C₂×C₅×C₃⋊D₃ #4390

order	180
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂×C₄×AGL(2,𝔽₃) (order 3456) · Inn(G) = C₃⋊D₃ · Out(G) = C₂×C₄×S₄
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₃

Aliases

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

	subgroup	G/H
center, hypercenter	C₂×C₅	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₅

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 90

A minimal faithful permutation representation — 3 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 35);