 20,3

class	1	2	4A	4B	5
size	1	5	5	5	4
ρ₁	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	linear of order 2
ρ₃	1	-1	-i	i	1	linear of order 4
ρ₄	1	-1	i	-i	1	linear of order 4
ρ₅	4	0	0	0	-1	orthogonal faithful

Character table of C₇⋊C₃

C₇⋊C₃: The semidirect product of C₇ and C₃ acting faithfully C7:C3 ID 21,1

class	1	3A	3B	7A	7B
size	1	7	7	3	3
ρ₁	1	1	1	1	1	trivial
ρ₂	1	ζ₃²	ζ₃	1	1	linear of order 3
ρ₃	1	ζ₃	ζ₃²	1	1	linear of order 3
ρ₄	3	0	0	^-1-√-7/₂	^-1+√-7/₂	complex faithful
ρ₅	3	0	0	^-1+√-7/₂	^-1-√-7/₂	complex faithful

Character table of S₄

S₄: Symmetric group on 4 letters; = PGL₂(𝔽₃) = Aut(Q₈) = Hol(C₂²) = tetrahedron symmetries = cube/octahedron rotations S4 ID 24,12

class	1	2A	2B	3	4
size	1	3	6	8	6
ρ₁	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	-1	linear of order 2
ρ₃	2	2	0	-1	0	orthogonal lifted from S₃
ρ₄	3	-1	-1	0	1	orthogonal faithful
ρ₅	3	-1	1	0	-1	orthogonal faithful

Character table of A₅

A₅: Alternating group on 5 letters; = SL₂(𝔽₄) = L₂(5) = L₂(4) = icosahedron/dodecahedron rotations; 1^st non-abelian simple A5 ID 60,5

class	1	2	3	5A	5B
size	1	15	20	12	12
ρ₁	1	1	1	1	1	trivial
ρ₂	3	-1	0	^1+√5/₂	^1-√5/₂	orthogonal faithful
ρ₃	3	-1	0	^1-√5/₂	^1+√5/₂	orthogonal faithful
ρ₄	4	0	1	-1	-1	orthogonal faithful
ρ₅	5	1	-1	0	0	orthogonal faithful

6x6 character tables

Character table of C₆

C₆: Cyclic group; = hexagon rotations C6 ID 6,2

class	1	2	3A	3B	6A	6B
size	1	1	1	1	1	1
ρ₁	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	-1	-1	linear of order 2
ρ₃	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	linear of order 3
ρ₄	1	-1	ζ₃²	ζ₃	ζ₆⁵	ζ₆	linear of order 6 faithful
ρ₅	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	linear of order 3
ρ₆	1	-1	ζ₃	ζ₃²	ζ₆	ζ₆⁵	linear of order 6 faithful

Character table of Dic₃

Dic₃: Dicyclic group; = C₃ ⋊C₄ Dic3 ID 12,1

class	1	2	3	4A	4B	6
size	1	1	2	3	3	2
ρ₁	1	1	1	1	1	1	trivial
ρ₂	1	1	1	-1	-1	1	linear of order 2
ρ₃	1	-1	1	i	-i	-1	linear of order 4
ρ₄	1	-1	1	-i	i	-1	linear of order 4
ρ₅	2	2	-1	0	0	-1	orthogonal lifted from S₃
ρ₆	2	-2	-1	0	0	1	symplectic faithful, Schur index 2

Character table of D₆

D₆: Dihedral group; = C₂×S₃ = hexagon symmetries D6 ID 12,4

class	1	2A	2B	2C	3	6
size	1	1	3	3	2	2
ρ₁	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	linear of order 2
ρ₃	1	-1	-1	1	1	-1	linear of order 2
ρ₄	1	-1	1	-1	1	-1	linear of order 2
ρ₅	2	2	0	0	-1	-1	orthogonal lifted from S₃
ρ₆	2	-2	0	0	-1	1	orthogonal faithful

Character table of D₉

D₉: Dihedral group D9 ID 18,1

class	1	2	3	9A	9B	9C
size	1	9	2	2	2	2
ρ₁	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	linear of order 2
ρ₃	2	0	2	-1	-1	-1	orthogonal lifted from S₃
ρ₄	2	0	-1	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	orthogonal faithful
ρ₅	2	0	-1	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	orthogonal faithful
ρ₆	2	0	-1	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	orthogonal faithful

Character table of C₃⋊S₃

C₃⋊S₃: The semidirect product of C₃ and S₃ acting via S₃/C₃=C₂ C3:S3 ID 18,4

class	1	2	3A	3B	3C	3D
size	1	9	2	2	2	2
ρ₁	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	linear of order 2
ρ₃	2	0	-1	-1	-1	2	orthogonal lifted from S₃
ρ₄	2	0	2	-1	-1	-1	orthogonal lifted from S₃
ρ₅	2	0	-1	2	-1	-1	orthogonal lifted from S₃
ρ₆	2	0	-1	-1	2	-1	orthogonal lifted from S₃

Character table of C₃²⋊C₄

C₃²⋊C₄: The semidirect product of C₃² and C₄ acting faithfully C3^2:C4 ID 36,9

class	1	2	3A	3B	4A	4B
size	1	9	4	4	9	9
ρ₁	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	linear of order 2
ρ₃	1	-1	1	1	i	-i	linear of order 4
ρ₄	1	-1	1	1	-i	i	linear of order 4
ρ₅	4	0	1	-2	0	0	orthogonal faithful
ρ₆	4	0	-2	1	0	0	orthogonal faithful

Character table of PSU₃(𝔽₂)

PSU₃(𝔽₂): Projective special unitary group on 𝔽₂³; = C₃² ⋊Q₈ = M₉ PSU(3,2) ID 72,41

class	1	2	3	4A	4B	4C
size	1	9	8	18	18	18
ρ₁	1	1	1	1	1	1	trivial
ρ₂	1	1	1	-1	1	-1	linear of order 2
ρ₃	1	1	1	1	-1	-1	linear of order 2
ρ₄	1	1	1	-1	-1	1	linear of order 2
ρ₅	2	-2	2	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₆	8	0	-1	0	0	0	orthogonal faithful

Character table of GL₃(𝔽₂)

GL₃(𝔽₂): General linear group on 𝔽₂³; = Aut(C₂³) = L₃(2) = L₂(7); 2^nd non-abelian simple GL(3,2) ID 168,42

class	1	2	3	4	7A	7B
size	1	21	56	42	24	24
ρ₁	1	1	1	1	1	1	trivial
ρ₂	3	-1	0	1	^-1+√-7/₂	^-1-√-7/₂	complex faithful
ρ₃	3	-1	0	1	^-1-√-7/₂	^-1+√-7/₂	complex faithful
ρ₄	6	2	0	0	-1	-1	orthogonal faithful
ρ₅	7	-1	1	-1	0	0	orthogonal faithful
ρ₆	8	0	-1	0	1	1	orthogonal faithful

7x7 character tables

Character table of C₇

C₇: Cyclic group C7 ID 7,1

class	1	7A	7B	7C	7D	7E	7F
size	1	1	1	1	1	1	1
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	ζ₇⁶	ζ₇²	ζ₇³	ζ₇⁴	ζ₇⁵	ζ₇	linear of order 7 faithful
ρ₃	1	ζ₇⁵	ζ₇⁴	ζ₇⁶	ζ₇	ζ₇³	ζ₇²	linear of order 7 faithful
ρ₄	1	ζ₇⁴	ζ₇⁶	ζ₇²	ζ₇⁵	ζ₇	ζ₇³	linear of order 7 faithful
ρ₅	1	ζ₇³	ζ₇	ζ₇⁵	ζ₇²	ζ₇⁶	ζ₇⁴	linear of order 7 faithful
ρ₆	1	ζ₇²	ζ₇³	ζ₇	ζ₇⁶	ζ₇⁴	ζ₇⁵	linear of order 7 faithful
ρ₇	1	ζ₇	ζ₇⁵	ζ₇⁴	ζ₇³	ζ₇²	ζ₇⁶	linear of order 7 faithful

Character table of D₈

D₈: Dihedral group D8 ID 16,7

class	1	2A	2B	2C	4	8A	8B
size	1	1	4	4	2	2	2
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	-1	-1	linear of order 2
ρ₃	1	1	1	-1	1	-1	-1	linear of order 2
ρ₄	1	1	-1	-1	1	1	1	linear of order 2
ρ₅	2	2	0	0	-2	0	0	orthogonal lifted from D₄
ρ₆	2	-2	0	0	0	√2	-√2	orthogonal faithful
ρ₇	2	-2	0	0	0	-√2	√2	orthogonal faithful

Character table of SD₁₆

SD₁₆: Semidihedral group; = Q₈ ⋊C₂ = QD₁₆ SD16 ID 16,8

class	1	2A	2B	4A	4B	8A	8B
size	1	1	4	2	4	2	2
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	-1	1	1	-1	-1	linear of order 2
ρ₄	1	1	-1	1	-1	1	1	linear of order 2
ρ₅	2	2	0	-2	0	0	0	orthogonal lifted from D₄
ρ₆	2	-2	0	0	0	-√-2	√-2	complex faithful
ρ₇	2	-2	0	0	0	√-2	-√-2	complex faithful

Character table of Q₁₆

Q₁₆: Generalised quaternion group; = C₈ .C₂ = Dic₄ Q16 ID 16,9

class	1	2	4A	4B	4C	8A	8B
size	1	1	2	4	4	2	2
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	-1	1	-1	-1	linear of order 2
ρ₄	1	1	1	-1	-1	1	1	linear of order 2
ρ₅	2	2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₆	2	-2	0	0	0	√2	-√2	symplectic faithful, Schur index 2
ρ₇	2	-2	0	0	0	-√2	√2	symplectic faithful, Schur index 2

Character table of D₁₁

D₁₁: Dihedral group D11 ID 22,1

class	1	2	11A	11B	11C	11D	11E
size	1	11	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	linear of order 2
ρ₃	2	0	ζ₁₁¹⁰+ζ₁₁	ζ₁₁⁷+ζ₁₁⁴	ζ₁₁⁹+ζ₁₁²	ζ₁₁⁸+ζ₁₁³	ζ₁₁⁶+ζ₁₁⁵	orthogonal faithful
ρ₄	2	0	ζ₁₁⁸+ζ₁₁³	ζ₁₁¹⁰+ζ₁₁	ζ₁₁⁶+ζ₁₁⁵	ζ₁₁⁹+ζ₁₁²	ζ₁₁⁷+ζ₁₁⁴	orthogonal faithful
ρ₅	2	0	ζ₁₁⁶+ζ₁₁⁵	ζ₁₁⁹+ζ₁₁²	ζ₁₁¹⁰+ζ₁₁	ζ₁₁⁷+ζ₁₁⁴	ζ₁₁⁸+ζ₁₁³	orthogonal faithful
ρ₆	2	0	ζ₁₁⁹+ζ₁₁²	ζ₁₁⁸+ζ₁₁³	ζ₁₁⁷+ζ₁₁⁴	ζ₁₁⁶+ζ₁₁⁵	ζ₁₁¹⁰+ζ₁₁	orthogonal faithful
ρ₇	2	0	ζ₁₁⁷+ζ₁₁⁴	ζ₁₁⁶+ζ₁₁⁵	ζ₁₁⁸+ζ₁₁³	ζ₁₁¹⁰+ζ₁₁	ζ₁₁⁹+ζ₁₁²	orthogonal faithful

Character table of SL₂(𝔽₃)

SL₂(𝔽₃): Special linear group on 𝔽₃²; = Q₈ ⋊C₃ = 2T = <2,3,3> = 1^st non-monomial group SL(2,3) ID 24,3

class	1	2	3A	3B	4	6A	6B
size	1	1	4	4	6	4	4
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	1	ζ₃²	ζ₃	1	ζ₃²	ζ₃	linear of order 3
ρ₃	1	1	ζ₃	ζ₃²	1	ζ₃	ζ₃²	linear of order 3
ρ₄	2	-2	-1	-1	0	1	1	symplectic faithful, Schur index 2
ρ₅	2	-2	ζ₆⁵	ζ₆	0	ζ₃	ζ₃²	complex faithful
ρ₆	2	-2	ζ₆	ζ₆⁵	0	ζ₃²	ζ₃	complex faithful
ρ₇	3	3	0	0	-1	0	0	orthogonal lifted from A₄

Character table of C₁₃⋊C₃

C₁₃⋊C₃: The semidirect product of C₁₃ and C₃ acting faithfully C13:C3 ID 39,1

class	1	3A	3B	13A	13B	13C	13D
size	1	13	13	3	3	3	3
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	ζ₃²	ζ₃	1	1	1	1	linear of order 3
ρ₃	1	ζ₃	ζ₃²	1	1	1	1	linear of order 3
ρ₄	3	0	0	ζ₁₃¹¹+ζ₁₃⁸+ζ₁₃⁷	ζ₁₃⁹+ζ₁₃³+ζ₁₃	ζ₁₃⁶+ζ₁₃⁵+ζ₁₃²	ζ₁₃¹²+ζ₁₃¹⁰+ζ₁₃⁴	complex faithful
ρ₅	3	0	0	ζ₁₃¹²+ζ₁₃¹⁰+ζ₁₃⁴	ζ₁₃¹¹+ζ₁₃⁸+ζ₁₃⁷	ζ₁₃⁹+ζ₁₃³+ζ₁₃	ζ₁₃⁶+ζ₁₃⁵+ζ₁₃²	complex faithful
ρ₆	3	0	0	ζ₁₃⁶+ζ₁₃⁵+ζ₁₃²	ζ₁₃¹²+ζ₁₃¹⁰+ζ₁₃⁴	ζ₁₃¹¹+ζ₁₃⁸+ζ₁₃⁷	ζ₁₃⁹+ζ₁₃³+ζ₁₃	complex faithful
ρ₇	3	0	0	ζ₁₃⁹+ζ₁₃³+ζ₁₃	ζ₁₃⁶+ζ₁₃⁵+ζ₁₃²	ζ₁₃¹²+ζ₁₃¹⁰+ζ₁₃⁴	ζ₁₃¹¹+ζ₁₃⁸+ζ₁₃⁷	complex faithful

