D₂₅⋊C₅ #6951

order	250
isoclinism	isoclinic class of D₂₅⋊C₅
automorphism	Aut(G) = AGL(1,ℤ/25) (order 500) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	D₂₅⋊C₅
unambiguous semidirect product	5₋¹⁺²⋊C₂

Fitting, Sylow	5₋¹⁺² ⋊ C₂
derived	C₂₅ ⋊ C₂×C₅

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂₅	C₂×C₅
Frattini, socle	C₅	C₅×D₅
Fitting	5₋¹⁺²	C₂
LCS	G ▷ C₂₅
UCS	C₁

14 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(250, 6);