(C₃⋊D₃)⋊D₅ #4385

order	180
isoclinism	isoclinic class of (C₃⋊D₃)⋊D₅
automorphism	Aut(G) = AGL(1,𝔽₅)×GO⁺(4,𝔽₂) (order 1440) · Inn(G) = G · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	(C₃⋊D₃)⋊D₅
unambiguous semidirect product	D₁₅⋊D₃

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

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

Polycyclic presentation:

2^2,3^2,5|c^a=c^2,e^a=e^4,c^b=c^2,d^b=d^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 30);