order	360
isoclinism	isoclinic class of (C₂²⋊C₉)⋊C₂
automorphism	Aut(G) = C₄×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 864) · Inn(G) = (C₂²⋊C₉)⋊C₂ · Out(G) = C₃×C₄
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₂

Aliases

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

Splittings

	subgroup	G/H
center, hypercenter	C₅	(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	C₂²⋊C₉	C₂×C₅
Frattini	C₃	C₅×S₄
Fitting, socle	C₂²×C₃×C₅	D₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₅

36 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 40);