C₂×C₇×#5935 #72229422

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

Aliases

Fitting	C₂×C₇×H(5,𝔽₂) ⋊ D₃
derived	#997 ⋊ C₂²×C₇

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

666 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

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

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2)(4,5,8)(7,10,9)(11,12,13,14,15,16,17);(3,8,4,5,6,9,7,10)

maximal subgroup of	C₂×ΓL(1,𝔽₈)×#5935, C₂×D₇×#5935
maximal normal subgroup of	C₂×ΓL(1,𝔽₈)×#5935, C₂×D₇×#5935

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

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