D₂₅ #253

order	50
isoclinism	isoclinic class of D₂₅
automorphism	Aut(G) = AGL(1,ℤ/25) (order 500) · Inn(G) = G · Out(G) = C₂×C₅
holomorph	G ⋊ Aut(G) = #72213951
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₂₅
unambiguous semidirect product	C₂₅⋊C₂

C₂₅ ⋊ C₂

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₅	C₂
Frattini, socle	C₅	D₅
LCS	G ▷ C₂₅
UCS	C₁

6 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(50, 1);