metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D69, C23⋊S3, C3⋊D23, C69⋊1C2, sometimes denoted D138 or Dih69 or Dih138, SmallGroup(138,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C69 — D69 |
Generators and relations for D69
G = < a,b | a69=b2=1, bab=a-1 >
Smallest permutation representation of D69
►On 69 points
Generators in S69
(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 63 64 65 66 67 68 69)
(1 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)
G:=sub<Sym(69)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,60),(11,59),(12,58),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36)]])
D69 is a maximal subgroup of
S3×D23 D207 C3⋊D69
D69 is a maximal quotient of Dic69 D207 C3⋊D69
36 conjugacy classes
| class | 1 | 2 | 3 | 23A | ··· | 23K | 69A | ··· | 69V |
| order | 1 | 2 | 3 | 23 | ··· | 23 | 69 | ··· | 69 |
| size | 1 | 69 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D23 | D69 |
| kernel | D69 | C69 | C23 | C3 | C1 |
| # reps | 1 | 1 | 1 | 11 | 22 |
Matrix representation of D69 ►in GL2(𝔽139) generated by
| 78 | 36 |
| 103 | 3 |
| 78 | 36 |
| 82 | 61 |
G:=sub<GL(2,GF(139))| [78,103,36,3],[78,82,36,61] >;
D69 in GAP, Magma, Sage, TeX
D_{69} % in TeX
G:=Group("D69"); // GroupNames label
G:=SmallGroup(138,3);
// by ID
G=gap.SmallGroup(138,3);
# by ID
G:=PCGroup([3,-2,-3,-23,25,1190]);
// Polycyclic
G:=Group<a,b|a^69=b^2=1,b*a*b=a^-1>;
// generators/relations
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