Character table of F₇

F₇: Frobenius group; = C₇ ⋊C₆ = AGL₁(𝔽₇) = Aut(D₇) = Hol(C₇) F7 ID 42,1

class	1	2	3A	3B	6A	6B	7
size	1	7	7	7	7	7	6
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	-1	-1	1	linear of order 2
ρ₃	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	1	linear of order 3
ρ₄	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	1	linear of order 3
ρ₅	1	-1	ζ₃	ζ₃²	ζ₆	ζ₆⁵	1	linear of order 6
ρ₆	1	-1	ζ₃²	ζ₃	ζ₆⁵	ζ₆	1	linear of order 6
ρ₇	6	0	0	0	0	0	-1	orthogonal faithful

Character table of C₁₃⋊C₄

C₁₃⋊C₄: The semidirect product of C₁₃ and C₄ acting faithfully C13:C4 ID 52,3

class	1	2	4A	4B	13A	13B	13C
size	1	13	13	13	4	4	4
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	linear of order 2
ρ₃	1	-1	-i	i	1	1	1	linear of order 4
ρ₄	1	-1	i	-i	1	1	1	linear of order 4
ρ₅	4	0	0	0	ζ₁₃⁹+ζ₁₃⁷+ζ₁₃⁶+ζ₁₃⁴	ζ₁₃¹²+ζ₁₃⁸+ζ₁₃⁵+ζ₁₃	ζ₁₃¹¹+ζ₁₃¹⁰+ζ₁₃³+ζ₁₃²	orthogonal faithful
ρ₆	4	0	0	0	ζ₁₃¹²+ζ₁₃⁸+ζ₁₃⁵+ζ₁₃	ζ₁₃¹¹+ζ₁₃¹⁰+ζ₁₃³+ζ₁₃²	ζ₁₃⁹+ζ₁₃⁷+ζ₁₃⁶+ζ₁₃⁴	orthogonal faithful
ρ₇	4	0	0	0	ζ₁₃¹¹+ζ₁₃¹⁰+ζ₁₃³+ζ₁₃²	ζ₁₃⁹+ζ₁₃⁷+ζ₁₃⁶+ζ₁₃⁴	ζ₁₃¹²+ζ₁₃⁸+ζ₁₃⁵+ζ₁₃	orthogonal faithful

Character table of C₁₁⋊C₅

C₁₁⋊C₅: The semidirect product of C₁₁ and C₅ acting faithfully C11:C5 ID 55,1

class	1	5A	5B	5C	5D	11A	11B
size	1	11	11	11	11	5	5
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	ζ₅	ζ₅³	ζ₅²	ζ₅⁴	1	1	linear of order 5
ρ₃	1	ζ₅⁴	ζ₅²	ζ₅³	ζ₅	1	1	linear of order 5
ρ₄	1	ζ₅²	ζ₅	ζ₅⁴	ζ₅³	1	1	linear of order 5
ρ₅	1	ζ₅³	ζ₅⁴	ζ₅	ζ₅²	1	1	linear of order 5
ρ₆	5	0	0	0	0	^-1+√-11/₂	^-1-√-11/₂	complex faithful
ρ₇	5	0	0	0	0	^-1-√-11/₂	^-1+√-11/₂	complex faithful

Character table of S₅

S₅: Symmetric group on 5 letters; = PGL₂(𝔽₅) = Aut(A₅) = 5-cell symmetries; almost simple S5 ID 120,34

class	1	2A	2B	3	4	5	6
size	1	10	15	20	30	24	20
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	-1	1	-1	linear of order 2
ρ₃	4	-2	0	1	0	-1	1	orthogonal faithful
ρ₄	4	2	0	1	0	-1	-1	orthogonal faithful
ρ₅	5	1	1	-1	-1	0	1	orthogonal faithful
ρ₆	5	-1	1	-1	1	0	-1	orthogonal faithful
ρ₇	6	0	-2	0	0	1	0	orthogonal faithful

Character table of A₆

A₆: Alternating group on 6 letters; = PSL₂(𝔽₉) = L₂(9); 3^rd non-abelian simple A6 ID 360,118

class	1	2	3A	3B	4	5A	5B
size	1	45	40	40	90	72	72
ρ₁	1	1	1	1	1	1	1	trivial
ρ₂	5	1	2	-1	-1	0	0	orthogonal faithful
ρ₃	5	1	-1	2	-1	0	0	orthogonal faithful
ρ₄	8	0	-1	-1	0	^1-√5/₂	^1+√5/₂	orthogonal faithful
ρ₅	8	0	-1	-1	0	^1+√5/₂	^1-√5/₂	orthogonal faithful
ρ₆	9	1	0	0	1	-1	-1	orthogonal faithful
ρ₇	10	-2	1	1	0	0	0	orthogonal faithful

8x8 character tables

Character table of C₈

C₈: Cyclic group C8 ID 8,1

class	1	2	4A	4B	8A	8B	8C	8D
size	1	1	1	1	1	1	1	1
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₃	1	-1	i	-i	ζ₈⁷	ζ₈⁵	ζ₈³	ζ₈	linear of order 8 faithful
ρ₄	1	1	-1	-1	-i	i	-i	i	linear of order 4
ρ₅	1	-1	-i	i	ζ₈⁵	ζ₈⁷	ζ₈	ζ₈³	linear of order 8 faithful
ρ₆	1	-1	i	-i	ζ₈³	ζ₈	ζ₈⁷	ζ₈⁵	linear of order 8 faithful
ρ₇	1	1	-1	-1	i	-i	i	-i	linear of order 4
ρ₈	1	-1	-i	i	ζ₈	ζ₈³	ζ₈⁵	ζ₈⁷	linear of order 8 faithful

Character table of C₂×C₄

C₂×C₄: Abelian group of type [2,4] C2xC4 ID 8,2

class	1	2A	2B	2C	4A	4B	4C	4D
size	1	1	1	1	1	1	1	1
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	-1	1	-1	linear of order 2
ρ₃	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	-1	-1	-1	1	-1	1	linear of order 2
ρ₅	1	-1	1	-1	i	i	-i	-i	linear of order 4
ρ₆	1	-1	-1	1	i	-i	-i	i	linear of order 4
ρ₇	1	-1	1	-1	-i	-i	i	i	linear of order 4
ρ₈	1	-1	-1	1	-i	i	i	-i	linear of order 4

Character table of C₂³

C₂³: Elementary abelian group of type [2,2,2] C2^3 ID 8,5

class	1	2A	2B	2C	2D	2E	2F	2G
size	1	1	1	1	1	1	1	1
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	-1	-1	-1	1	linear of order 2
ρ₃	1	-1	-1	-1	1	1	-1	1	linear of order 2
ρ₄	1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₅	1	-1	1	-1	1	-1	1	-1	linear of order 2
ρ₆	1	1	1	-1	-1	1	-1	-1	linear of order 2
ρ₇	1	1	-1	1	1	-1	-1	-1	linear of order 2
ρ₈	1	-1	-1	1	-1	1	1	-1	linear of order 2

Character table of Dic₅

Dic₅: Dicyclic group; = C₅ ⋊₂ C₄ Dic5 ID 20,1

class	1	2	4A	4B	5A	5B	10A	10B
size	1	1	5	5	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	linear of order 2
ρ₃	1	-1	-i	i	1	1	-1	-1	linear of order 4
ρ₄	1	-1	i	-i	1	1	-1	-1	linear of order 4
ρ₅	2	2	0	0	^-1+√5/₂	^-1-√5/₂	^-1-√5/₂	^-1+√5/₂	orthogonal lifted from D₅
ρ₆	2	2	0	0	^-1-√5/₂	^-1+√5/₂	^-1+√5/₂	^-1-√5/₂	orthogonal lifted from D₅
ρ₇	2	-2	0	0	^-1+√5/₂	^-1-√5/₂	^1+√5/₂	^1-√5/₂	symplectic faithful, Schur index 2
ρ₈	2	-2	0	0	^-1-√5/₂	^-1+√5/₂	^1-√5/₂	^1+√5/₂	symplectic faithful, Schur index 2

Character table of D₁₀

D₁₀: Dihedral group; = C₂×D₅ D10 ID 20,4

class	1	2A	2B	2C	5A	5B	10A	10B
size	1	1	5	5	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	linear of order 2
ρ₃	1	-1	-1	1	1	1	-1	-1	linear of order 2
ρ₄	1	-1	1	-1	1	1	-1	-1	linear of order 2
ρ₅	2	-2	0	0	^-1+√5/₂	^-1-√5/₂	^1+√5/₂	^1-√5/₂	orthogonal faithful
ρ₆	2	2	0	0	^-1+√5/₂	^-1-√5/₂	^-1-√5/₂	^-1+√5/₂	orthogonal lifted from D₅
ρ₇	2	2	0	0	^-1-√5/₂	^-1+√5/₂	^-1+√5/₂	^-1-√5/₂	orthogonal lifted from D₅
ρ₈	2	-2	0	0	^-1-√5/₂	^-1+√5/₂	^1-√5/₂	^1+√5/₂	orthogonal faithful

Character table of C₂×A₄

C₂×A₄: Direct product of C₂ and A₄; = AΣL₁(𝔽₈) C2xA4 ID 24,13

class	1	2A	2B	2C	3A	3B	6A	6B
size	1	1	3	3	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	-1	1	1	1	-1	-1	linear of order 2
ρ₃	1	1	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	linear of order 3
ρ₄	1	-1	-1	1	ζ₃²	ζ₃	ζ₆	ζ₆⁵	linear of order 6
ρ₅	1	-1	-1	1	ζ₃	ζ₃²	ζ₆⁵	ζ₆	linear of order 6
ρ₆	1	1	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	linear of order 3
ρ₇	3	3	-1	-1	0	0	0	0	orthogonal lifted from A₄
ρ₈	3	-3	1	-1	0	0	0	0	orthogonal faithful

Character table of D₁₃

D₁₃: Dihedral group D13 ID 26,1

class	1	2	13A	13B	13C	13D	13E	13F
size	1	13	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	1	linear of order 2
ρ₃	2	0	ζ₁₃⁹+ζ₁₃⁴	ζ₁₃⁷+ζ₁₃⁶	ζ₁₃⁸+ζ₁₃⁵	ζ₁₃¹⁰+ζ₁₃³	ζ₁₃¹²+ζ₁₃	ζ₁₃¹¹+ζ₁₃²	orthogonal faithful
ρ₄	2	0	ζ₁₃¹²+ζ₁₃	ζ₁₃⁸+ζ₁₃⁵	ζ₁₃¹¹+ζ₁₃²	ζ₁₃⁹+ζ₁₃⁴	ζ₁₃¹⁰+ζ₁₃³	ζ₁₃⁷+ζ₁₃⁶	orthogonal faithful
ρ₅	2	0	ζ₁₃¹⁰+ζ₁₃³	ζ₁₃¹¹+ζ₁₃²	ζ₁₃⁷+ζ₁₃⁶	ζ₁₃¹²+ζ₁₃	ζ₁₃⁹+ζ₁₃⁴	ζ₁₃⁸+ζ₁₃⁵	orthogonal faithful
ρ₆	2	0	ζ₁₃⁷+ζ₁₃⁶	ζ₁₃⁹+ζ₁₃⁴	ζ₁₃¹²+ζ₁₃	ζ₁₃¹¹+ζ₁₃²	ζ₁₃⁸+ζ₁₃⁵	ζ₁₃¹⁰+ζ₁₃³	orthogonal faithful
ρ₇	2	0	ζ₁₃¹¹+ζ₁₃²	ζ₁₃¹⁰+ζ₁₃³	ζ₁₃⁹+ζ₁₃⁴	ζ₁₃⁸+ζ₁₃⁵	ζ₁₃⁷+ζ₁₃⁶	ζ₁₃¹²+ζ₁₃	orthogonal faithful
ρ₈	2	0	ζ₁₃⁸+ζ₁₃⁵	ζ₁₃¹²+ζ₁₃	ζ₁₃¹⁰+ζ₁₃³	ζ₁₃⁷+ζ₁₃⁶	ζ₁₃¹¹+ζ₁₃²	ζ₁₃⁹+ζ₁₃⁴	orthogonal faithful

Character table of C₄²⋊C₃

C₄²⋊C₃: The semidirect product of C₄² and C₃ acting faithfully C4^2:C3 ID 48,3

class	1	2	3A	3B	4A	4B	4C	4D
size	1	3	16	16	3	3	3	3
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	ζ₃²	ζ₃	1	1	1	1	linear of order 3
ρ₃	1	1	ζ₃	ζ₃²	1	1	1	1	linear of order 3
ρ₄	3	3	0	0	-1	-1	-1	-1	orthogonal lifted from A₄
ρ₅	3	-1	0	0	1	-1+2i	-1-2i	1	complex faithful
ρ₆	3	-1	0	0	-1-2i	1	1	-1+2i	complex faithful
ρ₇	3	-1	0	0	1	-1-2i	-1+2i	1	complex faithful
ρ₈	3	-1	0	0	-1+2i	1	1	-1-2i	complex faithful

