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C₅×C₁₃ #587

order	65
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₃×C₄² (order 48) · Inn(G) = C₁ · Out(G) = C₃×C₄²
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×AGL(1,𝔽₁₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₆₅
abelian	C₅×C₁₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 65

A minimal faithful permutation representation — 1 generator on 18 points:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

center of	C₅×D₃×C₁₃, C₅×D₅×C₁₃, C₅×A₄×C₁₃, …
derived subgroup of	D₆₅, Dic₆₅, #63118, …
Frattini subgroup of	C₂₅×C₁₆₉, C₂²×C₂₅×C₁₆₉, C₂₅×#10621721, …
Fitting subgroup of	D₅×C₁₃, C₅×D₁₃, D₆₅, …
socle of	D₅×C₁₃, C₅×D₁₃, D₆₅, …
maximal subgroup of	D₅×C₁₃, C₅×D₁₃, D₆₅, …
maximal normal subgroup of	D₅×C₁₃, C₅×D₁₃, D₆₅, …
maximal quotient of	C₂×C₅×C₁₃, C₃×C₅×C₁₃, C₁₃×C₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(65, 1);