C₅²⋊C₉ #6568

order	225
isoclinism	isoclinic class of C₅²⋊C₃
automorphism	Aut(G) = C₃×AΓL(1,𝔽₂₅) (order 3600) · Inn(G) = C₅²⋊C₃ · Out(G) = C₃×M₁₆
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₅

Aliases

C₅²⋊C₉

C₅² ⋊ C₉

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₅²⋊C₃
commutator, hypocenter	C₅²	C₉
Fitting, socle, prime radical, cosocle	C₃×C₅²	C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃

10 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(225, 3);