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C₂×C₂₉ #298

order	58
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₄×C₇ (order 28) · Inn(G) = C₁ · Out(G) = C₄×C₇
holomorph	G ⋊ Aut(G) = C₂×AGL(1,𝔽₂₉)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₅₈
abelian	C₂×C₂₉
multiplicative	(ℤ/59)ˣ, (ℤ/118)ˣ

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) = 31
transitive degree	μ_t(G) = 58

A minimal faithful permutation representation — 1 generator on 31 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

center of	D₄×C₂₉, Q₈×C₂₉, Dic₃×C₂₉, …
derived subgroup of	Dic₅₈, D₁₁₆, C₂²⋊D₂₉, …
Frattini subgroup of	C₄×C₈₄₁, Dic₈₄₁, C₄×C₅×C₈₄₁, …
Fitting subgroup of	Dic₂₉, C₂×D₂₉, #6599, …
socle of	Dic₂₉, C₄×C₂₉, C₂×D₂₉, …
maximal subgroup of	Dic₂₉, C₄×C₂₉, C₂×D₂₉, …
maximal normal subgroup of	Dic₂₉, C₄×C₂₉, C₂×D₂₉, …
maximal quotient of	C₄×C₂₉, C₂²×C₂₉, D₃×C₂₉, …
automorphism group of	C₅₉, C₂×C₅₉
outer automorphism group of	C₅₉, C₂×C₅₉, Dic₅₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(58, 2);