C₂³×D₃×AGL(1,𝔽₈) #72244641

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, hypercenter	C₂³	D₃×AGL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂³×C₃	C₂⁴×C₇
Frattini	C₁	G
Fitting, socle	C₂⁶×C₃	C₂×C₇
LCS	G ▷ C₂³×C₃
UCS	C₁ ◁ C₂³

8844 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 8844 conjugacy classes of subgroups — too many to draw. Only its 249 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)(7,8,9)(11,13,17,16,12,15,14);(3,4)(10,15)(11,13)(12,17)(14,16);(1,2);(1,2)(3,4)(5,6)(8,9)

maximal subgroup of	D₃×AGL(1,𝔽₈)², C₂×D₃×C₂²⋊AΓL(1,𝔽₈)
maximal normal subgroup of	D₃×AGL(1,𝔽₈)², C₂×D₃×C₂²⋊AΓL(1,𝔽₈)

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

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