Character table of CSU₂(𝔽₃)

CSU₂(𝔽₃): Conformal special unitary group on 𝔽₃²; = Q₈ .S₃ = 2O = <2,3,4> CSU(2,3) ID 48,28

class	1	2	3	4A	4B	6	8A	8B
size	1	1	8	6	12	8	6	6
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	1	-1	-1	linear of order 2
ρ₃	2	2	-1	2	0	-1	0	0	orthogonal lifted from S₃
ρ₄	2	-2	-1	0	0	1	√2	-√2	symplectic faithful, Schur index 2
ρ₅	2	-2	-1	0	0	1	-√2	√2	symplectic faithful, Schur index 2
ρ₆	3	3	0	-1	-1	0	1	1	orthogonal lifted from S₄
ρ₇	3	3	0	-1	1	0	-1	-1	orthogonal lifted from S₄
ρ₈	4	-4	1	0	0	-1	0	0	symplectic faithful, Schur index 2

Character table of GL₂(𝔽₃)

GL₂(𝔽₃): General linear group on 𝔽₃²; = Q₈ ⋊S₃ = Aut(C₃²) GL(2,3) ID 48,29

class	1	2A	2B	3	4	6	8A	8B
size	1	1	12	8	6	8	6	6
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	1	-1	-1	linear of order 2
ρ₃	2	2	0	-1	2	-1	0	0	orthogonal lifted from S₃
ρ₄	2	-2	0	-1	0	1	-√-2	√-2	complex faithful
ρ₅	2	-2	0	-1	0	1	√-2	-√-2	complex faithful
ρ₆	3	3	-1	0	-1	0	1	1	orthogonal lifted from S₄
ρ₇	3	3	1	0	-1	0	-1	-1	orthogonal lifted from S₄
ρ₈	4	-4	0	1	0	-1	0	0	orthogonal faithful

Character table of C₂²⋊A₄

C₂²⋊A₄: The semidirect product of C₂² and A₄ acting via A₄/C₂²=C₃ C2^2:A4 ID 48,50

class	1	2A	2B	2C	2D	2E	3A	3B
size	1	3	3	3	3	3	16	16
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	ζ₃	ζ₃²	linear of order 3
ρ₃	1	1	1	1	1	1	ζ₃²	ζ₃	linear of order 3
ρ₄	3	3	-1	-1	-1	-1	0	0	orthogonal lifted from A₄
ρ₅	3	-1	-1	-1	3	-1	0	0	orthogonal lifted from A₄
ρ₆	3	-1	3	-1	-1	-1	0	0	orthogonal lifted from A₄
ρ₇	3	-1	-1	3	-1	-1	0	0	orthogonal lifted from A₄
ρ₈	3	-1	-1	-1	-1	3	0	0	orthogonal lifted from A₄

Character table of F₈

F₈: Frobenius group; = C₂³ ⋊C₇ = AGL₁(𝔽₈) F8 ID 56,11

class	1	2	7A	7B	7C	7D	7E	7F
size	1	7	8	8	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	ζ₇⁴	ζ₇⁶	ζ₇²	ζ₇⁵	ζ₇	ζ₇³	linear of order 7
ρ₃	1	1	ζ₇²	ζ₇³	ζ₇	ζ₇⁶	ζ₇⁴	ζ₇⁵	linear of order 7
ρ₄	1	1	ζ₇⁵	ζ₇⁴	ζ₇⁶	ζ₇	ζ₇³	ζ₇²	linear of order 7
ρ₅	1	1	ζ₇³	ζ₇	ζ₇⁵	ζ₇²	ζ₇⁶	ζ₇⁴	linear of order 7
ρ₆	1	1	ζ₇	ζ₇⁵	ζ₇⁴	ζ₇³	ζ₇²	ζ₇⁶	linear of order 7
ρ₇	1	1	ζ₇⁶	ζ₇²	ζ₇³	ζ₇⁴	ζ₇⁵	ζ₇	linear of order 7
ρ₈	7	-1	0	0	0	0	0	0	orthogonal faithful

Character table of C₁₇⋊C₄

C₁₇⋊C₄: The semidirect product of C₁₇ and C₄ acting faithfully C17:C4 ID 68,3

class	1	2	4A	4B	17A	17B	17C	17D
size	1	17	17	17	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	linear of order 2
ρ₃	1	-1	i	-i	1	1	1	1	linear of order 4
ρ₄	1	-1	-i	i	1	1	1	1	linear of order 4
ρ₅	4	0	0	0	ζ₁₇¹⁶+ζ₁₇¹³+ζ₁₇⁴+ζ₁₇	ζ₁₇¹⁴+ζ₁₇¹²+ζ₁₇⁵+ζ₁₇³	ζ₁₇¹¹+ζ₁₇¹⁰+ζ₁₇⁷+ζ₁₇⁶	ζ₁₇¹⁵+ζ₁₇⁹+ζ₁₇⁸+ζ₁₇²	orthogonal faithful
ρ₆	4	0	0	0	ζ₁₇¹⁵+ζ₁₇⁹+ζ₁₇⁸+ζ₁₇²	ζ₁₇¹¹+ζ₁₇¹⁰+ζ₁₇⁷+ζ₁₇⁶	ζ₁₇¹⁴+ζ₁₇¹²+ζ₁₇⁵+ζ₁₇³	ζ₁₇¹⁶+ζ₁₇¹³+ζ₁₇⁴+ζ₁₇	orthogonal faithful
ρ₇	4	0	0	0	ζ₁₇¹⁴+ζ₁₇¹²+ζ₁₇⁵+ζ₁₇³	ζ₁₇¹⁵+ζ₁₇⁹+ζ₁₇⁸+ζ₁₇²	ζ₁₇¹⁶+ζ₁₇¹³+ζ₁₇⁴+ζ₁₇	ζ₁₇¹¹+ζ₁₇¹⁰+ζ₁₇⁷+ζ₁₇⁶	orthogonal faithful
ρ₈	4	0	0	0	ζ₁₇¹¹+ζ₁₇¹⁰+ζ₁₇⁷+ζ₁₇⁶	ζ₁₇¹⁶+ζ₁₇¹³+ζ₁₇⁴+ζ₁₇	ζ₁₇¹⁵+ζ₁₇⁹+ζ₁₇⁸+ζ₁₇²	ζ₁₇¹⁴+ζ₁₇¹²+ζ₁₇⁵+ζ₁₇³	orthogonal faithful

Character table of C₁₃⋊C₆

C₁₃⋊C₆: The semidirect product of C₁₃ and C₆ acting faithfully C13:C6 ID 78,1

class	1	2	3A	3B	6A	6B	13A	13B
size	1	13	13	13	13	13	6	6
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	-1	-1	1	1	linear of order 2
ρ₃	1	-1	ζ₃²	ζ₃	ζ₆⁵	ζ₆	1	1	linear of order 6
ρ₄	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	1	1	linear of order 3
ρ₅	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	1	1	linear of order 3
ρ₆	1	-1	ζ₃	ζ₃²	ζ₆	ζ₆⁵	1	1	linear of order 6
ρ₇	6	0	0	0	0	0	^-1-√13/₂	^-1+√13/₂	orthogonal faithful
ρ₈	6	0	0	0	0	0	^-1+√13/₂	^-1-√13/₂	orthogonal faithful

Character table of C₂⁴⋊C₅

C₂⁴⋊C₅: The semidirect product of C₂⁴ and C₅ acting faithfully C2^4:C5 ID 80,49

class	1	2A	2B	2C	5A	5B	5C	5D
size	1	5	5	5	16	16	16	16
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	ζ₅²	ζ₅	ζ₅⁴	ζ₅³	linear of order 5
ρ₃	1	1	1	1	ζ₅³	ζ₅⁴	ζ₅	ζ₅²	linear of order 5
ρ₄	1	1	1	1	ζ₅⁴	ζ₅²	ζ₅³	ζ₅	linear of order 5
ρ₅	1	1	1	1	ζ₅	ζ₅³	ζ₅²	ζ₅⁴	linear of order 5
ρ₆	5	-3	1	1	0	0	0	0	orthogonal faithful
ρ₇	5	1	-3	1	0	0	0	0	orthogonal faithful
ρ₈	5	1	1	-3	0	0	0	0	orthogonal faithful

Character table of AΓL₁(𝔽₈)

AΓL₁(𝔽₈): Affine semilinear group on 𝔽₈¹; = F₈ ⋊C₃ = Aut(F₈) AGammaL(1,8) ID 168,43

class	1	2	3A	3B	6A	6B	7A	7B
size	1	7	28	28	28	28	24	24
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	1	linear of order 3
ρ₃	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	1	linear of order 3
ρ₄	3	3	0	0	0	0	^-1+√-7/₂	^-1-√-7/₂	complex lifted from C₇⋊C₃
ρ₅	3	3	0	0	0	0	^-1-√-7/₂	^-1+√-7/₂	complex lifted from C₇⋊C₃
ρ₆	7	-1	1	1	-1	-1	0	0	orthogonal faithful
ρ₇	7	-1	ζ₃	ζ₃²	ζ₆⁵	ζ₆	0	0	complex faithful
ρ₈	7	-1	ζ₃²	ζ₃	ζ₆	ζ₆⁵	0	0	complex faithful

Character table of C₅²⋊Q₈

C₅²⋊Q₈: The semidirect product of C₅² and Q₈ acting faithfully C5^2:Q8 ID 200,44

class	1	2	4A	4B	4C	5A	5B	5C
size	1	25	50	50	50	8	8	8
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	linear of order 2
ρ₃	1	1	1	-1	-1	1	1	1	linear of order 2
ρ₄	1	1	-1	1	-1	1	1	1	linear of order 2
ρ₅	2	-2	0	0	0	2	2	2	symplectic lifted from Q₈, Schur index 2
ρ₆	8	0	0	0	0	-2	3	-2	orthogonal faithful
ρ₇	8	0	0	0	0	3	-2	-2	orthogonal faithful
ρ₈	8	0	0	0	0	-2	-2	3	orthogonal faithful

Character table of C₅²⋊Dic₃

C₅²⋊Dic₃: The semidirect product of C₅² and Dic₃ acting faithfully C5^2:Dic3 ID 300,23

class	1	2	3	4A	4B	5A	5B	6
size	1	25	50	75	75	12	12	50
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	-1	-1	1	1	1	linear of order 2
ρ₃	1	-1	1	i	-i	1	1	-1	linear of order 4
ρ₄	1	-1	1	-i	i	1	1	-1	linear of order 4
ρ₅	2	2	-1	0	0	2	2	-1	orthogonal lifted from S₃
ρ₆	2	-2	-1	0	0	2	2	1	symplectic lifted from Dic₃, Schur index 2
ρ₇	12	0	0	0	0	2	-3	0	orthogonal faithful
ρ₈	12	0	0	0	0	-3	2	0	orthogonal faithful

9x9 character tables

Character table of C₉

C₉: Cyclic group C9 ID 9,1

class	1	3A	3B	9A	9B	9C	9D	9E	9F
size	1	1	1	1	1	1	1	1	1
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	ζ₃²	ζ₃	ζ₉⁷	ζ₉⁵	ζ₉	ζ₉⁴	ζ₉⁸	ζ₉²	linear of order 9 faithful
ρ₃	1	ζ₃	ζ₃²	ζ₉⁵	ζ₉	ζ₉²	ζ₉⁸	ζ₉⁷	ζ₉⁴	linear of order 9 faithful
ρ₄	1	1	1	ζ₃	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	linear of order 3
ρ₅	1	ζ₃²	ζ₃	ζ₉	ζ₉²	ζ₉⁴	ζ₉⁷	ζ₉⁵	ζ₉⁸	linear of order 9 faithful
ρ₆	1	ζ₃	ζ₃²	ζ₉⁸	ζ₉⁷	ζ₉⁵	ζ₉²	ζ₉⁴	ζ₉	linear of order 9 faithful
ρ₇	1	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	linear of order 3
ρ₈	1	ζ₃²	ζ₃	ζ₉⁴	ζ₉⁸	ζ₉⁷	ζ₉	ζ₉²	ζ₉⁵	linear of order 9 faithful
ρ₉	1	ζ₃	ζ₃²	ζ₉²	ζ₉⁴	ζ₉⁸	ζ₉⁵	ζ₉	ζ₉⁷	linear of order 9 faithful

Character table of C₃²

C₃²: Elementary abelian group of type [3,3] C3^2 ID 9,2

class	1	3A	3B	3C	3D	3E	3F	3G	3H
size	1	1	1	1	1	1	1	1	1
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	ζ₃²	1	ζ₃	ζ₃	ζ₃	ζ₃²	ζ₃²	1	linear of order 3
ρ₃	1	ζ₃	1	ζ₃²	ζ₃²	ζ₃²	ζ₃	ζ₃	1	linear of order 3
ρ₄	1	ζ₃²	ζ₃²	1	ζ₃	ζ₃²	1	ζ₃	ζ₃	linear of order 3
ρ₅	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	ζ₃²	1	ζ₃	linear of order 3
ρ₆	1	1	ζ₃²	ζ₃²	1	ζ₃	ζ₃	ζ₃²	ζ₃	linear of order 3
ρ₇	1	ζ₃	ζ₃	1	ζ₃²	ζ₃	1	ζ₃²	ζ₃²	linear of order 3
ρ₈	1	1	ζ₃	ζ₃	1	ζ₃²	ζ₃²	ζ₃	ζ₃²	linear of order 3
ρ₉	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	ζ₃	1	ζ₃²	linear of order 3

