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Cayley diagram

D61 #1240

order122
isoclinismisoclinic class of D61
automorphismAut(G) = AGL(1,𝔽61) (order 3660) · Inn(G) = G · Out(G) = C2×C3×C5
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD61
unambiguous semidirect productC61⋊C2

Splittings

Fitting, socle, derived, SylowC61 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC61C2
LCSG ▷ C61
UCSC1

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2,61|b^a=b^60

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6923, C2×D61, D61⋊C3, …
maximal normal subgroup of#6923, C2×D61, D61⋊C3, …
maximal quotient ofDic61, C2×D61, C3×D61, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(122, 1);