C₂³×C₅×D₃ #6845

order	240
isoclinism	isoclinic class of D₃
automorphism	Aut(G) order 32256
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product	C₂³×C₅×D₃
unambiguous semidirect product	C₃⋊(C₂⁴×C₅)
unambiguous semidirect product	(C₂×C₃)⋊(C₂³×C₅)

	subgroup	G/H
center, hypercenter	C₂³×C₅	D₃
commutator, cosocle, hypocenter	C₃	C₂⁴×C₅
Frattini	C₁	G
Fitting, socle	C₂³×C₃×C₅	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₅

268 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 4 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 206);