C₅×D₅² #92564

order	500
isoclinism	isoclinic class of D₅²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₅²
unambiguous semidirect product	(C₅×D₅)⋊(C₂×C₅)

Fitting, socle, Sylow	C₅³ ⋊ C₂²
derived	C₅² ⋊ C₂²×C₅

	subgroup	G/H
center, hypercenter	C₅	D₅²
commutator, cosocle, hypocenter	C₅²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₅³	C₂²
LCS	G ▷ C₅²
UCS	C₁ ◁ C₅

72 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 20

A minimal faithful permutation representation — 2 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 50);