((C₅⋊D₅)⋊C₅)⋊C₂ #92541

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

Aliases

unambiguous semidirect product	((C₅⋊D₅)⋊C₅)⋊C₂
unambiguous semidirect product	#6953⋊C₂

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	H(3,𝔽₅)	C₂²
Frattini, socle	C₅	D₅²
LCS	G ▷ H(3,𝔽₅)
UCS	C₁

43 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 27);