D₅×C₂²⋊C₉ #67112

order	360
isoclinism	isoclinic class of D₅×A₄
automorphism	Aut(G) = C₃×AGL(1,𝔽₅)×S₄ (order 1440) · Inn(G) = D₅×A₄ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₅×C₂²⋊C₉
unambiguous semidirect product	C₅⋊(C₂×C₂²⋊C₉)
unambiguous semidirect product	(C₂²×D₅)⋊C₉

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₅×A₄
commutator, hypocenter	C₂²×C₅	C₂×C₉
Fitting, socle, cosocle	C₂²×C₃×C₅	C₂×C₃
prime radical	C₂²×C₃×D₅	C₃
LCS	G ▷ C₂²×C₅
UCS	C₁ ◁ C₃

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 90

A minimal faithful permutation representation — 2 generators on 18 points:

(2,3,4)(6,7)(8,9)(10,14,18,12,17,13,15,11,16);(1,4)(2,3)(5,6,8,9,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 42);