C₅⋊(C₅⋊D₅) #6959

order	250
isoclinism	isoclinic class of C₅⋊(C₅⋊D₅)
automorphism	Aut(G) order 186000000
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₅³

Aliases

C₅⋊(C₅⋊D₅)

C₅³ ⋊ C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅³	C₂
LCS	G ▷ C₅³
UCS	C₁

128 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 125

A minimal faithful permutation representation — 4 generators on 15 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(250, 14);