order	150
isoclinism	isoclinic class of C₅⋊D₅
automorphism	Aut(G) = C₂×AGL(2,𝔽₅) (order 24000) · Inn(G) = C₅⋊D₅ · Out(G) = C₂×A₅⋊C₄
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₃×C₅⋊D₅
unambiguous semidirect product	C₅⋊(C₃×D₅)

Splittings

Fitting, socle, Hall	C₃×C₅² ⋊ C₂
derived, Sylow	C₅² ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₃	C₅⋊D₅
commutator, cosocle, hypocenter	C₅²	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₅²	C₂
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 75

A minimal faithful permutation representation — 3 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(150, 9);