Character tables

1x1 character tables

Character table of C1

C1: Trivial group C1 ID 1,1

 class 1
 size 1
ρ11    trivial faithful

2x2 character tables

Character table of C2

C2: Cyclic group C2 ID 2,1

 class 12
 size 11
ρ111    trivial
ρ21-1    linear of order 2 faithful

3x3 character tables

Character table of C3

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

 class 13A3B
 size 111
ρ1111    trivial
ρ21ζ3ζ32    linear of order 3 faithful
ρ31ζ32ζ3    linear of order 3 faithful

Character table of S3

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

 class 123
 size 132
ρ1111    trivial
ρ21-11    linear of order 2
ρ320-1    orthogonal faithful

4x4 character tables

Character table of C4

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

 class 124A4B
 size 1111
ρ11111    trivial
ρ211-1-1    linear of order 2
ρ31-1-ii    linear of order 4 faithful
ρ41-1i-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 12A2B2C
 size 1111
ρ11111    trivial
ρ211-1-1    linear of order 2
ρ31-11-1    linear of order 2
ρ41-1-11    linear of order 2

Character table of D5

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

 class 125A5B
 size 1522
ρ11111    trivial
ρ21-111    linear of order 2
ρ320-1-5/2-1+5/2    orthogonal faithful
ρ420-1+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 123A3B
 size 1344
ρ11111    trivial
ρ211ζ3ζ32    linear of order 3
ρ311ζ32ζ3    linear of order 3
ρ43-100    orthogonal faithful

5x5 character tables

Character table of C5

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

 class 15A5B5C5D
 size 11111
ρ111111    trivial
ρ21ζ5ζ52ζ53ζ54    linear of order 5 faithful
ρ31ζ52ζ54ζ5ζ53    linear of order 5 faithful
ρ41ζ53ζ5ζ54ζ52    linear of order 5 faithful
ρ51ζ54ζ53ζ52ζ5    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 12A2B2C4
 size 11222
ρ111111    trivial
ρ211-11-1    linear of order 2
ρ3111-1-1    linear of order 2
ρ411-1-11    linear of order 2
ρ52-2000    orthogonal faithful

Character table of Q8

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

 class 124A4B4C
 size 11222
ρ111111    trivial
ρ211-11-1    linear of order 2
ρ3111-1-1    linear of order 2
ρ411-1-11    linear of order 2
ρ52-2000    symplectic faithful, Schur index 2

Character table of D7

D7: Dihedral group D7 ID 14,1

 class 127A7B7C
 size 17222
ρ111111    trivial
ρ21-1111    linear of order 2
ρ320ζ767ζ7572ζ7473    orthogonal faithful
ρ420ζ7473ζ767ζ7572    orthogonal faithful
ρ520ζ7572ζ7473ζ767    orthogonal faithful

Character table of F5

F5: Frobenius group; = C5C4 = AGL1(𝔽5) = Aut(D5) = Hol(C5) = Sz(2) F5 ID 20,3

 class 124A4B5
 size 15554
ρ111111    trivial
ρ211-1-11    linear of order 2
ρ31-1-ii1    linear of order 4
ρ41-1i-i1    linear of order 4
ρ54000-1    orthogonal faithful

Character table of C7⋊C3

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

 class 13A3B7A7B
 size 17733
ρ111111    trivial
ρ21ζ32ζ311    linear of order 3
ρ31ζ3ζ3211    linear of order 3
ρ4300-1--7/2-1+-7/2    complex faithful
ρ5300-1+-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 12A2B34
 size 13686
ρ111111    trivial
ρ211-11-1    linear of order 2
ρ3220-10    orthogonal lifted from S3
ρ43-1-101    orthogonal faithful
ρ53-110-1    orthogonal faithful

Character table of A5

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

 class 1235A5B
 size 115201212
ρ111111    trivial
ρ23-101+5/21-5/2    orthogonal faithful
ρ33-101-5/21+5/2    orthogonal faithful
ρ4401-1-1    orthogonal faithful
ρ551-100    orthogonal faithful

6x6 character tables

Character table of C6

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

 class 123A3B6A6B
 size 111111
ρ1111111    trivial
ρ21-111-1-1    linear of order 2
ρ311ζ32ζ3ζ3ζ32    linear of order 3
ρ41-1ζ32ζ3ζ65ζ6    linear of order 6 faithful
ρ511ζ3ζ32ζ32ζ3    linear of order 3
ρ61-1ζ3ζ32ζ6ζ65    linear of order 6 faithful

Character table of Dic3

Dic3: Dicyclic group; = C3C4 Dic3 ID 12,1

 class 1234A4B6
 size 112332
ρ1111111    trivial
ρ2111-1-11    linear of order 2
ρ31-11i-i-1    linear of order 4
ρ41-11-ii-1    linear of order 4
ρ522-100-1    orthogonal lifted from S3
ρ62-2-1001    symplectic faithful, Schur index 2

Character table of D6

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

 class 12A2B2C36
 size 113322
ρ1111111    trivial
ρ211-1-111    linear of order 2
ρ31-1-111-1    linear of order 2
ρ41-11-11-1    linear of order 2
ρ52200-1-1    orthogonal lifted from S3
ρ62-200-11    orthogonal faithful

Character table of D9

D9: Dihedral group D9 ID 18,1

 class 1239A9B9C
 size 192222
ρ1111111    trivial
ρ21-11111    linear of order 2
ρ3202-1-1-1    orthogonal lifted from S3
ρ420-1ζ9594ζ9792ζ989    orthogonal faithful
ρ520-1ζ989ζ9594ζ9792    orthogonal faithful
ρ620-1ζ9792ζ989ζ9594    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 123A3B3C3D
 size 192222
ρ1111111    trivial
ρ21-11111    linear of order 2
ρ320-1-1-12    orthogonal lifted from S3
ρ4202-1-1-1    orthogonal lifted from S3
ρ520-12-1-1    orthogonal lifted from S3
ρ620-1-12-1    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 123A3B4A4B
 size 194499
ρ1111111    trivial
ρ21111-1-1    linear of order 2
ρ31-111i-i    linear of order 4
ρ41-111-ii    linear of order 4
ρ5401-200    orthogonal faithful
ρ640-2100    orthogonal faithful

Character table of PSU3(𝔽2)

PSU3(𝔽2): Projective special unitary group on 𝔽23; = C32Q8 = M9 PSU(3,2) ID 72,41

 class 1234A4B4C
 size 198181818
ρ1111111    trivial
ρ2111-11-1    linear of order 2
ρ31111-1-1    linear of order 2
ρ4111-1-11    linear of order 2
ρ52-22000    symplectic lifted from Q8, Schur index 2
ρ680-1000    orthogonal faithful

Character table of GL3(𝔽2)

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

 class 12347A7B
 size 12156422424
ρ1111111    trivial
ρ23-101-1+-7/2-1--7/2    complex faithful
ρ33-101-1--7/2-1+-7/2    complex faithful
ρ46200-1-1    orthogonal faithful
ρ57-11-100    orthogonal faithful
ρ680-1011    orthogonal faithful

7x7 character tables

Character table of C7

C7: Cyclic group C7 ID 7,1

 class 17A7B7C7D7E7F
 size 1111111
ρ11111111    trivial
ρ21ζ76ζ72ζ73ζ74ζ75ζ7    linear of order 7 faithful
ρ31ζ75ζ74ζ76ζ7ζ73ζ72    linear of order 7 faithful
ρ41ζ74ζ76ζ72ζ75ζ7ζ73    linear of order 7 faithful
ρ51ζ73ζ7ζ75ζ72ζ76ζ74    linear of order 7 faithful
ρ61ζ72ζ73ζ7ζ76ζ74ζ75    linear of order 7 faithful
ρ71ζ7ζ75ζ74ζ73ζ72ζ76    linear of order 7 faithful

