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Cayley diagram

C59 #299

order59
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C29 (order 58) · Inn(G) = C1 · Out(G) = C2×C29
holomorphG ⋊ Aut(G) = AGL(1,𝔽59)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC59

Subgroups

subgroupG/H
centerGC1
commutator, FrattiniC1G
LCSG ▷ C1
UCSC1 ◁ G
LECSG ▷ C1

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 59
transitive degreeμt(G) = 59

A minimal faithful permutation representation — 1 generator on 59 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59)

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

center ofD3×C59, D5×C59, A4×C59, …
derived subgroup ofD59, Dic59, C2×D59, …
Frattini subgroup ofC3481, C22×C3481, C2×D3481, …
Fitting subgroup ofD59, C59⋊C29, AGL(1,𝔽59), …
socle ofD59, C59⋊C29, AGL(1,𝔽59), …
maximal subgroup ofD59, C2×C59, C3×C59, …
maximal normal subgroup ofD59, C2×C59, C3×C59, …
maximal quotient ofC2×C59, C3×C59, C5×C59, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(59, 1);