C₅²×C₃⋊D₃ #90666

order	450
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅² · Schur multiplier (H₂) = C₃×C₅

Aliases

direct product	C₅²×C₃⋊D₃
unambiguous semidirect product	C₃⋊(C₅²×D₃)

Fitting, socle, Hall	C₃²×C₅² ⋊ C₂
derived, Sylow	C₃² ⋊ C₂×C₅²

	subgroup	G/H
center, hypercenter	C₅²	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₅²

96 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 225

A minimal faithful permutation representation — 3 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(450, 31);