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G = D59order 118 = 2·59

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D59, C59⋊C2, sometimes denoted D118 or Dih59 or Dih118, SmallGroup(118,1)

Series: Derived Chief Lower central Upper central

C1C59 — D59
C1C59 — D59
C59 — D59
C1

Generators and relations for D59
 G = < a,b | a59=b2=1, bab=a-1 >

59C2

Smallest permutation representation of D59
On 59 points: primitive
Generators in S59
(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)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)

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

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

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

D59 is a maximal subgroup of   D177
D59 is a maximal quotient of   Dic59  D177

31 conjugacy classes

class 1  2 59A···59AC
order1259···59
size1592···2

31 irreducible representations

dim112
type+++
imageC1C2D59
kernelD59C59C1
# reps1129

Matrix representation of D59 in GL2(𝔽709) generated by

82708
10
,
82708
342627
G:=sub<GL(2,GF(709))| [82,1,708,0],[82,342,708,627] >;

D59 in GAP, Magma, Sage, TeX

D_{59}
% in TeX

G:=Group("D59");
// GroupNames label

G:=SmallGroup(118,1);
// by ID

G=gap.SmallGroup(118,1);
# by ID

G:=PCGroup([2,-2,-59,465]);
// Polycyclic

G:=Group<a,b|a^59=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D59 in TeX

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