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C₁₃ #25

order	13
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₃×C₄ (order 12) · Inn(G) = C₁ · Out(G) = C₃×C₄
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₁₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₁₃

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 1 generator on 13 points:

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

Conjugacy classes & character table

13 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₁₃, D₅×C₁₃, A₄×C₁₃, …
derived subgroup of	D₁₃, C₁₃⋊C₃, Dic₁₃, …
Frattini subgroup of	C₁₆₉, D₁₆₉, C₂×C₁₆₉, …
Fitting subgroup of	D₁₃, C₁₃⋊C₃, #261, …
socle of	D₁₃, C₁₃⋊C₃, #261, …
maximal subgroup of	D₁₃, C₂×C₁₃, C₁₃⋊C₃, …
maximal normal subgroup of	D₁₃, C₂×C₁₃, C₁₃⋊C₃, …
maximal quotient of	C₂×C₁₃, C₃×C₁₃, C₅×C₁₃, …
outer automorphism group of	#6157

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13, 1);