Character table of D8

D8: Dihedral group D8 ID 16,7

 class 12A2B2C48A8B
 size 1144222
ρ11111111    trivial
ρ211-111-1-1    linear of order 2
ρ3111-11-1-1    linear of order 2
ρ411-1-1111    linear of order 2
ρ52200-200    orthogonal lifted from D4
ρ62-20002-2    orthogonal faithful
ρ72-2000-22    orthogonal faithful

Character table of SD16

SD16: Semidihedral group; = Q8C2 = QD16 SD16 ID 16,8

 class 12A2B4A4B8A8B
 size 1142422
ρ11111111    trivial
ρ21111-1-1-1    linear of order 2
ρ311-111-1-1    linear of order 2
ρ411-11-111    linear of order 2
ρ5220-2000    orthogonal lifted from D4
ρ62-2000--2-2    complex faithful
ρ72-2000-2--2    complex faithful

Character table of Q16

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

 class 124A4B4C8A8B
 size 1124422
ρ11111111    trivial
ρ21111-1-1-1    linear of order 2
ρ3111-11-1-1    linear of order 2
ρ4111-1-111    linear of order 2
ρ522-20000    orthogonal lifted from D4
ρ62-20002-2    symplectic faithful, Schur index 2
ρ72-2000-22    symplectic faithful, Schur index 2

Character table of D11

D11: Dihedral group D11 ID 22,1

 class 1211A11B11C11D11E
 size 11122222
ρ11111111    trivial
ρ21-111111    linear of order 2
ρ320ζ111011ζ117114ζ119112ζ118113ζ116115    orthogonal faithful
ρ420ζ118113ζ111011ζ116115ζ119112ζ117114    orthogonal faithful
ρ520ζ116115ζ119112ζ111011ζ117114ζ118113    orthogonal faithful
ρ620ζ119112ζ118113ζ117114ζ116115ζ111011    orthogonal faithful
ρ720ζ117114ζ116115ζ118113ζ111011ζ119112    orthogonal faithful

Character table of SL2(𝔽3)

SL2(𝔽3): Special linear group on 𝔽32; = Q8C3 = 2T = <2,3,3> = 1st non-monomial group SL(2,3) ID 24,3

 class 123A3B46A6B
 size 1144644
ρ11111111    trivial
ρ211ζ32ζ31ζ32ζ3    linear of order 3
ρ311ζ3ζ321ζ3ζ32    linear of order 3
ρ42-2-1-1011    symplectic faithful, Schur index 2
ρ52-2ζ65ζ60ζ3ζ32    complex faithful
ρ62-2ζ6ζ650ζ32ζ3    complex faithful
ρ73300-100    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 13A3B13A13B13C13D
 size 113133333
ρ11111111    trivial
ρ21ζ32ζ31111    linear of order 3
ρ31ζ3ζ321111    linear of order 3
ρ4300ζ1311138137ζ13913313ζ136135132ζ13121310134    complex faithful
ρ5300ζ13121310134ζ1311138137ζ13913313ζ136135132    complex faithful
ρ6300ζ136135132ζ13121310134ζ1311138137ζ13913313    complex faithful
ρ7300ζ13913313ζ136135132ζ13121310134ζ1311138137    complex faithful

Character table of F7

F7: Frobenius group; = C7C6 = AGL1(𝔽7) = Aut(D7) = Hol(C7) F7 ID 42,1

 class 123A3B6A6B7
 size 1777776
ρ11111111    trivial
ρ21-111-1-11    linear of order 2
ρ311ζ32ζ3ζ3ζ321    linear of order 3
ρ411ζ3ζ32ζ32ζ31    linear of order 3
ρ51-1ζ3ζ32ζ6ζ651    linear of order 6
ρ61-1ζ32ζ3ζ65ζ61    linear of order 6
ρ7600000-1    orthogonal faithful

Character table of C13⋊C4

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

 class 124A4B13A13B13C
 size 1131313444
ρ11111111    trivial
ρ211-1-1111    linear of order 2
ρ31-1-ii111    linear of order 4
ρ41-1i-i111    linear of order 4
ρ54000ζ139137136134ζ131213813513ζ13111310133132    orthogonal faithful
ρ64000ζ131213813513ζ13111310133132ζ139137136134    orthogonal faithful
ρ74000ζ13111310133132ζ139137136134ζ131213813513    orthogonal faithful

Character table of C11⋊C5

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

 class 15A5B5C5D11A11B
 size 11111111155
ρ11111111    trivial
ρ21ζ5ζ53ζ52ζ5411    linear of order 5
ρ31ζ54ζ52ζ53ζ511    linear of order 5
ρ41ζ52ζ5ζ54ζ5311    linear of order 5
ρ51ζ53ζ54ζ5ζ5211    linear of order 5
ρ650000-1+-11/2-1--11/2    complex faithful
ρ750000-1--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 12A2B3456
 size 1101520302420
ρ11111111    trivial
ρ21-111-11-1    linear of order 2
ρ34-2010-11    orthogonal faithful
ρ442010-1-1    orthogonal faithful
ρ5511-1-101    orthogonal faithful
ρ65-11-110-1    orthogonal faithful
ρ760-20010    orthogonal faithful

Character table of A6

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

 class 123A3B45A5B
 size 1454040907272
ρ11111111    trivial
ρ2512-1-100    orthogonal faithful
ρ351-12-100    orthogonal faithful
ρ480-1-101-5/21+5/2    orthogonal faithful
ρ580-1-101+5/21-5/2    orthogonal faithful
ρ691001-1-1    orthogonal faithful
ρ710-211000    orthogonal faithful

8x8 character tables

Character table of C8

C8: Cyclic group C8 ID 8,1

 class 124A4B8A8B8C8D
 size 11111111
ρ111111111    trivial
ρ21111-1-1-1-1    linear of order 2
ρ31-1i-iζ87ζ85ζ83ζ8    linear of order 8 faithful
ρ411-1-1-ii-ii    linear of order 4
ρ51-1-iiζ85ζ87ζ8ζ83    linear of order 8 faithful
ρ61-1i-iζ83ζ8ζ87ζ85    linear of order 8 faithful
ρ711-1-1i-ii-i    linear of order 4
ρ81-1-iiζ8ζ83ζ85ζ87    linear of order 8 faithful

Character table of C2×C4

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

 class 12A2B2C4A4B4C4D
 size 11111111
ρ111111111    trivial
ρ211-1-11-11-1    linear of order 2
ρ31111-1-1-1-1    linear of order 2
ρ411-1-1-11-11    linear of order 2
ρ51-11-1ii-i-i    linear of order 4
ρ61-1-11i-i-ii    linear of order 4
ρ71-11-1-i-iii    linear of order 4
ρ81-1-11-iii-i    linear of order 4

Character table of C23

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

 class 12A2B2C2D2E2F2G
 size 11111111
ρ111111111    trivial
ρ21-111-1-1-11    linear of order 2
ρ31-1-1-111-11    linear of order 2
ρ411-1-1-1-111    linear of order 2
ρ51-11-11-11-1    linear of order 2
ρ6111-1-11-1-1    linear of order 2
ρ711-111-1-1-1    linear of order 2
ρ81-1-11-111-1    linear of order 2

