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C₇×C₁₃ #784

order	91
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃²×C₄ (order 72) · Inn(G) = C₁ · Out(G) = C₂×C₃²×C₄
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₇)×AGL(1,𝔽₁₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₉₁
abelian	C₇×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) = 20
transitive degree	μ_t(G) = 91

A minimal faithful permutation representation — 1 generator on 20 points:

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

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₇×C₁₃, C₇×D₅×C₁₃, C₇×A₄×C₁₃, …
derived subgroup of	D₉₁, #63270, #63271, …
Frattini subgroup of	C₂×C₄₉×C₁₆₉, D₄₉×C₁₆₉, C₄₉×D₁₆₉, …
Fitting subgroup of	C₁₃×D₇, C₇×D₁₃, D₉₁, …
socle of	C₁₃×D₇, C₇×D₁₃, D₉₁, …
maximal subgroup of	C₁₃×D₇, C₇×D₁₃, D₉₁, …
maximal normal subgroup of	C₁₃×D₇, C₇×D₁₃, D₉₁, …
maximal quotient of	C₂×C₇×C₁₃, C₃×C₇×C₁₃, C₅×C₇×C₁₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(91, 1);