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
| C59 — D59 |
Generators and relations for D59
G = < a,b | a59=b2=1, bab=a-1 >
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 |
| order | 1 | 2 | 59 | ··· | 59 |
| size | 1 | 59 | 2 | ··· | 2 |
31 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D59 |
| kernel | D59 | C59 | C1 |
| # reps | 1 | 1 | 29 |
Matrix representation of D59 ►in GL2(𝔽709) generated by
| 82 | 708 |
| 1 | 0 |
| 82 | 708 |
| 342 | 627 |
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
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