AGL(1,𝔽₁₁)×ΓL(1,𝔽₈)⋊D₇ #44568549

order	32340
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	AGL(1,𝔽₁₁)×ΓL(1,𝔽₈)⋊D₇
unambiguous semidirect product	D₁₁⋊(C₅×ΓL(1,𝔽₈)⋊D₇)
unambiguous semidirect product	(ΓL(1,𝔽₈)×AGL(1,𝔽₁₁))⋊D₇
unambiguous semidirect product	(D₁₁×ΓL(1,𝔽₈)⋊D₇)⋊C₅

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₁₁	C₂²×C₃×C₅
LCS	G ▷ C₇²×C₁₁
UCS	C₁

200 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.

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

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 231

A minimal faithful permutation representation — 2 generators on 25 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 251);