C₃²×#477 #10593050

order	576
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

288 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.

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

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 4932);