AGL(1,𝔽₇)×#5933 #72259967

order	8064
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	AGL(1,𝔽₇)×#1020
commutator, cosocle, hypocenter	C₇×#997	C₂²×C₃
Fitting	C₇×H(5,𝔽₂)	C₂×C₃×D₃
socle	C₂×C₇	C₂×C₃×#1020
prime radical	AGL(1,𝔽₇)×#997	C₂
LCS	G ▷ C₇×#997
UCS	C₁ ◁ C₂

1152 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 44 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:

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

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

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