Character table of Dic5

Dic5: Dicyclic group; = C52C4 Dic5 ID 20,1

 class 124A4B5A5B10A10B
 size 11552222
ρ111111111    trivial
ρ211-1-11111    linear of order 2
ρ31-1-ii11-1-1    linear of order 4
ρ41-1i-i11-1-1    linear of order 4
ρ52200-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ62200-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ72-200-1+5/2-1-5/21+5/21-5/2    symplectic faithful, Schur index 2
ρ82-200-1-5/2-1+5/21-5/21+5/2    symplectic faithful, Schur index 2

Character table of D10

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

 class 12A2B2C5A5B10A10B
 size 11552222
ρ111111111    trivial
ρ211-1-11111    linear of order 2
ρ31-1-1111-1-1    linear of order 2
ρ41-11-111-1-1    linear of order 2
ρ52-200-1+5/2-1-5/21+5/21-5/2    orthogonal faithful
ρ62200-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ72200-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ82-200-1-5/2-1+5/21-5/21+5/2    orthogonal faithful

Character table of C2×A4

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

 class 12A2B2C3A3B6A6B
 size 11334444
ρ111111111    trivial
ρ21-1-1111-1-1    linear of order 2
ρ31111ζ32ζ3ζ32ζ3    linear of order 3
ρ41-1-11ζ32ζ3ζ6ζ65    linear of order 6
ρ51-1-11ζ3ζ32ζ65ζ6    linear of order 6
ρ61111ζ3ζ32ζ3ζ32    linear of order 3
ρ733-1-10000    orthogonal lifted from A4
ρ83-31-10000    orthogonal faithful

Character table of D13

D13: Dihedral group D13 ID 26,1

 class 1213A13B13C13D13E13F
 size 113222222
ρ111111111    trivial
ρ21-1111111    linear of order 2
ρ320ζ139134ζ137136ζ138135ζ1310133ζ131213ζ1311132    orthogonal faithful
ρ420ζ131213ζ138135ζ1311132ζ139134ζ1310133ζ137136    orthogonal faithful
ρ520ζ1310133ζ1311132ζ137136ζ131213ζ139134ζ138135    orthogonal faithful
ρ620ζ137136ζ139134ζ131213ζ1311132ζ138135ζ1310133    orthogonal faithful
ρ720ζ1311132ζ1310133ζ139134ζ138135ζ137136ζ131213    orthogonal faithful
ρ820ζ138135ζ131213ζ1310133ζ137136ζ1311132ζ139134    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 123A3B4A4B4C4D
 size 1316163333
ρ111111111    trivial
ρ211ζ32ζ31111    linear of order 3
ρ311ζ3ζ321111    linear of order 3
ρ43300-1-1-1-1    orthogonal lifted from A4
ρ53-1001-1+2i-1-2i1    complex faithful
ρ63-100-1-2i11-1+2i    complex faithful
ρ73-1001-1-2i-1+2i1    complex faithful
ρ83-100-1+2i11-1-2i    complex faithful

Character table of CSU2(𝔽3)

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

 class 1234A4B68A8B
 size 118612866
ρ111111111    trivial
ρ21111-11-1-1    linear of order 2
ρ322-120-100    orthogonal lifted from S3
ρ42-2-10012-2    symplectic faithful, Schur index 2
ρ52-2-1001-22    symplectic faithful, Schur index 2
ρ6330-1-1011    orthogonal lifted from S4
ρ7330-110-1-1    orthogonal lifted from S4
ρ84-4100-100    symplectic faithful, Schur index 2

Character table of GL2(𝔽3)

GL2(𝔽3): General linear group on 𝔽32; = Q8S3 = Aut(C32) = 2O = <2,3,4> GL(2,3) ID 48,29

 class 12A2B3468A8B
 size 111286866
ρ111111111    trivial
ρ211-1111-1-1    linear of order 2
ρ3220-12-100    orthogonal lifted from S3
ρ42-20-101--2-2    complex faithful
ρ52-20-101-2--2    complex faithful
ρ633-10-1011    orthogonal lifted from S4
ρ73310-10-1-1    orthogonal lifted from S4
ρ84-4010-100    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 12A2B2C2D2E3A3B
 size 1333331616
ρ111111111    trivial
ρ2111111ζ3ζ32    linear of order 3
ρ3111111ζ32ζ3    linear of order 3
ρ433-1-1-1-100    orthogonal lifted from A4
ρ53-1-1-13-100    orthogonal lifted from A4
ρ63-13-1-1-100    orthogonal lifted from A4
ρ73-1-13-1-100    orthogonal lifted from A4
ρ83-1-1-1-1300    orthogonal lifted from A4

Character table of F8

F8: Frobenius group; = C23C7 = AGL1(𝔽8) F8 ID 56,11

 class 127A7B7C7D7E7F
 size 17888888
ρ111111111    trivial
ρ211ζ74ζ76ζ72ζ75ζ7ζ73    linear of order 7
ρ311ζ72ζ73ζ7ζ76ζ74ζ75    linear of order 7
ρ411ζ75ζ74ζ76ζ7ζ73ζ72    linear of order 7
ρ511ζ73ζ7ζ75ζ72ζ76ζ74    linear of order 7
ρ611ζ7ζ75ζ74ζ73ζ72ζ76    linear of order 7
ρ711ζ76ζ72ζ73ζ74ζ75ζ7    linear of order 7
ρ87-1000000    orthogonal faithful

Character table of C17⋊C4

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

 class 124A4B17A17B17C17D
 size 11717174444
ρ111111111    trivial
ρ211-1-11111    linear of order 2
ρ31-1i-i1111    linear of order 4
ρ41-1-ii1111    linear of order 4
ρ54000ζ1716171317417ζ17141712175173ζ17111710177176ζ1715179178172    orthogonal faithful
ρ64000ζ1715179178172ζ17111710177176ζ17141712175173ζ1716171317417    orthogonal faithful
ρ74000ζ17141712175173ζ1715179178172ζ1716171317417ζ17111710177176    orthogonal faithful
ρ84000ζ17111710177176ζ1716171317417ζ1715179178172ζ17141712175173    orthogonal faithful

Character table of C13⋊C6

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

 class 123A3B6A6B13A13B
 size 1131313131366
ρ111111111    trivial
ρ21-111-1-111    linear of order 2
ρ31-1ζ32ζ3ζ65ζ611    linear of order 6
ρ411ζ3ζ32ζ32ζ311    linear of order 3
ρ511ζ32ζ3ζ3ζ3211    linear of order 3
ρ61-1ζ3ζ32ζ6ζ6511    linear of order 6
ρ7600000-1-13/2-1+13/2    orthogonal faithful
ρ8600000-1+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 12A2B2C5A5B5C5D
 size 155516161616
ρ111111111    trivial
ρ21111ζ52ζ5ζ54ζ53    linear of order 5
ρ31111ζ53ζ54ζ5ζ52    linear of order 5
ρ41111ζ54ζ52ζ53ζ5    linear of order 5
ρ51111ζ5ζ53ζ52ζ54    linear of order 5
ρ65-3110000    orthogonal faithful
ρ751-310000    orthogonal faithful
ρ8511-30000    orthogonal faithful

Character table of AΓL1(𝔽8)

