C₅×Q₈⋊C₉ #67084

order	360
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₃×C₄×S₄ (order 288) · Inn(G) = A₄ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₅×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×Q₈⋊C₉
unambiguous semidirect product	Q₈⋊(C₅×C₉)
unambiguous semidirect product	(C₅×Q₈)⋊C₉

derived, Sylow	Q₈ ⋊ C₅×C₉
Hall	C₅×Q₈ ⋊ C₉

	subgroup	G/H
center, socle, prime radical, hypercenter	C₂×C₃×C₅	A₄
commutator, hypocenter	Q₈	C₅×C₉
Frattini	C₂×C₃	C₅×A₄
Fitting	C₃×C₅×Q₈	C₃
cosocle	C₃×Q₈	C₃×C₅
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃×C₅

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 2 generators on 22 points:

(2,3,5)(6,8,7)(9,10,12,11,13,15,14,16,17)(18,19,20,21,22);(1,5,4,7)(2,8,6,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 14);