order	150
isoclinism	isoclinic class of C₅²⋊C₃
automorphism	Aut(G) = AΓL(1,𝔽₂₅) (order 1200) · Inn(G) = C₅²⋊C₃ · Out(G) = M₁₆
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₂×C₅²⋊C₃
unambiguous semidirect product	(C₂×C₅²)⋊C₃

Splittings

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

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

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 2 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(150, 7);