AΓL1(𝔽8): Affine semilinear group on 𝔽81; = F8C3 = Aut(F8) AGammaL(1,8) ID 168,43

 class 123A3B6A6B7A7B
 size 17282828282424
ρ111111111    trivial
ρ211ζ3ζ32ζ3ζ3211    linear of order 3
ρ311ζ32ζ3ζ32ζ311    linear of order 3
ρ4330000-1+-7/2-1--7/2    complex lifted from C7⋊C3
ρ5330000-1--7/2-1+-7/2    complex lifted from C7⋊C3
ρ67-111-1-100    orthogonal faithful
ρ77-1ζ3ζ32ζ65ζ600    complex faithful
ρ87-1ζ32ζ3ζ6ζ6500    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 124A4B4C5A5B5C
 size 125505050888
ρ111111111    trivial
ρ211-1-11111    linear of order 2
ρ3111-1-1111    linear of order 2
ρ411-11-1111    linear of order 2
ρ52-2000222    symplectic lifted from Q8, Schur index 2
ρ680000-23-2    orthogonal faithful
ρ7800003-2-2    orthogonal faithful
ρ880000-2-23    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 1234A4B5A5B6
 size 125507575121250
ρ111111111    trivial
ρ2111-1-1111    linear of order 2
ρ31-11i-i11-1    linear of order 4
ρ41-11-ii11-1    linear of order 4
ρ522-10022-1    orthogonal lifted from S3
ρ62-2-100221    symplectic lifted from Dic3, Schur index 2
ρ71200002-30    orthogonal faithful
ρ8120000-320    orthogonal faithful

9x9 character tables

Character table of C9

C9: Cyclic group C9 ID 9,1

 class 13A3B9A9B9C9D9E9F
 size 111111111
ρ1111111111    trivial
ρ21ζ32ζ3ζ97ζ95ζ9ζ94ζ98ζ92    linear of order 9 faithful
ρ31ζ3ζ32ζ95ζ9ζ92ζ98ζ97ζ94    linear of order 9 faithful
ρ4111ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ51ζ32ζ3ζ9ζ92ζ94ζ97ζ95ζ98    linear of order 9 faithful
ρ61ζ3ζ32ζ98ζ97ζ95ζ92ζ94ζ9    linear of order 9 faithful
ρ7111ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ81ζ32ζ3ζ94ζ98ζ97ζ9ζ92ζ95    linear of order 9 faithful
ρ91ζ3ζ32ζ92ζ94ζ98ζ95ζ9ζ97    linear of order 9 faithful

Character table of C32

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

 class 13A3B3C3D3E3F3G3H
 size 111111111
ρ1111111111    trivial
ρ21ζ321ζ3ζ3ζ3ζ32ζ321    linear of order 3
ρ31ζ31ζ32ζ32ζ32ζ3ζ31    linear of order 3
ρ41ζ32ζ321ζ3ζ321ζ3ζ3    linear of order 3
ρ51ζ3ζ32ζ3ζ321ζ321ζ3    linear of order 3
ρ611ζ32ζ321ζ3ζ3ζ32ζ3    linear of order 3
ρ71ζ3ζ31ζ32ζ31ζ32ζ32    linear of order 3
ρ811ζ3ζ31ζ32ζ32ζ3ζ32    linear of order 3
ρ91ζ32ζ3ζ32ζ31ζ31ζ32    linear of order 3

Character table of C3×S3

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

 class 123A3B3C3D3E6A6B
 size 131122233
ρ1111111111    trivial
ρ21-111111-1-1    linear of order 2
ρ31-1ζ32ζ31ζ32ζ3ζ65ζ6    linear of order 6
ρ41-1ζ3ζ321ζ3ζ32ζ6ζ65    linear of order 6
ρ511ζ3ζ321ζ3ζ32ζ32ζ3    linear of order 3
ρ611ζ32ζ31ζ32ζ3ζ3ζ32    linear of order 3
ρ72022-1-1-100    orthogonal lifted from S3
ρ820-1+-3-1--3-1ζ65ζ600    complex faithful
ρ920-1--3-1+-3-1ζ6ζ6500    complex faithful

Character table of Dic6

Dic6: Dicyclic group; = C3Q8 Dic6 ID 24,4

 class 1234A4B4C612A12B
 size 112266222
ρ1111111111    trivial
ρ21111-1-1111    linear of order 2
ρ3111-1-111-1-1    linear of order 2
ρ4111-11-11-1-1    linear of order 2
ρ522-1200-1-1-1    orthogonal lifted from S3
ρ622-1-200-111    orthogonal lifted from D6
ρ72-22000-200    symplectic lifted from Q8, Schur index 2
ρ82-2-100013-3    symplectic faithful, Schur index 2
ρ92-2-10001-33    symplectic faithful, Schur index 2

Character table of D12

D12: Dihedral group D12 ID 24,6

 class 12A2B2C34612A12B
 size 116622222
ρ1111111111    trivial
ρ211-1-111111    linear of order 2
ρ3111-11-11-1-1    linear of order 2
ρ411-111-11-1-1    linear of order 2
ρ52200-12-1-1-1    orthogonal lifted from S3
ρ62200-1-2-111    orthogonal lifted from D6
ρ72-20020-200    orthogonal lifted from D4
ρ82-200-1013-3    orthogonal faithful
ρ92-200-101-33    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 12A2B2C346A6B6C
 size 112626222
ρ1111111111    trivial
ρ2111-11-1111    linear of order 2
ρ311-111-1-1-11    linear of order 2
ρ411-1-111-1-11    linear of order 2
ρ522-20-1011-1    orthogonal lifted from D6
ρ62220-10-1-1-1    orthogonal lifted from S3
ρ72-2002000-2    orthogonal lifted from D4
ρ82-200-10--3-31    complex faithful
ρ92-200-10-3--31    complex faithful

Character table of D15

D15: Dihedral group D15 ID 30,3

 class 1235A5B15A15B15C15D
 size 1152222222
ρ1111111111    trivial
ρ21-11111111    linear of order 2
ρ320-122-1-1-1-1    orthogonal lifted from S3
ρ4202-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ5202-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ620-1-1-5/2-1+5/23ζ533ζ5253ζ3ζ533ζ5252ζ3ζ543ζ55ζ32ζ5432ζ55    orthogonal faithful
ρ720-1-1+5/2-1-5/2ζ32ζ5432ζ55ζ3ζ543ζ553ζ533ζ5253ζ3ζ533ζ5252    orthogonal faithful
ρ820-1-1-5/2-1+5/2ζ3ζ533ζ52523ζ533ζ5253ζ32ζ5432ζ55ζ3ζ543ζ55    orthogonal faithful
ρ920-1-1+5/2-1-5/2ζ3ζ543ζ55ζ32ζ5432ζ55ζ3ζ533ζ52523ζ533ζ5253    orthogonal faithful

Character table of S32

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

 class 12A2B2C3A3B3C6A6B
 size 133922466
ρ1111111111    trivial
ρ21-11-11111-1    linear of order 2
ρ311-1-1111-11    linear of order 2
ρ41-1-11111-1-1    linear of order 2
ρ522002-1-10-1    orthogonal lifted from S3
ρ62-2002-1-101    orthogonal lifted from D6
ρ72020-12-1-10    orthogonal lifted from S3
ρ820-20-12-110    orthogonal lifted from D6
ρ94000-2-2100    orthogonal faithful

