#226×GL(3,𝔽₂) #72228107

order	8064
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	S₄×GL(3,𝔽₂)
commutator, prime radical, hypocenter	SL(2,𝔽₃)×GL(3,𝔽₂)	C₂
Fitting	Q₈	D₃×GL(3,𝔽₂)
socle	C₂×GL(3,𝔽₂)	S₄
cosocle	SL(2,𝔽₃)	C₂×GL(3,𝔽₂)
LCS	G ▷ SL(2,𝔽₃)×GL(3,𝔽₂)
UCS	C₁ ◁ C₂

369 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 23 points:

(1,15,7,12,2,10,3,13)(4,8,9,14,5,6,11,16)(17,19)(18,23,22,20);(1,6,2,8)(3,10,7,15)(4,12,5,13)(9,16,11,14)(17,18,23,22,21,20,19)

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,15,7,12,2,10,3,13)(4,8,9,14,5,6,11,16)(17,19)(18,23,22,20), (1,6,2,8)(3,10,7,15)(4,12,5,13)(9,16,11,14)(17,18,23,22,21,20,19));