Character table of C₃×S₃

C₃×S₃: Direct product of C₃ and S₃; = U₂(𝔽₂) C3xS3 ID 18,3

class	1	2	3A	3B	3C	3D	3E	6A	6B
size	1	3	1	1	2	2	2	3	3
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	-1	-1	linear of order 2
ρ₃	1	-1	ζ₃²	ζ₃	1	ζ₃²	ζ₃	ζ₆⁵	ζ₆	linear of order 6
ρ₄	1	-1	ζ₃	ζ₃²	1	ζ₃	ζ₃²	ζ₆	ζ₆⁵	linear of order 6
ρ₅	1	1	ζ₃	ζ₃²	1	ζ₃	ζ₃²	ζ₃²	ζ₃	linear of order 3
ρ₆	1	1	ζ₃²	ζ₃	1	ζ₃²	ζ₃	ζ₃	ζ₃²	linear of order 3
ρ₇	2	0	2	2	-1	-1	-1	0	0	orthogonal lifted from S₃
ρ₈	2	0	-1+√-3	-1-√-3	-1	ζ₆⁵	ζ₆	0	0	complex faithful
ρ₉	2	0	-1-√-3	-1+√-3	-1	ζ₆	ζ₆⁵	0	0	complex faithful

Character table of Dic₆

Dic₆: Dicyclic group; = C₃ ⋊Q₈ Dic6 ID 24,4

class	1	2	3	4A	4B	4C	6	12A	12B
size	1	1	2	2	6	6	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	1	1	1	linear of order 2
ρ₃	1	1	1	-1	-1	1	1	-1	-1	linear of order 2
ρ₄	1	1	1	-1	1	-1	1	-1	-1	linear of order 2
ρ₅	2	2	-1	2	0	0	-1	-1	-1	orthogonal lifted from S₃
ρ₆	2	2	-1	-2	0	0	-1	1	1	orthogonal lifted from D₆
ρ₇	2	-2	2	0	0	0	-2	0	0	symplectic lifted from Q₈, Schur index 2
ρ₈	2	-2	-1	0	0	0	1	√3	-√3	symplectic faithful, Schur index 2
ρ₉	2	-2	-1	0	0	0	1	-√3	√3	symplectic faithful, Schur index 2

Character table of D₁₂

D₁₂: Dihedral group D12 ID 24,6

class	1	2A	2B	2C	3	4	6	12A	12B
size	1	1	6	6	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	1	linear of order 2
ρ₃	1	1	1	-1	1	-1	1	-1	-1	linear of order 2
ρ₄	1	1	-1	1	1	-1	1	-1	-1	linear of order 2
ρ₅	2	2	0	0	-1	2	-1	-1	-1	orthogonal lifted from S₃
ρ₆	2	2	0	0	-1	-2	-1	1	1	orthogonal lifted from D₆
ρ₇	2	-2	0	0	2	0	-2	0	0	orthogonal lifted from D₄
ρ₈	2	-2	0	0	-1	0	1	√3	-√3	orthogonal faithful
ρ₉	2	-2	0	0	-1	0	1	-√3	√3	orthogonal faithful

Character table of C₃⋊D₄

C₃⋊D₄: The semidirect product of C₃ and D₄ acting via D₄/C₂²=C₂ C3:D4 ID 24,8

class	1	2A	2B	2C	3	4	6A	6B	6C
size	1	1	2	6	2	6	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	-1	1	-1	1	1	1	linear of order 2
ρ₃	1	1	-1	1	1	-1	-1	-1	1	linear of order 2
ρ₄	1	1	-1	-1	1	1	-1	-1	1	linear of order 2
ρ₅	2	2	-2	0	-1	0	1	1	-1	orthogonal lifted from D₆
ρ₆	2	2	2	0	-1	0	-1	-1	-1	orthogonal lifted from S₃
ρ₇	2	-2	0	0	2	0	0	0	-2	orthogonal lifted from D₄
ρ₈	2	-2	0	0	-1	0	-√-3	√-3	1	complex faithful
ρ₉	2	-2	0	0	-1	0	√-3	-√-3	1	complex faithful

Character table of D₁₅

D₁₅: Dihedral group D15 ID 30,3

class	1	2	3	5A	5B	15A	15B	15C	15D
size	1	15	2	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	1	1	linear of order 2
ρ₃	2	0	-1	2	2	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₄	2	0	2	^-1+√5/₂	^-1-√5/₂	^-1+√5/₂	^-1+√5/₂	^-1-√5/₂	^-1-√5/₂	orthogonal lifted from D₅
ρ₅	2	0	2	^-1-√5/₂	^-1+√5/₂	^-1-√5/₂	^-1-√5/₂	^-1+√5/₂	^-1+√5/₂	orthogonal lifted from D₅
ρ₆	2	0	-1	^-1-√5/₂	^-1+√5/₂	-ζ₃ζ₅³+ζ₃ζ₅²-ζ₅³	ζ₃ζ₅³-ζ₃ζ₅²-ζ₅²	ζ₃ζ₅⁴-ζ₃ζ₅-ζ₅	ζ₃²ζ₅⁴-ζ₃²ζ₅-ζ₅	orthogonal faithful
ρ₇	2	0	-1	^-1+√5/₂	^-1-√5/₂	ζ₃²ζ₅⁴-ζ₃²ζ₅-ζ₅	ζ₃ζ₅⁴-ζ₃ζ₅-ζ₅	-ζ₃ζ₅³+ζ₃ζ₅²-ζ₅³	ζ₃ζ₅³-ζ₃ζ₅²-ζ₅²	orthogonal faithful
ρ₈	2	0	-1	^-1-√5/₂	^-1+√5/₂	ζ₃ζ₅³-ζ₃ζ₅²-ζ₅²	-ζ₃ζ₅³+ζ₃ζ₅²-ζ₅³	ζ₃²ζ₅⁴-ζ₃²ζ₅-ζ₅	ζ₃ζ₅⁴-ζ₃ζ₅-ζ₅	orthogonal faithful
ρ₉	2	0	-1	^-1+√5/₂	^-1-√5/₂	ζ₃ζ₅⁴-ζ₃ζ₅-ζ₅	ζ₃²ζ₅⁴-ζ₃²ζ₅-ζ₅	ζ₃ζ₅³-ζ₃ζ₅²-ζ₅²	-ζ₃ζ₅³+ζ₃ζ₅²-ζ₅³	orthogonal faithful

Character table of S₃²

S₃²: Direct product of S₃ and S₃; = Spin+₄(𝔽₂) = Hol(S₃) S3^2 ID 36,10

class	1	2A	2B	2C	3A	3B	3C	6A	6B
size	1	3	3	9	2	2	4	6	6
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	-1	1	1	1	1	-1	linear of order 2
ρ₃	1	1	-1	-1	1	1	1	-1	1	linear of order 2
ρ₄	1	-1	-1	1	1	1	1	-1	-1	linear of order 2
ρ₅	2	2	0	0	2	-1	-1	0	-1	orthogonal lifted from S₃
ρ₆	2	-2	0	0	2	-1	-1	0	1	orthogonal lifted from D₆
ρ₇	2	0	2	0	-1	2	-1	-1	0	orthogonal lifted from S₃
ρ₈	2	0	-2	0	-1	2	-1	1	0	orthogonal lifted from D₆
ρ₉	4	0	0	0	-2	-2	1	0	0	orthogonal faithful

Character table of C₁₉⋊C₃

C₁₉⋊C₃: The semidirect product of C₁₉ and C₃ acting faithfully C19:C3 ID 57,1

class	1	3A	3B	19A	19B	19C	19D	19E	19F
size	1	19	19	3	3	3	3	3	3
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	ζ₃	ζ₃²	1	1	1	1	1	1	linear of order 3
ρ₃	1	ζ₃²	ζ₃	1	1	1	1	1	1	linear of order 3
ρ₄	3	0	0	ζ₁₉¹⁸+ζ₁₉¹²+ζ₁₉⁸	ζ₁₉¹⁵+ζ₁₉¹³+ζ₁₉¹⁰	ζ₁₉¹⁷+ζ₁₉¹⁶+ζ₁₉⁵	ζ₁₉¹¹+ζ₁₉⁷+ζ₁₉	ζ₁₉¹⁴+ζ₁₉³+ζ₁₉²	ζ₁₉⁹+ζ₁₉⁶+ζ₁₉⁴	complex faithful
ρ₅	3	0	0	ζ₁₉¹⁵+ζ₁₉¹³+ζ₁₉¹⁰	ζ₁₉¹⁴+ζ₁₉³+ζ₁₉²	ζ₁₉¹¹+ζ₁₉⁷+ζ₁₉	ζ₁₉⁹+ζ₁₉⁶+ζ₁₉⁴	ζ₁₉¹⁸+ζ₁₉¹²+ζ₁₉⁸	ζ₁₉¹⁷+ζ₁₉¹⁶+ζ₁₉⁵	complex faithful
ρ₆	3	0	0	ζ₁₉¹⁴+ζ₁₉³+ζ₁₉²	ζ₁₉¹⁸+ζ₁₉¹²+ζ₁₉⁸	ζ₁₉⁹+ζ₁₉⁶+ζ₁₉⁴	ζ₁₉¹⁷+ζ₁₉¹⁶+ζ₁₉⁵	ζ₁₉¹⁵+ζ₁₉¹³+ζ₁₉¹⁰	ζ₁₉¹¹+ζ₁₉⁷+ζ₁₉	complex faithful
ρ₇	3	0	0	ζ₁₉⁹+ζ₁₉⁶+ζ₁₉⁴	ζ₁₉¹⁷+ζ₁₉¹⁶+ζ₁₉⁵	ζ₁₉¹⁸+ζ₁₉¹²+ζ₁₉⁸	ζ₁₉¹⁵+ζ₁₉¹³+ζ₁₉¹⁰	ζ₁₉¹¹+ζ₁₉⁷+ζ₁₉	ζ₁₉¹⁴+ζ₁₉³+ζ₁₉²	complex faithful
ρ₈	3	0	0	ζ₁₉¹¹+ζ₁₉⁷+ζ₁₉	ζ₁₉⁹+ζ₁₉⁶+ζ₁₉⁴	ζ₁₉¹⁴+ζ₁₉³+ζ₁₉²	ζ₁₉¹⁸+ζ₁₉¹²+ζ₁₉⁸	ζ₁₉¹⁷+ζ₁₉¹⁶+ζ₁₉⁵	ζ₁₉¹⁵+ζ₁₉¹³+ζ₁₉¹⁰	complex faithful
ρ₉	3	0	0	ζ₁₉¹⁷+ζ₁₉¹⁶+ζ₁₉⁵	ζ₁₉¹¹+ζ₁₉⁷+ζ₁₉	ζ₁₉¹⁵+ζ₁₉¹³+ζ₁₉¹⁰	ζ₁₉¹⁴+ζ₁₉³+ζ₁₉²	ζ₁₉⁹+ζ₁₉⁶+ζ₁₉⁴	ζ₁₉¹⁸+ζ₁₉¹²+ζ₁₉⁸	complex faithful

Character table of C₃⋊F₅

C₃⋊F₅: The semidirect product of C₃ and F₅ acting via F₅/D₅=C₂ C3:F5 ID 60,7

class	1	2	3	4A	4B	5	6	15A	15B
size	1	5	2	15	15	4	10	4	4
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	-1	-1	1	1	1	1	linear of order 2
ρ₃	1	-1	1	i	-i	1	-1	1	1	linear of order 4
ρ₄	1	-1	1	-i	i	1	-1	1	1	linear of order 4
ρ₅	2	2	-1	0	0	2	-1	-1	-1	orthogonal lifted from S₃
ρ₆	2	-2	-1	0	0	2	1	-1	-1	symplectic lifted from Dic₃, Schur index 2
ρ₇	4	0	4	0	0	-1	0	-1	-1	orthogonal lifted from F₅
ρ₈	4	0	-2	0	0	-1	0	^1-√-15/₂	^1+√-15/₂	complex faithful
ρ₉	4	0	-2	0	0	-1	0	^1+√-15/₂	^1-√-15/₂	complex faithful

Character table of C₃.S₄

C₃.S₄: The non-split extension by C₃ of S₄ acting via S₄/A₄=C₂ C3.S4 ID 72,15

class	1	2A	2B	3	4	6	9A	9B	9C
size	1	3	18	2	18	6	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	-1	1	1	1	1	linear of order 2
ρ₃	2	2	0	2	0	2	-1	-1	-1	orthogonal lifted from S₃
ρ₄	2	2	0	-1	0	-1	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₅	2	2	0	-1	0	-1	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₆	2	2	0	-1	0	-1	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₇	3	-1	-1	3	1	-1	0	0	0	orthogonal lifted from S₄
ρ₈	3	-1	1	3	-1	-1	0	0	0	orthogonal lifted from S₄
ρ₉	6	-2	0	-3	0	1	0	0	0	orthogonal faithful

Character table of F₉

F₉: Frobenius group; = C₃² ⋊C₈ = AGL₁(𝔽₉) F9 ID 72,39

class	1	2	3	4A	4B	8A	8B	8C	8D
size	1	9	8	9	9	9	9	9	9
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	-1	-1	-i	-i	i	i	linear of order 4
ρ₄	1	1	1	-1	-1	i	i	-i	-i	linear of order 4
ρ₅	1	-1	1	-i	i	ζ₈⁵	ζ₈	ζ₈⁷	ζ₈³	linear of order 8
ρ₆	1	-1	1	i	-i	ζ₈⁷	ζ₈³	ζ₈⁵	ζ₈	linear of order 8
ρ₇	1	-1	1	i	-i	ζ₈³	ζ₈⁷	ζ₈	ζ₈⁵	linear of order 8
ρ₈	1	-1	1	-i	i	ζ₈	ζ₈⁵	ζ₈³	ζ₈⁷	linear of order 8
ρ₉	8	0	-1	0	0	0	0	0	0	orthogonal faithful

