C₂×C₅×C₅⋊D₅ #92568

order	500
isoclinism	isoclinic class of C₅⋊D₅
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₅

Aliases

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

	subgroup	G/H
center, hypercenter	C₂×C₅	C₅⋊D₅
commutator, cosocle, hypocenter	C₅²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅³	C₂
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂×C₅

128 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 3 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 54);