direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D62, C2×D31, C62⋊C2, C31⋊C22, sometimes denoted D124 or Dih62 or Dih124, SmallGroup(124,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — D62 |
Generators and relations for D62
G = < a,b | a62=b2=1, bab=a-1 >
Smallest permutation representation of D62
►On 62 points
Generators in S62
(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 62)
(1 62)(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)
G:=sub<Sym(62)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62), (1,62)(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)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62), (1,62)(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) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62)], [(1,62),(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)]])
D62 is a maximal subgroup of
D124 C31⋊D4
D62 is a maximal quotient of Dic62 D124 C31⋊D4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 31A | ··· | 31O | 62A | ··· | 62O |
| order | 1 | 2 | 2 | 2 | 31 | ··· | 31 | 62 | ··· | 62 |
| size | 1 | 1 | 31 | 31 | 2 | ··· | 2 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D31 | D62 |
| kernel | D62 | D31 | C62 | C2 | C1 |
| # reps | 1 | 2 | 1 | 15 | 15 |
Matrix representation of D62 ►in GL3(𝔽311) generated by
| 310 | 0 | 0 |
| 0 | 239 | 211 |
| 0 | 100 | 100 |
| 1 | 0 | 0 |
| 0 | 239 | 211 |
| 0 | 198 | 72 |
G:=sub<GL(3,GF(311))| [310,0,0,0,239,100,0,211,100],[1,0,0,0,239,198,0,211,72] >;
D62 in GAP, Magma, Sage, TeX
D_{62} % in TeX
G:=Group("D62"); // GroupNames label
G:=SmallGroup(124,3);
// by ID
G=gap.SmallGroup(124,3);
# by ID
G:=PCGroup([3,-2,-2,-31,1082]);
// Polycyclic
G:=Group<a,b|a^62=b^2=1,b*a*b=a^-1>;
// generators/relations
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