C₃³×C₄⋊Q₈ #11724009

order	864
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃³ · Schur multiplier (H₂) = C₂²×C₃³

Aliases

direct product	C₃³×C₄⋊Q₈
unambiguous semidirect product	C₄⋊(C₃³×Q₈)

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

952 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 864

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 3976);