C₃×D₅×C₁₃ #87613

order	390
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₂×C₃×C₄×AGL(1,𝔽₅) (order 480) · Inn(G) = D₅ · Out(G) = C₂²×C₃×C₄
homology	Abelianization (H₁) = C₂×C₃×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₅×C₁₃
unambiguous semidirect product	C₅⋊(C₂×C₃×C₁₃)

	subgroup	G/H
center, hypercenter	C₃×C₁₃	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₁₃

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 195

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,3)(5,6)(7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21);(4,5,7,8,6)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(390, 6);