C₂²×C₃×D₅ #1234

order	120
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×S₄ (order 960) · Inn(G) = D₅ · Out(G) = C₂²×S₄
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂²×C₃×D₅
unambiguous semidirect product	C₅⋊(C₂³×C₃)
unambiguous semidirect product	(C₂×C₅)⋊(C₂²×C₃)

	subgroup	G/H
center, hypercenter	C₂²×C₃	D₅
commutator, cosocle, hypocenter	C₅	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₃

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 44);