Character table of C19⋊C3

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

 class 13A3B19A19B19C19D19E19F
 size 11919333333
ρ1111111111    trivial
ρ21ζ3ζ32111111    linear of order 3
ρ31ζ32ζ3111111    linear of order 3
ρ4300ζ19181912198ζ191519131910ζ19171916195ζ191119719ζ1914193192ζ199196194    complex faithful
ρ5300ζ191519131910ζ1914193192ζ191119719ζ199196194ζ19181912198ζ19171916195    complex faithful
ρ6300ζ1914193192ζ19181912198ζ199196194ζ19171916195ζ191519131910ζ191119719    complex faithful
ρ7300ζ199196194ζ19171916195ζ19181912198ζ191519131910ζ191119719ζ1914193192    complex faithful
ρ8300ζ191119719ζ199196194ζ1914193192ζ19181912198ζ19171916195ζ191519131910    complex faithful
ρ9300ζ19171916195ζ191119719ζ191519131910ζ1914193192ζ199196194ζ19181912198    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 1234A4B5615A15B
 size 152151541044
ρ1111111111    trivial
ρ2111-1-11111    linear of order 2
ρ31-11i-i1-111    linear of order 4
ρ41-11-ii1-111    linear of order 4
ρ522-1002-1-1-1    orthogonal lifted from S3
ρ62-2-10021-1-1    symplectic lifted from Dic3, Schur index 2
ρ740400-10-1-1    orthogonal lifted from F5
ρ840-200-101--15/21+-15/2    complex faithful
ρ940-200-101+-15/21--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 12A2B3469A9B9C
 size 13182186888
ρ1111111111    trivial
ρ211-11-11111    linear of order 2
ρ3220202-1-1-1    orthogonal lifted from S3
ρ4220-10-1ζ9594ζ9792ζ989    orthogonal lifted from D9
ρ5220-10-1ζ989ζ9594ζ9792    orthogonal lifted from D9
ρ6220-10-1ζ9792ζ989ζ9594    orthogonal lifted from D9
ρ73-1-131-1000    orthogonal lifted from S4
ρ83-113-1-1000    orthogonal lifted from S4
ρ96-20-301000    orthogonal faithful

Character table of F9

F9: Frobenius group; = C32C8 = AGL1(𝔽9) F9 ID 72,39

 class 1234A4B8A8B8C8D
 size 198999999
ρ1111111111    trivial
ρ211111-1-1-1-1    linear of order 2
ρ3111-1-1-i-iii    linear of order 4
ρ4111-1-1ii-i-i    linear of order 4
ρ51-11-iiζ85ζ8ζ87ζ83    linear of order 8
ρ61-11i-iζ87ζ83ζ85ζ8    linear of order 8
ρ71-11i-iζ83ζ87ζ8ζ85    linear of order 8
ρ81-11-iiζ8ζ85ζ83ζ87    linear of order 8
ρ980-1000000    orthogonal faithful

Character table of S3≀C2

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

 class 12A2B2C3A3B46A6B
 size 166944181212
ρ1111111111    trivial
ρ21-11111-11-1    linear of order 2
ρ311-1111-1-11    linear of order 2
ρ41-1-11111-1-1    linear of order 2
ρ5200-222000    orthogonal lifted from D4
ρ640-20-21010    orthogonal faithful
ρ74-2001-2001    orthogonal faithful
ρ84020-210-10    orthogonal faithful
ρ942001-200-1    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 12A2B3A3B3C3D46
 size 13182888186
ρ1111111111    trivial
ρ211-11111-11    linear of order 2
ρ32202-1-1-102    orthogonal lifted from S3
ρ4220-12-1-10-1    orthogonal lifted from S3
ρ5220-1-1-120-1    orthogonal lifted from S3
ρ6220-1-12-10-1    orthogonal lifted from S3
ρ73-1-130001-1    orthogonal lifted from S4
ρ83-113000-1-1    orthogonal lifted from S4
ρ96-20-300001    orthogonal faithful

Character table of C19⋊C6

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

 class 123A3B6A6B19A19B19C
 size 11919191919666
ρ1111111111    trivial
ρ21-111-1-1111    linear of order 2
ρ311ζ32ζ3ζ32ζ3111    linear of order 3
ρ41-1ζ32ζ3ζ6ζ65111    linear of order 6
ρ511ζ3ζ32ζ3ζ32111    linear of order 3
ρ61-1ζ3ζ32ζ65ζ6111    linear of order 6
ρ7600000ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194    orthogonal faithful
ρ8600000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192    orthogonal faithful
ρ9600000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719    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 12345A5B610A10B
 size 1120301212201212
ρ1111111111    trivial
ρ22-2-10-1+5/2-1-5/211+5/21-5/2    symplectic faithful, Schur index 2
ρ32-2-10-1-5/2-1+5/211-5/21+5/2    symplectic faithful, Schur index 2
ρ4330-11-5/21+5/201+5/21-5/2    orthogonal lifted from A5
ρ5330-11+5/21-5/201-5/21+5/2    orthogonal lifted from A5
ρ64410-1-11-1-1    orthogonal lifted from A5
ρ74-410-1-1-111    symplectic faithful, Schur index 2
ρ855-1100-100    orthogonal lifted from A5
ρ96-600110-1-1    symplectic faithful, Schur index 2

Character table of AΓL1(𝔽9)

AΓL1(𝔽9): Affine semilinear group on 𝔽91; = F9C2 = Aut(C32⋊C4) AGammaL(1,9) ID 144,182

 class 12A2B34A4B68A8B
 size 191281836241818
ρ1111111111    trivial
ρ211111-11-1-1    linear of order 2
ρ311-111-1-111    linear of order 2
ρ411-1111-1-1-1    linear of order 2
ρ52202-20000    orthogonal lifted from D4
ρ62-202000-2--2    complex lifted from SD16
ρ72-202000--2-2    complex lifted from SD16
ρ880-2-100100    orthogonal faithful
ρ9802-100-100    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 12A2B2C3A3B4A4B4C
 size 1312126464121212
ρ1111111111    trivial
ρ21111ζ32ζ3111    linear of order 3
ρ31111ζ3ζ32111    linear of order 3
ρ4333-100-1-1-1    orthogonal lifted from A4
ρ533-1-100-1-13    orthogonal lifted from A4
ρ633-1300-1-1-1    orthogonal lifted from A4
ρ733-1-1003-1-1    orthogonal lifted from A4
ρ833-1-100-13-1    orthogonal lifted from A4
ρ912-40000000    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 123A3B4A4B4C4D4E
 size 1364641212121212
ρ1111111111    trivial
ρ211ζ3ζ3211111    linear of order 3
ρ311ζ32ζ311111    linear of order 3
ρ43300-1-1-1-13    orthogonal lifted from A4
ρ53300-13-1-1-1    orthogonal lifted from A4
ρ633003-1-1-1-1    orthogonal lifted from A4
ρ73300-1-13-1-1    orthogonal lifted from A4
ρ83300-1-1-13-1    orthogonal lifted from A4
ρ912-40000000    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 12A2B34678A8B
 size 12128564256484242
ρ1111111111    trivial
ρ211-111-11-1-1    linear of order 2
ρ36-20020-100    orthogonal faithful
ρ4620000-1-22    orthogonal faithful
ρ5620000-12-2    orthogonal faithful
ρ67-111-110-1-1    orthogonal faithful
ρ77-1-11-1-1011    orthogonal faithful
ρ880-2-101100    orthogonal faithful
ρ9802-10-1100    orthogonal faithful

