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Cayley diagram

C2×C61 #1241

order122
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C3×C4×C5 (order 60) · Inn(G) = C1 · Out(G) = C3×C4×C5
holomorphG ⋊ Aut(G) = C2×AGL(1,𝔽61)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC122
abelianC2×C61

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 63
transitive degreeμt(G) = 122

A minimal faithful permutation representation — 1 generator on 63 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

center ofD4×C61, Q8×C61, Dic3×C61, …
derived subgroup ofDic122, D244, C22⋊D61, …
Frattini subgroup ofC4×C3721, Dic3721, D14884, …
Fitting subgroup ofDic61, C2×D61, C2×C61⋊C3, …
socle ofDic61, C4×C61, C2×D61, …
maximal subgroup ofDic61, C4×C61, C2×D61, …
maximal normal subgroup ofDic61, C4×C61, C2×D61, …
maximal quotient ofC4×C61, C22×C61, D3×C61, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(122, 2);