C₂⁵×D₅ #66301

order	320
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product	C₂⁵×D₅
unambiguous semidirect product	C₅⋊C₂⁶
unambiguous semidirect product	(C₂×C₅)⋊C₂⁵

Fitting, socle	C₂⁵×C₅ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂⁶

	subgroup	G/H
center, hypercenter	C₂⁵	D₅
commutator, cosocle, hypocenter	C₅	C₂⁶
Frattini	C₁	G
Fitting, socle	C₂⁵×C₅	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₂⁵

5650 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 6 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1639);