10x10 character tables

Character table of C10

C10: Cyclic group C10 ID 10,2

 class 125A5B5C5D10A10B10C10D
 size 1111111111
ρ11111111111    trivial
ρ21-11111-1-1-1-1    linear of order 2
ρ311ζ52ζ53ζ54ζ5ζ54ζ52ζ53ζ5    linear of order 5
ρ41-1ζ52ζ53ζ54ζ55452535    linear of order 10 faithful
ρ511ζ54ζ5ζ53ζ52ζ53ζ54ζ5ζ52    linear of order 5
ρ61-1ζ54ζ5ζ53ζ525354552    linear of order 10 faithful
ρ711ζ5ζ54ζ52ζ53ζ52ζ5ζ54ζ53    linear of order 5
ρ81-1ζ5ζ54ζ52ζ535255453    linear of order 10 faithful
ρ911ζ53ζ52ζ5ζ54ζ5ζ53ζ52ζ54    linear of order 5
ρ101-1ζ53ζ52ζ5ζ545535254    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 12A2B2C2D2E4A4B4C4D
 size 1111222222
ρ11111111111    trivial
ρ21111-1-1-111-1    linear of order 2
ρ31111-1-11-1-11    linear of order 2
ρ4111111-1-1-1-1    linear of order 2
ρ511-1-1-11i-ii-i    linear of order 4
ρ611-1-11-1-i-iii    linear of order 4
ρ711-1-1-11-ii-ii    linear of order 4
ρ811-1-11-1ii-i-i    linear of order 4
ρ92-2-22000000    orthogonal lifted from D4
ρ102-22-2000000    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 12A2B2C4A4B4C4D4E4F
 size 1111222222
ρ11111111111    trivial
ρ21111-11-1-11-1    linear of order 2
ρ31111-1-11-1-11    linear of order 2
ρ411111-1-11-1-1    linear of order 2
ρ511-1-1-1-ii1i-i    linear of order 4
ρ611-1-11-i-i-1ii    linear of order 4
ρ711-1-1-1i-i1-ii    linear of order 4
ρ811-1-11ii-1-i-i    linear of order 4
ρ92-22-2000000    orthogonal lifted from D4
ρ102-2-22000000    symplectic lifted from Q8, Schur index 2

Character table of M4(2)

M4(2): Modular maximal-cyclic group; = C83C2 M4(2) ID 16,6

 class 12A2B4A4B4C8A8B8C8D
 size 1121122222
ρ11111111111    trivial
ρ211-111-11-1-11    linear of order 2
ρ311-111-1-111-1    linear of order 2
ρ4111111-1-1-1-1    linear of order 2
ρ511-1-1-11i-ii-i    linear of order 4
ρ6111-1-1-1ii-i-i    linear of order 4
ρ711-1-1-11-ii-ii    linear of order 4
ρ8111-1-1-1-i-iii    linear of order 4
ρ92-202i-2i00000    complex faithful
ρ102-20-2i2i00000    complex faithful

Character table of C2×D4

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

 class 12A2B2C2D2E2F2G4A4B
 size 1111222222
ρ11111111111    trivial
ρ21-11-1-11-111-1    linear of order 2
ρ31-11-11-11-11-1    linear of order 2
ρ41111-1-1-1-111    linear of order 2
ρ51-11-11-1-11-11    linear of order 2
ρ61-11-1-111-1-11    linear of order 2
ρ71111-1-111-1-1    linear of order 2
ρ8111111-1-1-1-1    linear of order 2
ρ92-2-22000000    orthogonal lifted from D4
ρ1022-2-2000000    orthogonal lifted from D4

Character table of C2×Q8

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

 class 12A2B2C4A4B4C4D4E4F
 size 1111222222
ρ11111111111    trivial
ρ21-11-1-11-111-1    linear of order 2
ρ31-11-11-11-11-1    linear of order 2
ρ41111-1-1-1-111    linear of order 2
ρ51-11-11-1-11-11    linear of order 2
ρ61-11-1-111-1-11    linear of order 2
ρ71111-1-111-1-1    linear of order 2
ρ8111111-1-1-1-1    linear of order 2
ρ92-2-22000000    symplectic lifted from Q8, Schur index 2
ρ1022-2-2000000    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 12A2B2C2D4A4B4C4D4E
 size 1122211222
ρ11111111111    trivial
ρ2111-11-1-1-1-11    linear of order 2
ρ311-11-111-1-11    linear of order 2
ρ41111-1-1-11-1-1    linear of order 2
ρ511-1-11111-1-1    linear of order 2
ρ611-1-1-1-1-1111    linear of order 2
ρ7111-1-111-11-1    linear of order 2
ρ811-111-1-1-11-1    linear of order 2
ρ92-2000-2i2i000    complex faithful
ρ102-20002i-2i000    complex faithful

Character table of Dic7

Dic7: Dicyclic group; = C7C4 Dic7 ID 28,1

 class 124A4B7A7B7C14A14B14C
 size 1177222222
ρ11111111111    trivial
ρ211-1-1111111    linear of order 2
ρ31-1-ii111-1-1-1    linear of order 4
ρ41-1i-i111-1-1-1    linear of order 4
ρ52200ζ7572ζ7473ζ767ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ62200ζ767ζ7572ζ7473ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ72200ζ7473ζ767ζ7572ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ82-200ζ7473ζ767ζ757275727473767    symplectic faithful, Schur index 2
ρ92-200ζ7572ζ7473ζ76776775727473    symplectic faithful, Schur index 2
ρ102-200ζ767ζ7572ζ747374737677572    symplectic faithful, Schur index 2

Character table of D14

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

 class 12A2B2C7A7B7C14A14B14C
 size 1177222222
ρ11111111111    trivial
ρ21-1-11111-1-1-1    linear of order 2
ρ311-1-1111111    linear of order 2
ρ41-11-1111-1-1-1    linear of order 2
ρ52200ζ7572ζ7473ζ767ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ62200ζ767ζ7572ζ7473ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ72-200ζ7473ζ767ζ757275727473767    orthogonal faithful
ρ82200ζ7473ζ767ζ7572ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ92-200ζ7572ζ7473ζ76776775727473    orthogonal faithful
ρ102-200ζ767ζ7572ζ747374737677572    orthogonal faithful

