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Cayley diagram

C61 #313

order61
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C3×C4×C5 (order 60) · Inn(G) = C1 · Out(G) = C3×C4×C5
holomorphG ⋊ Aut(G) = AGL(1,𝔽61)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC61

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 61
transitive degreeμt(G) = 61

A minimal faithful permutation representation — 1 generator on 61 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61)

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

center ofD3×C61, D5×C61, A4×C61, …
derived subgroup ofD61, C61⋊C3, Dic61, …
Frattini subgroup ofC3721, C22×C3721, #28415637, …
Fitting subgroup ofD61, C61⋊C3, #6923, …
socle ofD61, C61⋊C3, #6923, …
maximal subgroup ofD61, C2×C61, C61⋊C3, …
maximal normal subgroup ofD61, C2×C61, C61⋊C3, …
maximal quotient ofC2×C61, C3×C61, C5×C61, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(61, 1);