C₃²×#468 #10593041

order	576
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃²
prime radical	C₃²×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ G

360 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

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

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 4923);