C₃³×#124 #11724005

order	864
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃³ · Schur multiplier (H₂) = C₂×C₃³×C₄

Aliases

	subgroup	G/H
center, socle	C₂²×C₃³	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃³
prime radical	C₂×C₃³×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃³ ◁ G

1064 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 3 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 3972);