C₂³×A₄×AGL(1,𝔽₇) #72244659

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁷×C₃

Aliases

	subgroup	G/H
center, hypercenter	C₂³	A₄×AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₂²×C₇	C₂⁴×C₃²
Frattini	C₁	G
Fitting, socle	C₂⁵×C₇	C₂×C₃²
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂³

5196 conjugacy classes of subgroups; only the 584 normal subgroups are shown.

normal subgroups only

This group has 5196 conjugacy classes of subgroups — too many to draw. Only its 584 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) = 168

A minimal faithful permutation representation — 4 generators on 17 points:

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

maximal subgroup of	C₂×A₄²×AGL(1,𝔽₇), C₂×AGL(1,𝔽₇)×GL(2,ℤ/4)
maximal normal subgroup of	C₂×A₄²×AGL(1,𝔽₇), C₂×AGL(1,𝔽₇)×GL(2,ℤ/4)

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

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