C₂×C₃²×#120 #10596322

order	576
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁷×C₃

Aliases

C₂×C₃²×#120

	subgroup	G/H
center, socle	C₂³×C₃²	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

1986 conjugacy classes of subgroups; only the 630 normal subgroups are shown.

normal subgroups only

This group has 1986 conjugacy classes of subgroups — too many to draw. Only its 630 normal subgroups are shown.

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

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 8204);