ΓL(1,𝔽₈)×#5936 #72276614

order	4032
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

derived	C₇×C₂²⋊A₄ ⋊ C₃×C₄
Sylow	C₇ ⋊ C₃×#5936

	subgroup	G/H
center, Frattini, hypercenter	C₂	ΓL(1,𝔽₈)×#1020
commutator, hypocenter	C₇×C₂²⋊A₄	C₃×C₄
Fitting, socle	C₂⁵×C₇	C₃×D₃
prime radical	C₂×ΓL(1,𝔽₈)×C₂²⋊A₄	C₂
cosocle	C₂×C₇×C₂²⋊A₄	C₂×C₃
LCS	G ▷ C₇×C₂²⋊A₄
UCS	C₁ ◁ C₂

628 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.

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

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 19 points:

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

maximal subgroup of	AGL(1,𝔽₇)×#5936, #72276701, ΓL(1,𝔽₈)×#87524, …
maximal normal subgroup of	AGL(1,𝔽₇)×#5936, #72276701, ΓL(1,𝔽₈)×#87524, …

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

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