C₂⁴×D₅ #4162

order	160
isoclinism	isoclinic class of D₅
automorphism	Aut(G) order 6451200
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product	C₂⁴×D₅
unambiguous semidirect product	C₅⋊C₂⁵
unambiguous semidirect product	(C₂×C₅)⋊C₂⁴

Fitting, socle	C₂⁴×C₅ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂⁵

	subgroup	G/H
center, hypercenter	C₂⁴	D₅
commutator, cosocle, hypocenter	C₅	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂⁴×C₅	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₂⁴

748 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 5 generators on 13 points:

(1,2)(3,4)(5,6)(7,8)(10,11)(12,13);(7,8)(9,10,12,13,11);(5,6);(3,4);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 237);