C₅×C₅²⋊C₃ #67333

order	375
isoclinism	isoclinic class of C₅²⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅×C₅²⋊C₃
unambiguous semidirect product	C₅³⋊C₃

Fitting, socle, Sylow	C₅³ ⋊ C₃
derived	C₅² ⋊ C₃×C₅

	subgroup	G/H
center, hypercenter	C₅	C₅²⋊C₃
commutator, cosocle, hypocenter	C₅²	C₃×C₅
Frattini	C₁	G
Fitting, socle	C₅³	C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₅

28 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 15

A minimal faithful permutation representation — 2 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(375, 6);