C₃²×C₅×C₄⋊Q₈ #13088237

order	1440
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃²×C₅ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product	C₃²×C₅×C₄⋊Q₈
unambiguous semidirect product	C₄⋊(C₃²×C₅×Q₈)

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²×C₅	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃²×C₅ ◁ G

408 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 1440

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3)(7,8,9,10,11)(13,15)(16,17,20,23)(18,22,19,21);(4,5,6)(16,18,20,19)(17,21,23,22);(12,13,14,15)(16,19,20,18)(17,22,23,21)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4554);