Character table of D17

D17: Dihedral group D17 ID 34,1

 class 1217A17B17C17D17E17F17G17H
 size 11722222222
ρ11111111111    trivial
ρ21-111111111    linear of order 2
ρ320ζ1715172ζ1714173ζ1713174ζ1712175ζ1711176ζ1710177ζ179178ζ171617    orthogonal faithful
ρ420ζ1710177ζ1715172ζ1714173ζ179178ζ1713174ζ171617ζ1711176ζ1712175    orthogonal faithful
ρ520ζ179178ζ1712175ζ171617ζ1714173ζ1710177ζ1711176ζ1715172ζ1713174    orthogonal faithful
ρ620ζ1714173ζ1713174ζ1711176ζ171617ζ179178ζ1715172ζ1712175ζ1710177    orthogonal faithful
ρ720ζ1711176ζ179178ζ1712175ζ1715172ζ171617ζ1713174ζ1710177ζ1714173    orthogonal faithful
ρ820ζ1712175ζ171617ζ1710177ζ1713174ζ1715172ζ179178ζ1714173ζ1711176    orthogonal faithful
ρ920ζ1713174ζ1711176ζ179178ζ1710177ζ1712175ζ1714173ζ171617ζ1715172    orthogonal faithful
ρ1020ζ171617ζ1710177ζ1715172ζ1711176ζ1714173ζ1712175ζ1713174ζ179178    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 124A4B58A8B8C8D10
 size 1155455554
ρ11111111111    trivial
ρ211111-1-1-1-11    linear of order 2
ρ311-1-11i-ii-i1    linear of order 4
ρ411-1-11-ii-ii1    linear of order 4
ρ51-1i-i1ζ85ζ87ζ8ζ83-1    linear of order 8
ρ61-1-ii1ζ83ζ8ζ87ζ85-1    linear of order 8
ρ71-1i-i1ζ8ζ83ζ85ζ87-1    linear of order 8
ρ81-1-ii1ζ87ζ85ζ83ζ8-1    linear of order 8
ρ94400-10000-1    orthogonal lifted from F5
ρ104-400-100001    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 12A2B2C4A4B4C4D510
 size 1155555544
ρ11111111111    trivial
ρ21111-1-1-1-111    linear of order 2
ρ31-1-11-111-11-1    linear of order 2
ρ41-1-111-1-111-1    linear of order 2
ρ51-11-1ii-i-i1-1    linear of order 4
ρ61-11-1-i-iii1-1    linear of order 4
ρ711-1-1-ii-ii11    linear of order 4
ρ811-1-1i-ii-i11    linear of order 4
ρ94-4000000-11    orthogonal faithful
ρ1044000000-1-1    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 123A3B6A6B7A7B14A14B
 size 1177773333
ρ11111111111    trivial
ρ21-111-1-111-1-1    linear of order 2
ρ311ζ32ζ3ζ3ζ321111    linear of order 3
ρ411ζ3ζ32ζ32ζ31111    linear of order 3
ρ51-1ζ3ζ32ζ6ζ6511-1-1    linear of order 6
ρ61-1ζ32ζ3ζ65ζ611-1-1    linear of order 6
ρ7330000-1+-7/2-1--7/2-1--7/2-1+-7/2    complex lifted from C7⋊C3
ρ83-30000-1+-7/2-1--7/21+-7/21--7/2    complex faithful
ρ93-30000-1--7/2-1+-7/21--7/21+-7/2    complex faithful
ρ10330000-1--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 12A2B2C34A4B4C4D6
 size 1133866668
ρ11111111111    trivial
ρ211111-1-1-1-11    linear of order 2
ρ31-11-11-ii-ii-1    linear of order 4
ρ41-11-11i-ii-i-1    linear of order 4
ρ52222-10000-1    orthogonal lifted from S3
ρ62-22-2-100001    symplectic lifted from Dic3, Schur index 2
ρ733-1-1011-1-10    orthogonal lifted from S4
ρ833-1-10-1-1110    orthogonal lifted from S4
ρ93-3-110i-i-ii0    complex faithful
ρ103-3-110-iii-i0    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 12A2B2C2D2E34A4B6
 size 1133668668
ρ11111111111    trivial
ρ21-11-11-11-11-1    linear of order 2
ρ31111-1-11-1-11    linear of order 2
ρ41-11-1-1111-1-1    linear of order 2
ρ52-22-200-1001    orthogonal lifted from D6
ρ6222200-100-1    orthogonal lifted from S3
ρ733-1-1110-1-10    orthogonal lifted from S4
ρ83-3-11-110-110    orthogonal faithful
ρ93-3-111-101-10    orthogonal faithful
ρ1033-1-1-1-10110    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 123A3B3C3D3E3F6A6B
 size 1923366699
ρ11111111111    trivial
ρ21-1111111-1-1    linear of order 2
ρ3111ζ3ζ321ζ3ζ32ζ32ζ3    linear of order 3
ρ41-11ζ3ζ321ζ3ζ32ζ6ζ65    linear of order 6
ρ5111ζ32ζ31ζ32ζ3ζ3ζ32    linear of order 3
ρ61-11ζ32ζ31ζ32ζ3ζ65ζ6    linear of order 6
ρ720222-1-1-100    orthogonal lifted from S3
ρ8202-1--3-1+-3-1ζ6ζ6500    complex lifted from C3×S3
ρ9202-1+-3-1--3-1ζ65ζ600    complex lifted from C3×S3
ρ1060-30000000    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 123A3B3C6A6B9A9B9C
 size 1923399666
ρ11111111111    trivial
ρ21-1111-1-1111    linear of order 2
ρ31-11ζ3ζ32ζ6ζ65ζ321ζ3    linear of order 6
ρ4111ζ32ζ3ζ3ζ32ζ31ζ32    linear of order 3
ρ5111ζ3ζ32ζ32ζ3ζ321ζ3    linear of order 3
ρ61-11ζ32ζ3ζ65ζ6ζ31ζ32    linear of order 6
ρ72022200-1-1-1    orthogonal lifted from S3
ρ8202-1+-3-1--300ζ6-1ζ65    complex lifted from C3×S3
ρ9202-1--3-1+-300ζ65-1ζ6    complex lifted from C3×S3
ρ1060-30000000    orthogonal faithful

Character table of He3⋊C2

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

 class 123A3B3C3D3E3F6A6B
 size 1911666699
ρ11111111111    trivial
ρ21-1111111-1-1    linear of order 2
ρ32022-1-12-100    orthogonal lifted from S3
ρ420222-1-1-100    orthogonal lifted from S3
ρ52022-1-1-1200    orthogonal lifted from S3
ρ62022-12-1-100    orthogonal lifted from S3
ρ731-3-3-3/2-3+3-3/20000ζ3ζ32    complex faithful
ρ831-3+3-3/2-3-3-3/20000ζ32ζ3    complex faithful
ρ93-1-3+3-3/2-3-3-3/20000ζ6ζ65    complex faithful
ρ103-1-3-3-3/2-3+3-3/20000ζ65ζ6    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 12A2B34A4B4C4D8A8B
 size 131232336121212
ρ11111111111    trivial
ρ211-11111-1-1-1    linear of order 2
ρ3220-1222000    orthogonal lifted from S3
ρ433-10-1-1-1-111    orthogonal lifted from S4
ρ53310-1-1-11-1-1    orthogonal lifted from S4
ρ63-110-1+2i-1-2i1-1-ii    complex faithful
ρ73-1-10-1+2i-1-2i11i-i    complex faithful
ρ83-1-10-1-2i-1+2i11-ii    complex faithful
ρ93-110-1-2i-1+2i1-1i-i    complex faithful
ρ106-20022-2000    orthogonal faithful

Character table of C24⋊C6

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

 class 12A2B2C2D3A3B46A6B
 size 134661616121616
ρ11111111111    trivial
ρ211-11111-1-1-1    linear of order 2
ρ311111ζ3ζ321ζ32ζ3    linear of order 3
ρ411-111ζ32ζ3-1ζ65ζ6    linear of order 6
ρ511-111ζ3ζ32-1ζ6ζ65    linear of order 6
ρ611111ζ32ζ31ζ3ζ32    linear of order 3
ρ7333-1-100-100    orthogonal lifted from A4
ρ833-3-1-100100    orthogonal lifted from C2×A4
ρ96-202-200000    orthogonal faithful
ρ106-20-2200000    orthogonal faithful

Character table of C42⋊C6

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

 class 12A2B3A3B4A4B4C6A6B
 size 134161666121616
ρ11111111111    trivial
ρ211-11111-1-1-1    linear of order 2
ρ311-1ζ3ζ3211-1ζ6ζ65    linear of order 6
ρ411-1ζ32ζ311-1ζ65ζ6    linear of order 6
ρ5111ζ3ζ32111ζ32ζ3    linear of order 3
ρ6111ζ32ζ3111ζ3ζ32    linear of order 3
ρ733300-1-1-100    orthogonal lifted from A4
ρ833-300-1-1100    orthogonal lifted from C2×A4
ρ96-2000-2i2i000    complex faithful
ρ106-20002i-2i000    complex faithful

Character table of C23.A4

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

 class 12A2B2C3A3B4A4B6A6B
 size 134121616661616
ρ11111111111    trivial
ρ211-1-11111-1-1    linear of order 2
ρ31111ζ32ζ311ζ3ζ32    linear of order 3
ρ41111ζ3ζ3211ζ32ζ3    linear of order 3
ρ511-1-1ζ3ζ3211ζ6ζ65    linear of order 6
ρ611-1-1ζ32ζ311ζ65ζ6    linear of order 6
ρ7333-100-1-100    orthogonal lifted from A4
ρ833-3100-1-100    orthogonal lifted from C2×A4
ρ96-200002-200    orthogonal faithful
ρ106-20000-2200    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 12A2B2C2D2E34A4B4C
 size 133361232121212
ρ11111111111    trivial
ρ211111-11-1-1-1    linear of order 2
ρ3222220-1000    orthogonal lifted from S3
ρ43-13-1-1-101-11    orthogonal lifted from S4
ρ53-1-13-110-1-11    orthogonal lifted from S4
ρ633-1-1-1-10-111    orthogonal lifted from S4
ρ73-13-1-110-11-1    orthogonal lifted from S4
ρ83-1-13-1-1011-1    orthogonal lifted from S4
ρ933-1-1-1101-1-1    orthogonal lifted from S4
ρ106-2-2-2200000    orthogonal faithful

