C₂×D₃×D₅ #1232

order	120
isoclinism	isoclinic class of D₃×D₅
automorphism	Aut(G) = C₂²×D₃×AGL(1,𝔽₅) (order 480) · Inn(G) = D₃×D₅ · Out(G) = C₂³
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂×D₃×D₅
unambiguous semidirect product	(C₅×D₃)⋊C₂²
unambiguous semidirect product	(C₃×D₅)⋊C₂²
unambiguous semidirect product	D₁₅⋊C₂²

	subgroup	G/H
center, hypercenter	C₂	D₃×D₅
commutator, cosocle, hypocenter	C₃×C₅	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂

64 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 3 generators on 10 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 42);