C₃×#32931 #10651538

order	768
nilpotency class	3
isoclinism	isoclinic class of #32929
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂⁵×C₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂⁴×C₃×C₄ ◁ G

1630 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 366 normal subgroups are shown.

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

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2,3)(4,5)(6,7)(8,9)(10,11)(12,13,16,19)(14,20,23,27)(15,24,21,18)(17,25,26,22);(12,14,21,25,16,23,15,22)(13,17,24,27,19,26,18,20);(5,7);(9,11)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 25919);