C₅×D₃×D₅ #64494

order	300
isoclinism	isoclinic class of D₃×D₅
automorphism	Aut(G) = C₄×D₃×AGL(1,𝔽₅) (order 480) · Inn(G) = D₃×D₅ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₃×D₅
unambiguous semidirect product	(C₅×D₃)⋊(C₂×C₅)
unambiguous semidirect product	(C₃×D₅)⋊(C₂×C₅)
unambiguous semidirect product	D₁₅⋊(C₂×C₅)

	subgroup	G/H
center, hypercenter	C₅	D₃×D₅
commutator, cosocle, hypocenter	C₃×C₅	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅²	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₅

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 37);