Character table of S₃≀C₂

S₃≀C₂: Wreath product of S₃ by C₂; = SO+₄(𝔽₂) S3wrC2 ID 72,40

class	1	2A	2B	2C	3A	3B	4	6A	6B
size	1	6	6	9	4	4	18	12	12
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	-1	1	-1	linear of order 2
ρ₃	1	1	-1	1	1	1	-1	-1	1	linear of order 2
ρ₄	1	-1	-1	1	1	1	1	-1	-1	linear of order 2
ρ₅	2	0	0	-2	2	2	0	0	0	orthogonal lifted from D₄
ρ₆	4	0	-2	0	-2	1	0	1	0	orthogonal faithful
ρ₇	4	-2	0	0	1	-2	0	0	1	orthogonal faithful
ρ₈	4	0	2	0	-2	1	0	-1	0	orthogonal faithful
ρ₉	4	2	0	0	1	-2	0	0	-1	orthogonal faithful

Character table of C₃⋊S₄

C₃⋊S₄: The semidirect product of C₃ and S₄ acting via S₄/A₄=C₂ C3:S4 ID 72,43

class	1	2A	2B	3A	3B	3C	3D	4	6
size	1	3	18	2	8	8	8	18	6
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	1	1	-1	1	linear of order 2
ρ₃	2	2	0	2	-1	-1	-1	0	2	orthogonal lifted from S₃
ρ₄	2	2	0	-1	2	-1	-1	0	-1	orthogonal lifted from S₃
ρ₅	2	2	0	-1	-1	-1	2	0	-1	orthogonal lifted from S₃
ρ₆	2	2	0	-1	-1	2	-1	0	-1	orthogonal lifted from S₃
ρ₇	3	-1	-1	3	0	0	0	1	-1	orthogonal lifted from S₄
ρ₈	3	-1	1	3	0	0	0	-1	-1	orthogonal lifted from S₄
ρ₉	6	-2	0	-3	0	0	0	0	1	orthogonal faithful

Character table of C₁₉⋊C₆

C₁₉⋊C₆: The semidirect product of C₁₉ and C₆ acting faithfully C19:C6 ID 114,1

class	1	2	3A	3B	6A	6B	19A	19B	19C
size	1	19	19	19	19	19	6	6	6
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	-1	-1	1	1	1	linear of order 2
ρ₃	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	1	1	linear of order 3
ρ₄	1	-1	ζ₃²	ζ₃	ζ₆	ζ₆⁵	1	1	1	linear of order 6
ρ₅	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	1	1	linear of order 3
ρ₆	1	-1	ζ₃	ζ₃²	ζ₆⁵	ζ₆	1	1	1	linear of order 6
ρ₇	6	0	0	0	0	0	ζ₁₉¹⁸+ζ₁₉¹²+ζ₁₉¹¹+ζ₁₉⁸+ζ₁₉⁷+ζ₁₉	ζ₁₉¹⁷+ζ₁₉¹⁶+ζ₁₉¹⁴+ζ₁₉⁵+ζ₁₉³+ζ₁₉²	ζ₁₉¹⁵+ζ₁₉¹³+ζ₁₉¹⁰+ζ₁₉⁹+ζ₁₉⁶+ζ₁₉⁴	orthogonal faithful
ρ₈	6	0	0	0	0	0	ζ₁₉¹⁵+ζ₁₉¹³+ζ₁₉¹⁰+ζ₁₉⁹+ζ₁₉⁶+ζ₁₉⁴	ζ₁₉¹⁸+ζ₁₉¹²+ζ₁₉¹¹+ζ₁₉⁸+ζ₁₉⁷+ζ₁₉	ζ₁₉¹⁷+ζ₁₉¹⁶+ζ₁₉¹⁴+ζ₁₉⁵+ζ₁₉³+ζ₁₉²	orthogonal faithful
ρ₉	6	0	0	0	0	0	ζ₁₉¹⁷+ζ₁₉¹⁶+ζ₁₉¹⁴+ζ₁₉⁵+ζ₁₉³+ζ₁₉²	ζ₁₉¹⁵+ζ₁₉¹³+ζ₁₉¹⁰+ζ₁₉⁹+ζ₁₉⁶+ζ₁₉⁴	ζ₁₉¹⁸+ζ₁₉¹²+ζ₁₉¹¹+ζ₁₉⁸+ζ₁₉⁷+ζ₁₉	orthogonal faithful

Character table of SL₂(𝔽₅)

SL₂(𝔽₅): Special linear group on 𝔽₅²; = C₂ .A₅ = 2I = <2,3,5> SL(2,5) ID 120,5

class	1	2	3	4	5A	5B	6	10A	10B
size	1	1	20	30	12	12	20	12	12
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	2	-2	-1	0	^-1+√5/₂	^-1-√5/₂	1	^1+√5/₂	^1-√5/₂	symplectic faithful, Schur index 2
ρ₃	2	-2	-1	0	^-1-√5/₂	^-1+√5/₂	1	^1-√5/₂	^1+√5/₂	symplectic faithful, Schur index 2
ρ₄	3	3	0	-1	^1-√5/₂	^1+√5/₂	0	^1+√5/₂	^1-√5/₂	orthogonal lifted from A₅
ρ₅	3	3	0	-1	^1+√5/₂	^1-√5/₂	0	^1-√5/₂	^1+√5/₂	orthogonal lifted from A₅
ρ₆	4	4	1	0	-1	-1	1	-1	-1	orthogonal lifted from A₅
ρ₇	4	-4	1	0	-1	-1	-1	1	1	symplectic faithful, Schur index 2
ρ₈	5	5	-1	1	0	0	-1	0	0	orthogonal lifted from A₅
ρ₉	6	-6	0	0	1	1	0	-1	-1	symplectic faithful, Schur index 2

Character table of AΓL₁(𝔽₉)

AΓL₁(𝔽₉): Affine semilinear group on 𝔽₉¹; = F₉ ⋊C₂ = Aut(C₃²⋊C₄) AGammaL(1,9) ID 144,182

class	1	2A	2B	3	4A	4B	6	8A	8B
size	1	9	12	8	18	36	24	18	18
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	-1	1	-1	-1	linear of order 2
ρ₃	1	1	-1	1	1	-1	-1	1	1	linear of order 2
ρ₄	1	1	-1	1	1	1	-1	-1	-1	linear of order 2
ρ₅	2	2	0	2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₆	2	-2	0	2	0	0	0	√-2	-√-2	complex lifted from SD₁₆
ρ₇	2	-2	0	2	0	0	0	-√-2	√-2	complex lifted from SD₁₆
ρ₈	8	0	-2	-1	0	0	1	0	0	orthogonal faithful
ρ₉	8	0	2	-1	0	0	-1	0	0	orthogonal faithful

Character table of C₄²⋊A₄

C₄²⋊A₄: The semidirect product of C₄² and A₄ acting faithfully C4^2:A4 ID 192,1023

class	1	2A	2B	2C	3A	3B	4A	4B	4C
size	1	3	12	12	64	64	12	12	12
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	ζ₃²	ζ₃	1	1	1	linear of order 3
ρ₃	1	1	1	1	ζ₃	ζ₃²	1	1	1	linear of order 3
ρ₄	3	3	3	-1	0	0	-1	-1	-1	orthogonal lifted from A₄
ρ₅	3	3	-1	-1	0	0	-1	-1	3	orthogonal lifted from A₄
ρ₆	3	3	-1	3	0	0	-1	-1	-1	orthogonal lifted from A₄
ρ₇	3	3	-1	-1	0	0	3	-1	-1	orthogonal lifted from A₄
ρ₈	3	3	-1	-1	0	0	-1	3	-1	orthogonal lifted from A₄
ρ₉	12	-4	0	0	0	0	0	0	0	orthogonal faithful

Character table of C₄².A₄

C₄².A₄: The non-split extension by C₄² of A₄ acting faithfully C4^2.A4 ID 192,1025

class	1	2	3A	3B	4A	4B	4C	4D	4E
size	1	3	64	64	12	12	12	12	12
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	ζ₃	ζ₃²	1	1	1	1	1	linear of order 3
ρ₃	1	1	ζ₃²	ζ₃	1	1	1	1	1	linear of order 3
ρ₄	3	3	0	0	-1	-1	-1	-1	3	orthogonal lifted from A₄
ρ₅	3	3	0	0	-1	3	-1	-1	-1	orthogonal lifted from A₄
ρ₆	3	3	0	0	3	-1	-1	-1	-1	orthogonal lifted from A₄
ρ₇	3	3	0	0	-1	-1	3	-1	-1	orthogonal lifted from A₄
ρ₈	3	3	0	0	-1	-1	-1	3	-1	orthogonal lifted from A₄
ρ₉	12	-4	0	0	0	0	0	0	0	symplectic faithful, Schur index 2

Character table of PGL₂(𝔽₇)

PGL₂(𝔽₇): Projective linear group on 𝔽₇²; = GL₃(𝔽₂)⋊C₂ = Aut(GL₃(𝔽₂)); almost simple PGL(2,7) ID 336,208

class	1	2A	2B	3	4	6	7	8A	8B
size	1	21	28	56	42	56	48	42	42
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	-1	1	-1	-1	linear of order 2
ρ₃	6	-2	0	0	2	0	-1	0	0	orthogonal faithful
ρ₄	6	2	0	0	0	0	-1	-√2	√2	orthogonal faithful
ρ₅	6	2	0	0	0	0	-1	√2	-√2	orthogonal faithful
ρ₆	7	-1	1	1	-1	1	0	-1	-1	orthogonal faithful
ρ₇	7	-1	-1	1	-1	-1	0	1	1	orthogonal faithful
ρ₈	8	0	-2	-1	0	1	1	0	0	orthogonal faithful
ρ₉	8	0	2	-1	0	-1	1	0	0	orthogonal faithful

10x10 character tables

Character table of C₁₀

C₁₀: Cyclic group C10 ID 10,2

class	1	2	5A	5B	5C	5D	10A	10B	10C	10D
size	1	1	1	1	1	1	1	1	1	1
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	ζ₅²	ζ₅³	ζ₅⁴	ζ₅	ζ₅⁴	ζ₅²	ζ₅³	ζ₅	linear of order 5
ρ₄	1	-1	ζ₅²	ζ₅³	ζ₅⁴	ζ₅	-ζ₅⁴	-ζ₅²	-ζ₅³	-ζ₅	linear of order 10 faithful
ρ₅	1	1	ζ₅⁴	ζ₅	ζ₅³	ζ₅²	ζ₅³	ζ₅⁴	ζ₅	ζ₅²	linear of order 5
ρ₆	1	-1	ζ₅⁴	ζ₅	ζ₅³	ζ₅²	-ζ₅³	-ζ₅⁴	-ζ₅	-ζ₅²	linear of order 10 faithful
ρ₇	1	1	ζ₅	ζ₅⁴	ζ₅²	ζ₅³	ζ₅²	ζ₅	ζ₅⁴	ζ₅³	linear of order 5
ρ₈	1	-1	ζ₅	ζ₅⁴	ζ₅²	ζ₅³	-ζ₅²	-ζ₅	-ζ₅⁴	-ζ₅³	linear of order 10 faithful
ρ₉	1	1	ζ₅³	ζ₅²	ζ₅	ζ₅⁴	ζ₅	ζ₅³	ζ₅²	ζ₅⁴	linear of order 5
ρ₁₀	1	-1	ζ₅³	ζ₅²	ζ₅	ζ₅⁴	-ζ₅	-ζ₅³	-ζ₅²	-ζ₅⁴	linear of order 10 faithful

Character table of C₂²⋊C₄

C₂²⋊C₄: The semidirect product of C₂² and C₄ acting via C₄/C₂=C₂ C2^2:C4 ID 16,3

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D
size	1	1	1	1	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	-1	1	1	-1	linear of order 2
ρ₃	1	1	1	1	-1	-1	1	-1	-1	1	linear of order 2
ρ₄	1	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₅	1	1	-1	-1	-1	1	i	-i	i	-i	linear of order 4
ρ₆	1	1	-1	-1	1	-1	-i	-i	i	i	linear of order 4
ρ₇	1	1	-1	-1	-1	1	-i	i	-i	i	linear of order 4
ρ₈	1	1	-1	-1	1	-1	i	i	-i	-i	linear of order 4
ρ₉	2	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄

Character table of C₄⋊C₄

C₄⋊C₄: The semidirect product of C₄ and C₄ acting via C₄/C₂=C₂ C4:C4 ID 16,4

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F
size	1	1	1	1	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	1	-1	-1	1	-1	linear of order 2
ρ₃	1	1	1	1	-1	-1	1	-1	-1	1	linear of order 2
ρ₄	1	1	1	1	1	-1	-1	1	-1	-1	linear of order 2
ρ₅	1	1	-1	-1	-1	-i	i	1	i	-i	linear of order 4
ρ₆	1	1	-1	-1	1	-i	-i	-1	i	i	linear of order 4
ρ₇	1	1	-1	-1	-1	i	-i	1	-i	i	linear of order 4
ρ₈	1	1	-1	-1	1	i	i	-1	-i	-i	linear of order 4
ρ₉	2	-2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	-2	2	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2

Character table of M₄(2)

