C₃×D₅² #64493

order	300
isoclinism	isoclinic class of D₅²
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)²⋊C₂ (order 1600) · Inn(G) = D₅² · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₃×D₅²
unambiguous semidirect product	(C₅×D₅)⋊(C₂×C₃)

Fitting, socle, Hall	C₃×C₅² ⋊ C₂²
derived, Sylow	C₅² ⋊ C₂²×C₃

	subgroup	G/H
center, hypercenter	C₃	D₅²
commutator, cosocle, hypocenter	C₅²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₅²	C₂²
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 2 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 36);