C₅²×D₅ #6957

order	250
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = AGL(1,𝔽₅)×GL(2,𝔽₅) (order 9600) · Inn(G) = D₅ · Out(G) = C₂×GL(2,𝔽₅)
homology	Abelianization (H₁) = C₂×C₅² · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅²×D₅
unambiguous semidirect product	C₅⋊(C₂×C₅²)

Fitting, socle, Sylow	C₅³ ⋊ C₂
derived	C₅ ⋊ C₂×C₅²

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

56 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 50

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(250, 12);