C₃³×AGL(1,ℤ/8) #11724017

order	864
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃³ · Schur multiplier (H₂) = C₂²×C₃³

Aliases

	subgroup	G/H
center, socle	C₂×C₃³	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂³×C₃³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃³ ◁ C₂×C₃³×C₄ ◁ G

952 conjugacy classes of subgroups; only the 560 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 560 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) = 216

A minimal faithful permutation representation — 3 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 3984);