order	480
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×2₋¹⁺⁴⋊C₃
unambiguous semidirect product	(C₅×2₋¹⁺⁴)⋊C₃

Splittings

Fitting, Hall	C₅×2₋¹⁺⁴ ⋊ C₃
Sylow	2₋¹⁺⁴ ⋊ C₃×C₅

	subgroup	G/H
center, socle	C₂×C₅	C₂²×A₄
commutator, cosocle, hypocenter	Q₈	C₂²×C₃×C₅
Frattini	C₂	C₂²×C₅×A₄
Fitting	C₅×2₋¹⁺⁴	C₃
hypercenter	C₅×D₄	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₅ ◁ C₅×D₄

92 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1132);