#13062013

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂	C₂²×GL(3,𝔽₂)
commutator, hypocenter	SL(2,𝔽₇)	C₂²
Fitting, hypercenter	D₄	GL(3,𝔽₂)
LCS	G ▷ SL(2,𝔽₇)
UCS	C₁ ◁ C₂ ◁ D₄

157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,13,29,26,30,31,28,19,11,23,16,8,4,2)(3,21,25,17,20,12,6)(5,22,15,24,27,18,10)(7,14);(1,10,11,5,2,9,4,25,26,17,8,15,7,3)(6,28,32,31,24,16,27,30,21,14,22,19,12,13)(18,23)(20,29)

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

maximal quotient of

D₄×SL(2,𝔽₇)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11302);