C₂²×#72214032 #72276996

order	8064
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, hypercenter	C₂²	#72214032
commutator, cosocle, hypocenter	C₇×C₂²⋊A₄	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂⁶×C₇	C₃×D₃
LCS	G ▷ C₇×C₂²⋊A₄
UCS	C₁ ◁ C₂²

5778 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 5778 conjugacy classes of subgroups — too many to draw. Only its 112 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 — 3 generators on 19 points:

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

maximal subgroup of	C₂²×AGL(1,𝔽₇)×#1020
maximal normal subgroup of	C₂²×AGL(1,𝔽₇)×#1020

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

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