order	225
isoclinism	isoclinic class of C₅²⋊C₃
automorphism	Aut(G) = D₃×AΓL(1,𝔽₂₅) (order 7200) · Inn(G) = C₅²⋊C₃ · Out(G) = D₃×M₁₆
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₃×C₅

Aliases

direct product	C₃×C₅²⋊C₃
unambiguous semidirect product	C₅²⋊C₃²
unambiguous semidirect product	(C₃×C₅²)⋊C₃

Splittings

Fitting, socle	C₃×C₅² ⋊ C₃
derived, Sylow	C₅² ⋊ C₃²

	subgroup	G/H
center, hypercenter	C₃	C₅²⋊C₃
commutator, cosocle, hypocenter	C₅²	C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅²	C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 2 generators on 18 points:

(4,5,7)(6,10,12)(8,13,16)(9,15,18)(11,14,17);(1,2,3)(4,6,11,16,18)(5,10,14,8,9)(7,13,12,15,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(225, 5);