D21 - GroupNames

class	1	2	3	7A	7B	7C	21A	21B	21C	21D	21E	21F
size	1	21	2	2	2	2	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	1	1	1	1	1	linear of order 2
ρ₃	2	0	-1	2	2	2	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₄	2	0	2	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	orthogonal lifted from D₇
ρ₅	2	0	2	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	orthogonal lifted from D₇
ρ₆	2	0	2	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	orthogonal lifted from D₇
ρ₇	2	0	-1	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₃ζ₇⁵-ζ₃ζ₇²-ζ₇²	-ζ₃²ζ₇⁴+ζ₃²ζ₇³-ζ₇⁴	ζ₃²ζ₇⁴-ζ₃²ζ₇³-ζ₇³	ζ₃²ζ₇⁵-ζ₃²ζ₇²-ζ₇²	-ζ₃ζ₇⁶+ζ₃ζ₇-ζ₇⁶	ζ₃ζ₇⁶-ζ₃ζ₇-ζ₇	orthogonal faithful
ρ₈	2	0	-1	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	-ζ₃ζ₇⁶+ζ₃ζ₇-ζ₇⁶	ζ₃ζ₇⁵-ζ₃ζ₇²-ζ₇²	ζ₃²ζ₇⁵-ζ₃²ζ₇²-ζ₇²	ζ₃ζ₇⁶-ζ₃ζ₇-ζ₇	ζ₃²ζ₇⁴-ζ₃²ζ₇³-ζ₇³	-ζ₃²ζ₇⁴+ζ₃²ζ₇³-ζ₇⁴	orthogonal faithful
ρ₉	2	0	-1	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₃²ζ₇⁴-ζ₃²ζ₇³-ζ₇³	-ζ₃ζ₇⁶+ζ₃ζ₇-ζ₇⁶	ζ₃ζ₇⁶-ζ₃ζ₇-ζ₇	-ζ₃²ζ₇⁴+ζ₃²ζ₇³-ζ₇⁴	ζ₃²ζ₇⁵-ζ₃²ζ₇²-ζ₇²	ζ₃ζ₇⁵-ζ₃ζ₇²-ζ₇²	orthogonal faithful
ρ₁₀	2	0	-1	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₃ζ₇⁶-ζ₃ζ₇-ζ₇	ζ₃²ζ₇⁵-ζ₃²ζ₇²-ζ₇²	ζ₃ζ₇⁵-ζ₃ζ₇²-ζ₇²	-ζ₃ζ₇⁶+ζ₃ζ₇-ζ₇⁶	-ζ₃²ζ₇⁴+ζ₃²ζ₇³-ζ₇⁴	ζ₃²ζ₇⁴-ζ₃²ζ₇³-ζ₇³	orthogonal faithful
ρ₁₁	2	0	-1	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₃²ζ₇⁵-ζ₃²ζ₇²-ζ₇²	ζ₃²ζ₇⁴-ζ₃²ζ₇³-ζ₇³	-ζ₃²ζ₇⁴+ζ₃²ζ₇³-ζ₇⁴	ζ₃ζ₇⁵-ζ₃ζ₇²-ζ₇²	ζ₃ζ₇⁶-ζ₃ζ₇-ζ₇	-ζ₃ζ₇⁶+ζ₃ζ₇-ζ₇⁶	orthogonal faithful
ρ₁₂	2	0	-1	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	-ζ₃²ζ₇⁴+ζ₃²ζ₇³-ζ₇⁴	ζ₃ζ₇⁶-ζ₃ζ₇-ζ₇	-ζ₃ζ₇⁶+ζ₃ζ₇-ζ₇⁶	ζ₃²ζ₇⁴-ζ₃²ζ₇³-ζ₇³	ζ₃ζ₇⁵-ζ₃ζ₇²-ζ₇²	ζ₃²ζ₇⁵-ζ₃²ζ₇²-ζ₇²	orthogonal faithful

Permutation representations of D₂₁
►On 21 points - transitive group 21T5

Generators in S₂₁

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21)

(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)

G:=sub<Sym(21)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12)]])

G:=TransitiveGroup(21,5);

D₂₁ is a maximal subgroup of
S₃×D₇ D₆₃ C₃⋊F₇ C₃⋊D₂₁ C₇⋊S₄ D₁₀₅ D₁₄₇ C₇⋊D₂₁ D₂₃₁
D₂₁ is a maximal quotient of
Dic₂₁ D₆₃ C₃⋊D₂₁ C₇⋊S₄ D₁₀₅ D₁₄₇ C₇⋊D₂₁ D₂₃₁

Matrix representation of D₂₁ ►in GL₂(𝔽₄₁) generated by

40	20
21	30

30	35
20	11

G:=sub<GL(2,GF(41))| [40,21,20,30],[30,20,35,11] >;

D₂₁ in GAP, Magma, Sage, TeX

D_{21}

% in TeX

G:=Group("D21");

// GroupNames label

G:=SmallGroup(42,5);

// by ID

G=gap.SmallGroup(42,5);

# by ID

G:=PCGroup([3,-2,-3,-7,25,326]);

// Polycyclic

G:=Group<a,b|a^21=b^2=1,b*a*b=a^-1>;

// generators/relations

Export

Subgroup lattice of D₂₁ in TeX
Character table of D₂₁ in TeX

G = D21 order 42 = 2·3·7

Dihedral group

G = D₂₁ order 42 = 2·3·7