C₃²×#1654 #11768703

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

	subgroup	G/H
center, socle	C₂²×C₃²	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
prime radical	C₃²×#547	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

720 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2714);