Character table of C25⋊C4

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

 class 124A4B525A25B25C25D25E
 size 1252525444444
ρ11111111111    trivial
ρ211-1-1111111    linear of order 2
ρ31-1-ii111111    linear of order 4
ρ41-1i-i111111    linear of order 4
ρ540004-1-1-1-1-1    orthogonal lifted from F5
ρ64000-1ζ251625132512259ζ2524251825725ζ25222521254253ζ25192517258256ζ252325142511252    orthogonal faithful
ρ74000-1ζ2524251825725ζ252325142511252ζ25192517258256ζ251625132512259ζ25222521254253    orthogonal faithful
ρ84000-1ζ25192517258256ζ251625132512259ζ252325142511252ζ25222521254253ζ2524251825725    orthogonal faithful
ρ94000-1ζ25222521254253ζ25192517258256ζ2524251825725ζ252325142511252ζ251625132512259    orthogonal faithful
ρ104000-1ζ252325142511252ζ25222521254253ζ251625132512259ζ2524251825725ζ25192517258256    orthogonal faithful

Character table of C5⋊F5

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

 class 124A4B5A5B5C5D5E5F
 size 1252525444444
ρ11111111111    trivial
ρ211-1-1111111    linear of order 2
ρ31-1-ii111111    linear of order 4
ρ41-1i-i111111    linear of order 4
ρ54000-1-14-1-1-1    orthogonal lifted from F5
ρ64000-14-1-1-1-1    orthogonal lifted from F5
ρ74000-1-1-14-1-1    orthogonal lifted from F5
ρ840004-1-1-1-1-1    orthogonal lifted from F5
ρ94000-1-1-1-1-14    orthogonal lifted from F5
ρ104000-1-1-1-14-1    orthogonal lifted from F5

Character table of C52⋊C4

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

 class 124A4B5A5B5C5D5E5F
 size 1252525444444
ρ11111111111    trivial
ρ211-1-1111111    linear of order 2
ρ31-1-ii111111    linear of order 4
ρ41-1i-i111111    linear of order 4
ρ54000-1-1-14-1-1    orthogonal lifted from F5
ρ64000-1-1-1-14-1    orthogonal lifted from F5
ρ74000-1-5-1+53+5/2-1-13-5/2    orthogonal faithful
ρ840003+5/23-5/2-1+5-1-1-1-5    orthogonal faithful
ρ94000-1+5-1-53-5/2-1-13+5/2    orthogonal faithful
ρ1040003-5/23+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 12A2B2C35A5B610A10B
 size 111515201212201212
ρ11111111111    trivial
ρ21-1-11111-1-1-1    linear of order 2
ρ333-1-101+5/21-5/201-5/21+5/2    orthogonal lifted from A5
ρ433-1-101-5/21+5/201+5/21-5/2    orthogonal lifted from A5
ρ53-31-101+5/21-5/20-1+5/2-1-5/2    orthogonal faithful
ρ63-31-101-5/21+5/20-1-5/2-1+5/2    orthogonal faithful
ρ744001-1-11-1-1    orthogonal lifted from A5
ρ84-4001-1-1-111    orthogonal faithful
ρ95511-100-100    orthogonal lifted from A5
ρ105-5-11-100100    orthogonal faithful

Character table of C17⋊C8

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

 class 124A4B8A8B8C8D17A17B
 size 11717171717171788
ρ11111111111    trivial
ρ21111-1-1-1-111    linear of order 2
ρ311-1-1i-i-ii11    linear of order 4
ρ411-1-1-iii-i11    linear of order 4
ρ51-1-iiζ87ζ8ζ85ζ8311    linear of order 8
ρ61-1-iiζ83ζ85ζ8ζ8711    linear of order 8
ρ71-1i-iζ8ζ87ζ83ζ8511    linear of order 8
ρ81-1i-iζ85ζ83ζ87ζ811    linear of order 8
ρ980000000-1+17/2-1-17/2    orthogonal faithful
ρ1080000000-1-17/2-1+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 123A3B5A5B5C5D6A6B
 size 125252566662525
ρ11111111111    trivial
ρ21-1111111-1-1    linear of order 2
ρ311ζ32ζ31111ζ32ζ3    linear of order 3
ρ41-1ζ32ζ31111ζ6ζ65    linear of order 6
ρ51-1ζ3ζ321111ζ65ζ6    linear of order 6
ρ611ζ3ζ321111ζ3ζ32    linear of order 3
ρ76000-3-5/21-51+5-3+5/200    orthogonal faithful
ρ860001+5-3-5/2-3+5/21-500    orthogonal faithful
ρ96000-3+5/21+51-5-3-5/200    orthogonal faithful
ρ1060001-5-3+5/2-3-5/21+500    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 12A2B2C2D4A4B4C5A5B
 size 1555202020203232
ρ11111111111    trivial
ρ21111-1-1-1-111    linear of order 2
ρ322220000-1+5/2-1-5/2    orthogonal lifted from D5
ρ422220000-1-5/2-1+5/2    orthogonal lifted from D5
ρ55-311-111-100    orthogonal faithful
ρ65-3111-1-1100    orthogonal faithful
ρ751-3111-1-100    orthogonal faithful
ρ851-31-1-11100    orthogonal faithful
ρ9511-31-11-100    orthogonal faithful
ρ10511-3-11-1100    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 123A3B3C3D3E46A6B
 size 19812122424543636
ρ11111111111    trivial
ρ2111ζ32ζ3ζ32ζ31ζ32ζ3    linear of order 3
ρ3111ζ3ζ32ζ3ζ321ζ3ζ32    linear of order 3
ρ42-22-1-1-1-1011    symplectic lifted from SL2(𝔽3), Schur index 2
ρ52-22ζ6ζ65ζ6ζ650ζ32ζ3    complex lifted from SL2(𝔽3)
ρ62-22ζ65ζ6ζ65ζ60ζ3ζ32    complex lifted from SL2(𝔽3)
ρ73330000-100    orthogonal lifted from A4
ρ880-122-1-1000    orthogonal faithful
ρ980-1-1--3-1+-3ζ6ζ65000    complex faithful
ρ1080-1-1+-3-1--3ζ65ζ6000    complex faithful

Character table of C23.F8

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

 class 124A4B7A7B7C7D7E7F
 size 172828646464646464
ρ11111111111    trivial
ρ21111ζ74ζ76ζ72ζ75ζ7ζ73    linear of order 7
ρ31111ζ72ζ73ζ7ζ76ζ74ζ75    linear of order 7
ρ41111ζ75ζ74ζ76ζ7ζ73ζ72    linear of order 7
ρ51111ζ73ζ7ζ75ζ72ζ76ζ74    linear of order 7
ρ61111ζ7ζ75ζ74ζ73ζ72ζ76    linear of order 7
ρ71111ζ76ζ72ζ73ζ74ζ75ζ7    linear of order 7
ρ877-1-1000000    orthogonal lifted from F8
ρ914-2-2i2i000000    complex faithful
ρ1014-22i-2i000000    complex faithful
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