D₇×B(4,𝔽₂)⋊C₃ #72276707

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, hypercenter	C₂	D₇×#120⋊C₃
commutator, cosocle, hypocenter	C₇×H(5,𝔽₂)	C₂²×C₃
Frattini	C₂³	C₂×A₄×D₇
Fitting	C₇×B(4,𝔽₂)	C₂×C₃
socle	C₂×C₇	C₂×#120⋊C₃
LCS	G ▷ C₇×H(5,𝔽₂)
UCS	C₁ ◁ C₂

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

normal subgroups only

This group has 486 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:

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

maximal subgroup of	AGL(1,𝔽₇)×B(4,𝔽₂)⋊C₃, D₇×#73026
maximal normal subgroup of	AGL(1,𝔽₇)×B(4,𝔽₂)⋊C₃, D₇×#73026

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

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