M₄(2): Modular maximal-cyclic group; = C₈ ⋊₃ C₂ M4(2) ID 16,6

class	1	2A	2B	4A	4B	4C	8A	8B	8C	8D
size	1	1	2	1	1	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	-1	1	-1	-1	1	linear of order 2
ρ₃	1	1	-1	1	1	-1	-1	1	1	-1	linear of order 2
ρ₄	1	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₅	1	1	-1	-1	-1	1	i	-i	i	-i	linear of order 4
ρ₆	1	1	1	-1	-1	-1	i	i	-i	-i	linear of order 4
ρ₇	1	1	-1	-1	-1	1	-i	i	-i	i	linear of order 4
ρ₈	1	1	1	-1	-1	-1	-i	-i	i	i	linear of order 4
ρ₉	2	-2	0	2i	-2i	0	0	0	0	0	complex faithful
ρ₁₀	2	-2	0	-2i	2i	0	0	0	0	0	complex faithful

Character table of C₂×D₄

C₂×D₄: Direct product of C₂ and D₄ C2xD4 ID 16,11

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B
size	1	1	1	1	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	-1	-1	1	-1	1	1	-1	linear of order 2
ρ₃	1	-1	1	-1	1	-1	1	-1	1	-1	linear of order 2
ρ₄	1	1	1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₅	1	-1	1	-1	1	-1	-1	1	-1	1	linear of order 2
ρ₆	1	-1	1	-1	-1	1	1	-1	-1	1	linear of order 2
ρ₇	1	1	1	1	-1	-1	1	1	-1	-1	linear of order 2
ρ₈	1	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₉	2	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	-2	-2	0	0	0	0	0	0	orthogonal lifted from D₄

Character table of C₂×Q₈

C₂×Q₈: Direct product of C₂ and Q₈ C2xQ8 ID 16,12

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F
size	1	1	1	1	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	-1	-1	1	-1	1	1	-1	linear of order 2
ρ₃	1	-1	1	-1	1	-1	1	-1	1	-1	linear of order 2
ρ₄	1	1	1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₅	1	-1	1	-1	1	-1	-1	1	-1	1	linear of order 2
ρ₆	1	-1	1	-1	-1	1	1	-1	-1	1	linear of order 2
ρ₇	1	1	1	1	-1	-1	1	1	-1	-1	linear of order 2
ρ₈	1	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₉	2	-2	-2	2	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₀	2	2	-2	-2	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2

Character table of C₄○D₄

C₄○D₄: Pauli group = central product of C₄ and D₄ C4oD4 ID 16,13

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E
size	1	1	2	2	2	1	1	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	-1	1	-1	-1	-1	-1	1	linear of order 2
ρ₃	1	1	-1	1	-1	1	1	-1	-1	1	linear of order 2
ρ₄	1	1	1	1	-1	-1	-1	1	-1	-1	linear of order 2
ρ₅	1	1	-1	-1	1	1	1	1	-1	-1	linear of order 2
ρ₆	1	1	-1	-1	-1	-1	-1	1	1	1	linear of order 2
ρ₇	1	1	1	-1	-1	1	1	-1	1	-1	linear of order 2
ρ₈	1	1	-1	1	1	-1	-1	-1	1	-1	linear of order 2
ρ₉	2	-2	0	0	0	-2i	2i	0	0	0	complex faithful
ρ₁₀	2	-2	0	0	0	2i	-2i	0	0	0	complex faithful

Character table of Dic₇

Dic₇: Dicyclic group; = C₇ ⋊C₄ Dic7 ID 28,1

class	1	2	4A	4B	7A	7B	7C	14A	14B	14C
size	1	1	7	7	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	1	1	linear of order 2
ρ₃	1	-1	-i	i	1	1	1	-1	-1	-1	linear of order 4
ρ₄	1	-1	i	-i	1	1	1	-1	-1	-1	linear of order 4
ρ₅	2	2	0	0	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	orthogonal lifted from D₇
ρ₆	2	2	0	0	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	orthogonal lifted from D₇
ρ₇	2	2	0	0	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	orthogonal lifted from D₇
ρ₈	2	-2	0	0	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	-ζ₇⁵-ζ₇²	-ζ₇⁴-ζ₇³	-ζ₇⁶-ζ₇	symplectic faithful, Schur index 2
ρ₉	2	-2	0	0	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	-ζ₇⁶-ζ₇	-ζ₇⁵-ζ₇²	-ζ₇⁴-ζ₇³	symplectic faithful, Schur index 2
ρ₁₀	2	-2	0	0	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	-ζ₇⁴-ζ₇³	-ζ₇⁶-ζ₇	-ζ₇⁵-ζ₇²	symplectic faithful, Schur index 2

Character table of D₁₄

D₁₄: Dihedral group; = C₂×D₇ D14 ID 28,3

class	1	2A	2B	2C	7A	7B	7C	14A	14B	14C
size	1	1	7	7	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	-1	1	1	1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	-1	-1	1	1	1	1	1	1	linear of order 2
ρ₄	1	-1	1	-1	1	1	1	-1	-1	-1	linear of order 2
ρ₅	2	2	0	0	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	orthogonal lifted from D₇
ρ₆	2	2	0	0	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	orthogonal lifted from D₇
ρ₇	2	-2	0	0	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	-ζ₇⁵-ζ₇²	-ζ₇⁴-ζ₇³	-ζ₇⁶-ζ₇	orthogonal faithful
ρ₈	2	2	0	0	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	orthogonal lifted from D₇
ρ₉	2	-2	0	0	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	-ζ₇⁶-ζ₇	-ζ₇⁵-ζ₇²	-ζ₇⁴-ζ₇³	orthogonal faithful
ρ₁₀	2	-2	0	0	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	-ζ₇⁴-ζ₇³	-ζ₇⁶-ζ₇	-ζ₇⁵-ζ₇²	orthogonal faithful

Character table of D₁₇

D₁₇: Dihedral group D17 ID 34,1

class	1	2	17A	17B	17C	17D	17E	17F	17G	17H
size	1	17	2	2	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	1	1	1	linear of order 2
ρ₃	2	0	ζ₁₇¹⁵+ζ₁₇²	ζ₁₇¹⁴+ζ₁₇³	ζ₁₇¹³+ζ₁₇⁴	ζ₁₇¹²+ζ₁₇⁵	ζ₁₇¹¹+ζ₁₇⁶	ζ₁₇¹⁰+ζ₁₇⁷	ζ₁₇⁹+ζ₁₇⁸	ζ₁₇¹⁶+ζ₁₇	orthogonal faithful
ρ₄	2	0	ζ₁₇¹⁰+ζ₁₇⁷	ζ₁₇¹⁵+ζ₁₇²	ζ₁₇¹⁴+ζ₁₇³	ζ₁₇⁹+ζ₁₇⁸	ζ₁₇¹³+ζ₁₇⁴	ζ₁₇¹⁶+ζ₁₇	ζ₁₇¹¹+ζ₁₇⁶	ζ₁₇¹²+ζ₁₇⁵	orthogonal faithful
ρ₅	2	0	ζ₁₇⁹+ζ₁₇⁸	ζ₁₇¹²+ζ₁₇⁵	ζ₁₇¹⁶+ζ₁₇	ζ₁₇¹⁴+ζ₁₇³	ζ₁₇¹⁰+ζ₁₇⁷	ζ₁₇¹¹+ζ₁₇⁶	ζ₁₇¹⁵+ζ₁₇²	ζ₁₇¹³+ζ₁₇⁴	orthogonal faithful
ρ₆	2	0	ζ₁₇¹⁴+ζ₁₇³	ζ₁₇¹³+ζ₁₇⁴	ζ₁₇¹¹+ζ₁₇⁶	ζ₁₇¹⁶+ζ₁₇	ζ₁₇⁹+ζ₁₇⁸	ζ₁₇¹⁵+ζ₁₇²	ζ₁₇¹²+ζ₁₇⁵	ζ₁₇¹⁰+ζ₁₇⁷	orthogonal faithful
ρ₇	2	0	ζ₁₇¹¹+ζ₁₇⁶	ζ₁₇⁹+ζ₁₇⁸	ζ₁₇¹²+ζ₁₇⁵	ζ₁₇¹⁵+ζ₁₇²	ζ₁₇¹⁶+ζ₁₇	ζ₁₇¹³+ζ₁₇⁴	ζ₁₇¹⁰+ζ₁₇⁷	ζ₁₇¹⁴+ζ₁₇³	orthogonal faithful
ρ₈	2	0	ζ₁₇¹²+ζ₁₇⁵	ζ₁₇¹⁶+ζ₁₇	ζ₁₇¹⁰+ζ₁₇⁷	ζ₁₇¹³+ζ₁₇⁴	ζ₁₇¹⁵+ζ₁₇²	ζ₁₇⁹+ζ₁₇⁸	ζ₁₇¹⁴+ζ₁₇³	ζ₁₇¹¹+ζ₁₇⁶	orthogonal faithful
ρ₉	2	0	ζ₁₇¹³+ζ₁₇⁴	ζ₁₇¹¹+ζ₁₇⁶	ζ₁₇⁹+ζ₁₇⁸	ζ₁₇¹⁰+ζ₁₇⁷	ζ₁₇¹²+ζ₁₇⁵	ζ₁₇¹⁴+ζ₁₇³	ζ₁₇¹⁶+ζ₁₇	ζ₁₇¹⁵+ζ₁₇²	orthogonal faithful
ρ₁₀	2	0	ζ₁₇¹⁶+ζ₁₇	ζ₁₇¹⁰+ζ₁₇⁷	ζ₁₇¹⁵+ζ₁₇²	ζ₁₇¹¹+ζ₁₇⁶	ζ₁₇¹⁴+ζ₁₇³	ζ₁₇¹²+ζ₁₇⁵	ζ₁₇¹³+ζ₁₇⁴	ζ₁₇⁹+ζ₁₇⁸	orthogonal faithful

Character table of C₅⋊C₈

C₅⋊C₈: The semidirect product of C₅ and C₈ acting via C₈/C₂=C₄ C5:C8 ID 40,3

class	1	2	4A	4B	5	8A	8B	8C	8D	10
size	1	1	5	5	4	5	5	5	5	4
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	-1	-1	-1	-1	1	linear of order 2
ρ₃	1	1	-1	-1	1	i	-i	i	-i	1	linear of order 4
ρ₄	1	1	-1	-1	1	-i	i	-i	i	1	linear of order 4
ρ₅	1	-1	i	-i	1	ζ₈⁵	ζ₈⁷	ζ₈	ζ₈³	-1	linear of order 8
ρ₆	1	-1	-i	i	1	ζ₈³	ζ₈	ζ₈⁷	ζ₈⁵	-1	linear of order 8
ρ₇	1	-1	i	-i	1	ζ₈	ζ₈³	ζ₈⁵	ζ₈⁷	-1	linear of order 8
ρ₈	1	-1	-i	i	1	ζ₈⁷	ζ₈⁵	ζ₈³	ζ₈	-1	linear of order 8
ρ₉	4	4	0	0	-1	0	0	0	0	-1	orthogonal lifted from F₅
ρ₁₀	4	-4	0	0	-1	0	0	0	0	1	symplectic faithful, Schur index 2

Character table of C₂×F₅

C₂×F₅: Direct product of C₂ and F₅; = Aut(D₁₀) = Hol(C₁₀) C2xF5 ID 40,12

class	1	2A	2B	2C	4A	4B	4C	4D	5	10
size	1	1	5	5	5	5	5	5	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₃	1	-1	-1	1	-1	1	1	-1	1	-1	linear of order 2
ρ₄	1	-1	-1	1	1	-1	-1	1	1	-1	linear of order 2
ρ₅	1	-1	1	-1	i	i	-i	-i	1	-1	linear of order 4
ρ₆	1	-1	1	-1	-i	-i	i	i	1	-1	linear of order 4
ρ₇	1	1	-1	-1	-i	i	-i	i	1	1	linear of order 4
ρ₈	1	1	-1	-1	i	-i	i	-i	1	1	linear of order 4
ρ₉	4	-4	0	0	0	0	0	0	-1	1	orthogonal faithful
ρ₁₀	4	4	0	0	0	0	0	0	-1	-1	orthogonal lifted from F₅

Character table of C₂×C₇⋊C₃

C₂×C₇⋊C₃: Direct product of C₂ and C₇⋊C₃ C2xC7:C3 ID 42,2

class	1	2	3A	3B	6A	6B	7A	7B	14A	14B
size	1	1	7	7	7	7	3	3	3	3
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	-1	-1	1	1	-1	-1	linear of order 2
ρ₃	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	1	1	1	1	linear of order 3
ρ₄	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	1	1	1	1	linear of order 3
ρ₅	1	-1	ζ₃	ζ₃²	ζ₆	ζ₆⁵	1	1	-1	-1	linear of order 6
ρ₆	1	-1	ζ₃²	ζ₃	ζ₆⁵	ζ₆	1	1	-1	-1	linear of order 6
ρ₇	3	3	0	0	0	0	^-1+√-7/₂	^-1-√-7/₂	^-1-√-7/₂	^-1+√-7/₂	complex lifted from C₇⋊C₃
ρ₈	3	-3	0	0	0	0	^-1+√-7/₂	^-1-√-7/₂	^1+√-7/₂	^1-√-7/₂	complex faithful
ρ₉	3	-3	0	0	0	0	^-1-√-7/₂	^-1+√-7/₂	^1-√-7/₂	^1+√-7/₂	complex faithful
ρ₁₀	3	3	0	0	0	0	^-1-√-7/₂	^-1+√-7/₂	^-1+√-7/₂	^-1-√-7/₂	complex lifted from C₇⋊C₃

