(C₅⋊D₅)⋊C₅ #6950

order	250
isoclinism	isoclinic class of (C₅⋊D₅)⋊C₅
automorphism	Aut(G) = #14522510 (order 2000) · Inn(G) = G · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₅

Aliases

(C₅⋊D₅)⋊C₅

Fitting, Sylow	H(3,𝔽₅) ⋊ C₂
derived	C₅² ⋊ C₂×C₅

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₅²	C₂×C₅
Frattini, socle	C₅	C₅×D₅
Fitting	H(3,𝔽₅)	C₂
LCS	G ▷ C₅²
UCS	C₁

18 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 25

A minimal faithful permutation representation — 2 generators on 25 points:

(2,9,11,18,15,4,5,17,19,10)(3,7)(6,23,24,20,21,13,14,22,12,25)(8,16);(1,2,6,13,4)(3,5,12,22,10)(7,15,24,20,9)(8,11,14,25,18)(16,19,21,23,17)

Polycyclic presentation:

2,5^3|c^a=c^4,d^a=d^4,c^b=cd

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(250, 5);