C₂×C₃²×#123 #10596325

order	576
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

C₂×C₃²×#123

	subgroup	G/H
center, socle	C₂³×C₃²	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃²
prime radical	C₂²×C₃²×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

1026 conjugacy classes of subgroups; only the 534 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 534 normal subgroups are shown.

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

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 8207);