Character table of A₄⋊C₄

A₄⋊C₄: The semidirect product of A₄ and C₄ acting via C₄/C₂=C₂; = SL₂(ℤ/4ℤ) A4:C4 ID 48,30

class	1	2A	2B	2C	3	4A	4B	4C	4D	6
size	1	1	3	3	8	6	6	6	6	8
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	-1	-1	-1	-1	1	linear of order 2
ρ₃	1	-1	1	-1	1	-i	i	-i	i	-1	linear of order 4
ρ₄	1	-1	1	-1	1	i	-i	i	-i	-1	linear of order 4
ρ₅	2	2	2	2	-1	0	0	0	0	-1	orthogonal lifted from S₃
ρ₆	2	-2	2	-2	-1	0	0	0	0	1	symplectic lifted from Dic₃, Schur index 2
ρ₇	3	3	-1	-1	0	1	1	-1	-1	0	orthogonal lifted from S₄
ρ₈	3	3	-1	-1	0	-1	-1	1	1	0	orthogonal lifted from S₄
ρ₉	3	-3	-1	1	0	i	-i	-i	i	0	complex faithful
ρ₁₀	3	-3	-1	1	0	-i	i	i	-i	0	complex faithful

Character table of C₂×S₄

C₂×S₄: Direct product of C₂ and S₄; = O₃(𝔽₃) = cube/octahedron symmetries C2xS4 ID 48,48

class	1	2A	2B	2C	2D	2E	3	4A	4B	6
size	1	1	3	3	6	6	8	6	6	8
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	-1	1	-1	1	-1	1	-1	linear of order 2
ρ₃	1	1	1	1	-1	-1	1	-1	-1	1	linear of order 2
ρ₄	1	-1	1	-1	-1	1	1	1	-1	-1	linear of order 2
ρ₅	2	-2	2	-2	0	0	-1	0	0	1	orthogonal lifted from D₆
ρ₆	2	2	2	2	0	0	-1	0	0	-1	orthogonal lifted from S₃
ρ₇	3	3	-1	-1	1	1	0	-1	-1	0	orthogonal lifted from S₄
ρ₈	3	-3	-1	1	-1	1	0	-1	1	0	orthogonal faithful
ρ₉	3	-3	-1	1	1	-1	0	1	-1	0	orthogonal faithful
ρ₁₀	3	3	-1	-1	-1	-1	0	1	1	0	orthogonal lifted from S₄

Character table of C₃²⋊C₆

C₃²⋊C₆: The semidirect product of C₃² and C₆ acting faithfully C3^2:C6 ID 54,5

class	1	2	3A	3B	3C	3D	3E	3F	6A	6B
size	1	9	2	3	3	6	6	6	9	9
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	1	-1	-1	linear of order 2
ρ₃	1	1	1	ζ₃	ζ₃²	1	ζ₃	ζ₃²	ζ₃²	ζ₃	linear of order 3
ρ₄	1	-1	1	ζ₃	ζ₃²	1	ζ₃	ζ₃²	ζ₆	ζ₆⁵	linear of order 6
ρ₅	1	1	1	ζ₃²	ζ₃	1	ζ₃²	ζ₃	ζ₃	ζ₃²	linear of order 3
ρ₆	1	-1	1	ζ₃²	ζ₃	1	ζ₃²	ζ₃	ζ₆⁵	ζ₆	linear of order 6
ρ₇	2	0	2	2	2	-1	-1	-1	0	0	orthogonal lifted from S₃
ρ₈	2	0	2	-1-√-3	-1+√-3	-1	ζ₆	ζ₆⁵	0	0	complex lifted from C₃×S₃
ρ₉	2	0	2	-1+√-3	-1-√-3	-1	ζ₆⁵	ζ₆	0	0	complex lifted from C₃×S₃
ρ₁₀	6	0	-3	0	0	0	0	0	0	0	orthogonal faithful

Character table of C₉⋊C₆

C₉⋊C₆: The semidirect product of C₉ and C₆ acting faithfully; = Aut(D₉) = Hol(C₉) C9:C6 ID 54,6

class	1	2	3A	3B	3C	6A	6B	9A	9B	9C
size	1	9	2	3	3	9	9	6	6	6
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	-1	-1	1	1	1	linear of order 2
ρ₃	1	-1	1	ζ₃	ζ₃²	ζ₆	ζ₆⁵	ζ₃²	1	ζ₃	linear of order 6
ρ₄	1	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃	1	ζ₃²	linear of order 3
ρ₅	1	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃²	1	ζ₃	linear of order 3
ρ₆	1	-1	1	ζ₃²	ζ₃	ζ₆⁵	ζ₆	ζ₃	1	ζ₃²	linear of order 6
ρ₇	2	0	2	2	2	0	0	-1	-1	-1	orthogonal lifted from S₃
ρ₈	2	0	2	-1+√-3	-1-√-3	0	0	ζ₆	-1	ζ₆⁵	complex lifted from C₃×S₃
ρ₉	2	0	2	-1-√-3	-1+√-3	0	0	ζ₆⁵	-1	ζ₆	complex lifted from C₃×S₃
ρ₁₀	6	0	-3	0	0	0	0	0	0	0	orthogonal faithful

Character table of He₃⋊C₂

He₃⋊C₂: 2^nd semidirect product of He₃ and C₂ acting faithfully; = Aut(3_-¹⁺²) He3:C2 ID 54,8

class	1	2	3A	3B	3C	3D	3E	3F	6A	6B
size	1	9	1	1	6	6	6	6	9	9
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	1	-1	-1	linear of order 2
ρ₃	2	0	2	2	-1	-1	2	-1	0	0	orthogonal lifted from S₃
ρ₄	2	0	2	2	2	-1	-1	-1	0	0	orthogonal lifted from S₃
ρ₅	2	0	2	2	-1	-1	-1	2	0	0	orthogonal lifted from S₃
ρ₆	2	0	2	2	-1	2	-1	-1	0	0	orthogonal lifted from S₃
ρ₇	3	1	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	0	ζ₃	ζ₃²	complex faithful
ρ₈	3	1	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	0	ζ₃²	ζ₃	complex faithful
ρ₉	3	-1	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	0	ζ₆	ζ₆⁵	complex faithful
ρ₁₀	3	-1	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	0	ζ₆⁵	ζ₆	complex faithful

Character table of C₄²⋊S₃

C₄²⋊S₃: The semidirect product of C₄² and S₃ acting faithfully C4^2:S3 ID 96,64

class	1	2A	2B	3	4A	4B	4C	4D	8A	8B
size	1	3	12	32	3	3	6	12	12	12
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	1	1	-1	-1	-1	linear of order 2
ρ₃	2	2	0	-1	2	2	2	0	0	0	orthogonal lifted from S₃
ρ₄	3	3	-1	0	-1	-1	-1	-1	1	1	orthogonal lifted from S₄
ρ₅	3	3	1	0	-1	-1	-1	1	-1	-1	orthogonal lifted from S₄
ρ₆	3	-1	1	0	-1+2i	-1-2i	1	-1	-i	i	complex faithful
ρ₇	3	-1	-1	0	-1+2i	-1-2i	1	1	i	-i	complex faithful
ρ₈	3	-1	-1	0	-1-2i	-1+2i	1	1	-i	i	complex faithful
ρ₉	3	-1	1	0	-1-2i	-1+2i	1	-1	i	-i	complex faithful
ρ₁₀	6	-2	0	0	2	2	-2	0	0	0	orthogonal faithful

Character table of C₂⁴⋊C₆

C₂⁴⋊C₆: 1^st semidirect product of C₂⁴ and C₆ acting faithfully C2^4:C6 ID 96,70

class	1	2A	2B	2C	2D	3A	3B	4	6A	6B
size	1	3	4	6	6	16	16	12	16	16
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	ζ₃	ζ₃²	1	ζ₃²	ζ₃	linear of order 3
ρ₄	1	1	-1	1	1	ζ₃²	ζ₃	-1	ζ₆⁵	ζ₆	linear of order 6
ρ₅	1	1	-1	1	1	ζ₃	ζ₃²	-1	ζ₆	ζ₆⁵	linear of order 6
ρ₆	1	1	1	1	1	ζ₃²	ζ₃	1	ζ₃	ζ₃²	linear of order 3
ρ₇	3	3	3	-1	-1	0	0	-1	0	0	orthogonal lifted from A₄
ρ₈	3	3	-3	-1	-1	0	0	1	0	0	orthogonal lifted from C₂×A₄
ρ₉	6	-2	0	2	-2	0	0	0	0	0	orthogonal faithful
ρ₁₀	6	-2	0	-2	2	0	0	0	0	0	orthogonal faithful

Character table of C₄²⋊C₆

C₄²⋊C₆: 1^st semidirect product of C₄² and C₆ acting faithfully C4^2:C6 ID 96,71

class	1	2A	2B	3A	3B	4A	4B	4C	6A	6B
size	1	3	4	16	16	6	6	12	16	16
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	-1	ζ₃	ζ₃²	1	1	-1	ζ₆	ζ₆⁵	linear of order 6
ρ₄	1	1	-1	ζ₃²	ζ₃	1	1	-1	ζ₆⁵	ζ₆	linear of order 6
ρ₅	1	1	1	ζ₃	ζ₃²	1	1	1	ζ₃²	ζ₃	linear of order 3
ρ₆	1	1	1	ζ₃²	ζ₃	1	1	1	ζ₃	ζ₃²	linear of order 3
ρ₇	3	3	3	0	0	-1	-1	-1	0	0	orthogonal lifted from A₄
ρ₈	3	3	-3	0	0	-1	-1	1	0	0	orthogonal lifted from C₂×A₄
ρ₉	6	-2	0	0	0	-2i	2i	0	0	0	complex faithful
ρ₁₀	6	-2	0	0	0	2i	-2i	0	0	0	complex faithful

Character table of C₂³.A₄

C₂³.A₄: 2^nd non-split extension by C₂³ of A₄ acting faithfully C2^3.A4 ID 96,72

class	1	2A	2B	2C	3A	3B	4A	4B	6A	6B
size	1	3	4	12	16	16	6	6	16	16
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	-1	-1	linear of order 2
ρ₃	1	1	1	1	ζ₃²	ζ₃	1	1	ζ₃	ζ₃²	linear of order 3
ρ₄	1	1	1	1	ζ₃	ζ₃²	1	1	ζ₃²	ζ₃	linear of order 3
ρ₅	1	1	-1	-1	ζ₃	ζ₃²	1	1	ζ₆	ζ₆⁵	linear of order 6
ρ₆	1	1	-1	-1	ζ₃²	ζ₃	1	1	ζ₆⁵	ζ₆	linear of order 6
ρ₇	3	3	3	-1	0	0	-1	-1	0	0	orthogonal lifted from A₄
ρ₈	3	3	-3	1	0	0	-1	-1	0	0	orthogonal lifted from C₂×A₄
ρ₉	6	-2	0	0	0	0	2	-2	0	0	orthogonal faithful
ρ₁₀	6	-2	0	0	0	0	-2	2	0	0	orthogonal faithful

Character table of C₂²⋊S₄

C₂²⋊S₄: The semidirect product of C₂² and S₄ acting via S₄/C₂²=S₃ C2^2:S4 ID 96,227

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C
size	1	3	3	3	6	12	32	12	12	12
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	-1	1	-1	-1	-1	linear of order 2
ρ₃	2	2	2	2	2	0	-1	0	0	0	orthogonal lifted from S₃
ρ₄	3	-1	3	-1	-1	-1	0	1	-1	1	orthogonal lifted from S₄
ρ₅	3	-1	-1	3	-1	1	0	-1	-1	1	orthogonal lifted from S₄
ρ₆	3	3	-1	-1	-1	-1	0	-1	1	1	orthogonal lifted from S₄
ρ₇	3	-1	3	-1	-1	1	0	-1	1	-1	orthogonal lifted from S₄
ρ₈	3	-1	-1	3	-1	-1	0	1	1	-1	orthogonal lifted from S₄
ρ₉	3	3	-1	-1	-1	1	0	1	-1	-1	orthogonal lifted from S₄
ρ₁₀	6	-2	-2	-2	2	0	0	0	0	0	orthogonal faithful

