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Cayley diagram

D59 #1188

order118
isoclinismisoclinic class of D59
automorphismAut(G) = AGL(1,𝔽59) (order 3422) · Inn(G) = G · Out(G) = C29
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD59
unambiguous semidirect productC59⋊C2

Splittings

Fitting, socle, derived, SylowC59 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC59C2
LCSG ▷ C59
UCSC1

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2,59|b^a=b^58

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D59, C3×D59, D177, …
maximal normal subgroup ofC2×D59, C3×D59, C5×D59, …
maximal quotient ofDic59, C2×D59, C3×D59, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(118, 1);