C₂×ΓL(1,𝔽₈)⋊(C₇⋊AGL(1,𝔽₁₁)) #44568570

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

Aliases

direct product	C₂×ΓL(1,𝔽₈)⋊(C₇⋊AGL(1,𝔽₁₁))
unambiguous semidirect product	(C₂×ΓL(1,𝔽₈))⋊(C₇⋊AGL(1,𝔽₁₁))
unambiguous semidirect product	(C₂×C₁₁⋊C₅)⋊(ΓL(1,𝔽₈)⋊D₇)
unambiguous semidirect product	(C₂×C₁₁⋊(ΓL(1,𝔽₈)⋊D₇))⋊C₅

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)⋊(C₇⋊AGL(1,𝔽₁₁))
commutator, cosocle, hypocenter	C₇²×C₁₁	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₁₁	C₂×C₃×C₅
LCS	G ▷ C₇²×C₁₁
UCS	C₁ ◁ C₂

200 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.

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

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 462

A minimal faithful permutation representation — 2 generators on 27 points:

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

290 conjugacy classes. The character table is too large to display.

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 272);