C₂²×C₅×D₃ #1235

order	120
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₄×D₃×S₄ (order 576) · Inn(G) = D₃ · Out(G) = C₄×S₄
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂²×C₅×D₃
unambiguous semidirect product	C₃⋊(C₂³×C₅)
unambiguous semidirect product	(C₂×C₃)⋊(C₂²×C₅)

	subgroup	G/H
center, hypercenter	C₂²×C₅	D₃
commutator, cosocle, hypocenter	C₃	C₂³×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₅

64 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 3 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 45);