#1043⋊C₅ #92531

order	500
isoclinism	isoclinic class of #1043⋊C₅
automorphism	Aut(G) = #14522510 (order 2000) · Inn(G) = G · Out(G) = C₄
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

Fitting, Sylow	H(3,𝔽₅) ⋊ C₄
derived	C₅² ⋊ C₄×C₅

23 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 17);