Character table of C₂₅⋊C₄

C₂₅⋊C₄: The semidirect product of C₂₅ and C₄ acting faithfully C25:C4 ID 100,3

class	1	2	4A	4B	5	25A	25B	25C	25D	25E
size	1	25	25	25	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	1	1	linear of order 2
ρ₃	1	-1	-i	i	1	1	1	1	1	1	linear of order 4
ρ₄	1	-1	i	-i	1	1	1	1	1	1	linear of order 4
ρ₅	4	0	0	0	4	-1	-1	-1	-1	-1	orthogonal lifted from F₅
ρ₆	4	0	0	0	-1	ζ₂₅¹⁶+ζ₂₅¹³+ζ₂₅¹²+ζ₂₅⁹	ζ₂₅²⁴+ζ₂₅¹⁸+ζ₂₅⁷+ζ₂₅	ζ₂₅²²+ζ₂₅²¹+ζ₂₅⁴+ζ₂₅³	ζ₂₅¹⁹+ζ₂₅¹⁷+ζ₂₅⁸+ζ₂₅⁶	ζ₂₅²³+ζ₂₅¹⁴+ζ₂₅¹¹+ζ₂₅²	orthogonal faithful
ρ₇	4	0	0	0	-1	ζ₂₅²⁴+ζ₂₅¹⁸+ζ₂₅⁷+ζ₂₅	ζ₂₅²³+ζ₂₅¹⁴+ζ₂₅¹¹+ζ₂₅²	ζ₂₅¹⁹+ζ₂₅¹⁷+ζ₂₅⁸+ζ₂₅⁶	ζ₂₅¹⁶+ζ₂₅¹³+ζ₂₅¹²+ζ₂₅⁹	ζ₂₅²²+ζ₂₅²¹+ζ₂₅⁴+ζ₂₅³	orthogonal faithful
ρ₈	4	0	0	0	-1	ζ₂₅¹⁹+ζ₂₅¹⁷+ζ₂₅⁸+ζ₂₅⁶	ζ₂₅¹⁶+ζ₂₅¹³+ζ₂₅¹²+ζ₂₅⁹	ζ₂₅²³+ζ₂₅¹⁴+ζ₂₅¹¹+ζ₂₅²	ζ₂₅²²+ζ₂₅²¹+ζ₂₅⁴+ζ₂₅³	ζ₂₅²⁴+ζ₂₅¹⁸+ζ₂₅⁷+ζ₂₅	orthogonal faithful
ρ₉	4	0	0	0	-1	ζ₂₅²²+ζ₂₅²¹+ζ₂₅⁴+ζ₂₅³	ζ₂₅¹⁹+ζ₂₅¹⁷+ζ₂₅⁸+ζ₂₅⁶	ζ₂₅²⁴+ζ₂₅¹⁸+ζ₂₅⁷+ζ₂₅	ζ₂₅²³+ζ₂₅¹⁴+ζ₂₅¹¹+ζ₂₅²	ζ₂₅¹⁶+ζ₂₅¹³+ζ₂₅¹²+ζ₂₅⁹	orthogonal faithful
ρ₁₀	4	0	0	0	-1	ζ₂₅²³+ζ₂₅¹⁴+ζ₂₅¹¹+ζ₂₅²	ζ₂₅²²+ζ₂₅²¹+ζ₂₅⁴+ζ₂₅³	ζ₂₅¹⁶+ζ₂₅¹³+ζ₂₅¹²+ζ₂₅⁹	ζ₂₅²⁴+ζ₂₅¹⁸+ζ₂₅⁷+ζ₂₅	ζ₂₅¹⁹+ζ₂₅¹⁷+ζ₂₅⁸+ζ₂₅⁶	orthogonal faithful

Character table of C₅⋊F₅

C₅⋊F₅: 1^st semidirect product of C₅ and F₅ acting via F₅/C₅=C₄ C5:F5 ID 100,11

class	1	2	4A	4B	5A	5B	5C	5D	5E	5F
size	1	25	25	25	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	1	1	linear of order 2
ρ₃	1	-1	-i	i	1	1	1	1	1	1	linear of order 4
ρ₄	1	-1	i	-i	1	1	1	1	1	1	linear of order 4
ρ₅	4	0	0	0	-1	-1	4	-1	-1	-1	orthogonal lifted from F₅
ρ₆	4	0	0	0	-1	4	-1	-1	-1	-1	orthogonal lifted from F₅
ρ₇	4	0	0	0	-1	-1	-1	4	-1	-1	orthogonal lifted from F₅
ρ₈	4	0	0	0	4	-1	-1	-1	-1	-1	orthogonal lifted from F₅
ρ₉	4	0	0	0	-1	-1	-1	-1	-1	4	orthogonal lifted from F₅
ρ₁₀	4	0	0	0	-1	-1	-1	-1	4	-1	orthogonal lifted from F₅

Character table of C₅²⋊C₄

C₅²⋊C₄: 4^th semidirect product of C₅² and C₄ acting faithfully C5^2:C4 ID 100,12

class	1	2	4A	4B	5A	5B	5C	5D	5E	5F
size	1	25	25	25	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	-1	1	1	1	1	1	1	linear of order 2
ρ₃	1	-1	-i	i	1	1	1	1	1	1	linear of order 4
ρ₄	1	-1	i	-i	1	1	1	1	1	1	linear of order 4
ρ₅	4	0	0	0	-1	-1	-1	4	-1	-1	orthogonal lifted from F₅
ρ₆	4	0	0	0	-1	-1	-1	-1	4	-1	orthogonal lifted from F₅
ρ₇	4	0	0	0	-1-√5	-1+√5	^3+√5/₂	-1	-1	^3-√5/₂	orthogonal faithful
ρ₈	4	0	0	0	^3+√5/₂	^3-√5/₂	-1+√5	-1	-1	-1-√5	orthogonal faithful
ρ₉	4	0	0	0	-1+√5	-1-√5	^3-√5/₂	-1	-1	^3+√5/₂	orthogonal faithful
ρ₁₀	4	0	0	0	^3-√5/₂	^3+√5/₂	-1-√5	-1	-1	-1+√5	orthogonal faithful

Character table of C₂×A₅

C₂×A₅: Direct product of C₂ and A₅; = icosahedron/dodecahedron symmetries C2xA5 ID 120,35

class	1	2A	2B	2C	3	5A	5B	6	10A	10B
size	1	1	15	15	20	12	12	20	12	12
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	-1	1	1	1	1	-1	-1	-1	linear of order 2
ρ₃	3	3	-1	-1	0	^1+√5/₂	^1-√5/₂	0	^1-√5/₂	^1+√5/₂	orthogonal lifted from A₅
ρ₄	3	3	-1	-1	0	^1-√5/₂	^1+√5/₂	0	^1+√5/₂	^1-√5/₂	orthogonal lifted from A₅
ρ₅	3	-3	1	-1	0	^1+√5/₂	^1-√5/₂	0	^-1+√5/₂	^-1-√5/₂	orthogonal faithful
ρ₆	3	-3	1	-1	0	^1-√5/₂	^1+√5/₂	0	^-1-√5/₂	^-1+√5/₂	orthogonal faithful
ρ₇	4	4	0	0	1	-1	-1	1	-1	-1	orthogonal lifted from A₅
ρ₈	4	-4	0	0	1	-1	-1	-1	1	1	orthogonal faithful
ρ₉	5	5	1	1	-1	0	0	-1	0	0	orthogonal lifted from A₅
ρ₁₀	5	-5	-1	1	-1	0	0	1	0	0	orthogonal faithful

Character table of C₁₇⋊C₈

C₁₇⋊C₈: The semidirect product of C₁₇ and C₈ acting faithfully C17:C8 ID 136,12

class	1	2	4A	4B	8A	8B	8C	8D	17A	17B
size	1	17	17	17	17	17	17	17	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₃	1	1	-1	-1	i	-i	-i	i	1	1	linear of order 4
ρ₄	1	1	-1	-1	-i	i	i	-i	1	1	linear of order 4
ρ₅	1	-1	-i	i	ζ₈⁷	ζ₈	ζ₈⁵	ζ₈³	1	1	linear of order 8
ρ₆	1	-1	-i	i	ζ₈³	ζ₈⁵	ζ₈	ζ₈⁷	1	1	linear of order 8
ρ₇	1	-1	i	-i	ζ₈	ζ₈⁷	ζ₈³	ζ₈⁵	1	1	linear of order 8
ρ₈	1	-1	i	-i	ζ₈⁵	ζ₈³	ζ₈⁷	ζ₈	1	1	linear of order 8
ρ₉	8	0	0	0	0	0	0	0	^-1+√17/₂	^-1-√17/₂	orthogonal faithful
ρ₁₀	8	0	0	0	0	0	0	0	^-1-√17/₂	^-1+√17/₂	orthogonal faithful

Character table of C₅²⋊C₆

C₅²⋊C₆: The semidirect product of C₅² and C₆ acting faithfully C5^2:C6 ID 150,6

class	1	2	3A	3B	5A	5B	5C	5D	6A	6B
size	1	25	25	25	6	6	6	6	25	25
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	1	-1	-1	linear of order 2
ρ₃	1	1	ζ₃²	ζ₃	1	1	1	1	ζ₃²	ζ₃	linear of order 3
ρ₄	1	-1	ζ₃²	ζ₃	1	1	1	1	ζ₆	ζ₆⁵	linear of order 6
ρ₅	1	-1	ζ₃	ζ₃²	1	1	1	1	ζ₆⁵	ζ₆	linear of order 6
ρ₆	1	1	ζ₃	ζ₃²	1	1	1	1	ζ₃	ζ₃²	linear of order 3
ρ₇	6	0	0	0	^-3-√5/₂	1-√5	1+√5	^-3+√5/₂	0	0	orthogonal faithful
ρ₈	6	0	0	0	1+√5	^-3-√5/₂	^-3+√5/₂	1-√5	0	0	orthogonal faithful
ρ₉	6	0	0	0	^-3+√5/₂	1+√5	1-√5	^-3-√5/₂	0	0	orthogonal faithful
ρ₁₀	6	0	0	0	1-√5	^-3+√5/₂	^-3-√5/₂	1+√5	0	0	orthogonal faithful

Character table of C₂⁴⋊D₅

C₂⁴⋊D₅: The semidirect product of C₂⁴ and D₅ acting faithfully C2^4:D5 ID 160,234

class	1	2A	2B	2C	2D	4A	4B	4C	5A	5B
size	1	5	5	5	20	20	20	20	32	32
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₃	2	2	2	2	0	0	0	0	^-1+√5/₂	^-1-√5/₂	orthogonal lifted from D₅
ρ₄	2	2	2	2	0	0	0	0	^-1-√5/₂	^-1+√5/₂	orthogonal lifted from D₅
ρ₅	5	-3	1	1	-1	1	1	-1	0	0	orthogonal faithful
ρ₆	5	-3	1	1	1	-1	-1	1	0	0	orthogonal faithful
ρ₇	5	1	-3	1	1	1	-1	-1	0	0	orthogonal faithful
ρ₈	5	1	-3	1	-1	-1	1	1	0	0	orthogonal faithful
ρ₉	5	1	1	-3	1	-1	1	-1	0	0	orthogonal faithful
ρ₁₀	5	1	1	-3	-1	1	-1	1	0	0	orthogonal faithful

Character table of ASL₂(𝔽₃)

ASL₂(𝔽₃): Hessian group = Affine special linear group on 𝔽₃²; = PSU₃(𝔽₂)⋊C₃ ASL(2,3) ID 216,153

class	1	2	3A	3B	3C	3D	3E	4	6A	6B
size	1	9	8	12	12	24	24	54	36	36
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	ζ₃²	ζ₃	linear of order 3
ρ₃	1	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	ζ₃	ζ₃²	linear of order 3
ρ₄	2	-2	2	-1	-1	-1	-1	0	1	1	symplectic lifted from SL₂(𝔽₃), Schur index 2
ρ₅	2	-2	2	ζ₆	ζ₆⁵	ζ₆	ζ₆⁵	0	ζ₃²	ζ₃	complex lifted from SL₂(𝔽₃)
ρ₆	2	-2	2	ζ₆⁵	ζ₆	ζ₆⁵	ζ₆	0	ζ₃	ζ₃²	complex lifted from SL₂(𝔽₃)
ρ₇	3	3	3	0	0	0	0	-1	0	0	orthogonal lifted from A₄
ρ₈	8	0	-1	2	2	-1	-1	0	0	0	orthogonal faithful
ρ₉	8	0	-1	-1-√-3	-1+√-3	ζ₆	ζ₆⁵	0	0	0	complex faithful
ρ₁₀	8	0	-1	-1+√-3	-1-√-3	ζ₆⁵	ζ₆	0	0	0	complex faithful

Character table of C₂³.F₈

C₂³.F₈: 2^nd non-split extension by C₂³ of F₈ acting via F₈/C₂³=C₇ C2^3.F8 ID 448,179

class	1	2	4A	4B	7A	7B	7C	7D	7E	7F
size	1	7	28	28	64	64	64	64	64	64
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	ζ₇⁴	ζ₇⁶	ζ₇²	ζ₇⁵	ζ₇	ζ₇³	linear of order 7
ρ₃	1	1	1	1	ζ₇²	ζ₇³	ζ₇	ζ₇⁶	ζ₇⁴	ζ₇⁵	linear of order 7
ρ₄	1	1	1	1	ζ₇⁵	ζ₇⁴	ζ₇⁶	ζ₇	ζ₇³	ζ₇²	linear of order 7
ρ₅	1	1	1	1	ζ₇³	ζ₇	ζ₇⁵	ζ₇²	ζ₇⁶	ζ₇⁴	linear of order 7
ρ₆	1	1	1	1	ζ₇	ζ₇⁵	ζ₇⁴	ζ₇³	ζ₇²	ζ₇⁶	linear of order 7
ρ₇	1	1	1	1	ζ₇⁶	ζ₇²	ζ₇³	ζ₇⁴	ζ₇⁵	ζ₇	linear of order 7
ρ₈	7	7	-1	-1	0	0	0	0	0	0	orthogonal lifted from F₈
ρ₉	14	-2	-2i	2i	0	0	0	0	0	0	complex faithful
ρ₁₀	14	-2	2i	-2i	0	0	0	0	0	0	complex faithful

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ℤ

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ℚ

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Character tables

1x1 character tables

Character table of C1

2x2 character tables

Character table of C2

3x3 character tables

Character table of C3

Character table of S3

4x4 character tables

Character table of C4

Character table of C22

Character table of D5

Character table of A4

5x5 character tables

Character table of C5

Character table of D4

Character table of Q8

Character table of D7

Character table of F5

Character table of C7⋊C3

Character table of S4

Character table of A5

6x6 character tables

Character table of C6

Character table of Dic3

Character table of D6

Character table of D9

Character table of C3⋊S3

Character table of C32⋊C4

Character table of PSU3(𝔽2)

Character table of GL3(𝔽2)

7x7 character tables

Character table of C7

Character table of D8

Character table of SD16

Character table of Q16

Character table of D11

Character table of SL2(𝔽3)

Character table of C13⋊C3

Character table of F7

Character table of C13⋊C4

Character table of C11⋊C5

Character table of S5

Character table of A6