C₅×D₃² #4383

order	180
isoclinism	isoclinic class of D₃²
automorphism	Aut(G) = C₄×GO⁺(4,𝔽₂) (order 288) · Inn(G) = D₃² · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₃²
unambiguous semidirect product	(C₃×D₃)⋊(C₂×C₅)
unambiguous semidirect product	(C₃⋊D₃)⋊(C₂×C₅)

Fitting, socle, Hall	C₃²×C₅ ⋊ C₂²
derived, Sylow	C₃² ⋊ C₂²×C₅

	subgroup	G/H
center, hypercenter	C₅	D₃²
commutator, cosocle, hypocenter	C₃²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₅

44 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 2 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 28);