direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D58, C2×D29, C58⋊C2, C29⋊C22, sometimes denoted D116 or Dih58 or Dih116, SmallGroup(116,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — D58 |
Generators and relations for D58
G = < a,b | a58=b2=1, bab=a-1 >
Smallest permutation representation of D58
►On 58 points
Generators in S58
(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)
(1 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)
G:=sub<Sym(58)| (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), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)>;
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), (1,58)(2,57)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30) );
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)], [(1,58),(2,57),(3,56),(4,55),(5,54),(6,53),(7,52),(8,51),(9,50),(10,49),(11,48),(12,47),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,40),(20,39),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30)]])
D58 is a maximal subgroup of
D116 C29⋊D4
D58 is a maximal quotient of Dic58 D116 C29⋊D4
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 29A | ··· | 29N | 58A | ··· | 58N |
| order | 1 | 2 | 2 | 2 | 29 | ··· | 29 | 58 | ··· | 58 |
| size | 1 | 1 | 29 | 29 | 2 | ··· | 2 | 2 | ··· | 2 |
32 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D29 | D58 |
| kernel | D58 | D29 | C58 | C2 | C1 |
| # reps | 1 | 2 | 1 | 14 | 14 |
Matrix representation of D58 ►in GL2(𝔽59) generated by
| 46 | 11 |
| 30 | 29 |
| 45 | 22 |
| 26 | 14 |
G:=sub<GL(2,GF(59))| [46,30,11,29],[45,26,22,14] >;
D58 in GAP, Magma, Sage, TeX
D_{58} % in TeX
G:=Group("D58"); // GroupNames label
G:=SmallGroup(116,4);
// by ID
G=gap.SmallGroup(116,4);
# by ID
G:=PCGroup([3,-2,-2,-29,1010]);
// Polycyclic
G:=Group<a,b|a^58=b^2=1,b*a*b=a^-1>;
// generators/relations
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