C₇×#87516 #72259722

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂×C₇	C₂×#1020
commutator, cosocle, hypocenter	#997	C₂²×C₇
Frattini	C₂	C₂×C₇×#1020
Fitting	C₂×C₇×H(5,𝔽₂)	D₃
hypercenter	C₂²×C₇	#1020
LCS	G ▷ #997
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇

552 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 32 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:

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

maximal subgroup of	ΓL(1,𝔽₈)×#87516
maximal normal subgroup of	ΓL(1,𝔽₈)×#87516

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

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