ΓL(1,𝔽₈)×B(4,𝔽₂)⋊C₃ #72276694

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)×#120⋊C₃
commutator, cosocle, hypocenter	C₇×H(5,𝔽₂)	C₂×C₃²
Frattini	C₂³	C₂×A₄×ΓL(1,𝔽₈)
Fitting	C₇×B(4,𝔽₂)	C₃²
socle	C₂×C₇	C₃×#120⋊C₃
LCS	G ▷ C₇×H(5,𝔽₂)
UCS	C₁ ◁ C₂

292 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.

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

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2,4,6,3,5,7)(9,15,10,12,13,14);(2,3,4)(5,7,6)(8,13)(9,10)(11,15)(12,14)

maximal subgroup of	AGL(1,𝔽₇)×B(4,𝔽₂)⋊C₃, ΓL(1,𝔽₈)×#73026, #72276831
maximal normal subgroup of	AGL(1,𝔽₇)×B(4,𝔽₂)⋊C₃, ΓL(1,𝔽₈)×#73026, #72276831

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

gap> G := Group((1,2,4,6,3,5,7)(9,15,10,12,13,14), (2,3,4)(5,7,6)(8,13)(9,10)(11,15)(12,14));