metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D83, C83⋊C2, sometimes denoted D166 or Dih83 or Dih166, SmallGroup(166,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C83 — D83 |
Generators and relations for D83
G = < a,b | a83=b2=1, bab=a-1 >
Smallest permutation representation of D83
►On 83 points: primitive
Generators in S83
(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 70 71 72 73 74 75 76 77 78 79 80 81 82 83)
(1 83)(2 82)(3 81)(4 80)(5 79)(6 78)(7 77)(8 76)(9 75)(10 74)(11 73)(12 72)(13 71)(14 70)(15 69)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 60)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)
G:=sub<Sym(83)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83), (1,83)(2,82)(3,81)(4,80)(5,79)(6,78)(7,77)(8,76)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83), (1,83)(2,82)(3,81)(4,80)(5,79)(6,78)(7,77)(8,76)(9,75)(10,74)(11,73)(12,72)(13,71)(14,70)(15,69)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83)], [(1,83),(2,82),(3,81),(4,80),(5,79),(6,78),(7,77),(8,76),(9,75),(10,74),(11,73),(12,72),(13,71),(14,70),(15,69),(16,68),(17,67),(18,66),(19,65),(20,64),(21,63),(22,62),(23,61),(24,60),(25,59),(26,58),(27,57),(28,56),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43)]])
D83 is a maximal subgroup of
D249
D83 is a maximal quotient of Dic83 D249
43 conjugacy classes
| class | 1 | 2 | 83A | ··· | 83AO |
| order | 1 | 2 | 83 | ··· | 83 |
| size | 1 | 83 | 2 | ··· | 2 |
43 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D83 |
| kernel | D83 | C83 | C1 |
| # reps | 1 | 1 | 41 |
Matrix representation of D83 ►in GL2(𝔽167) generated by
| 4 | 166 |
| 1 | 0 |
| 4 | 166 |
| 15 | 163 |
G:=sub<GL(2,GF(167))| [4,1,166,0],[4,15,166,163] >;
D83 in GAP, Magma, Sage, TeX
D_{83} % in TeX
G:=Group("D83"); // GroupNames label
G:=SmallGroup(166,1);
// by ID
G=gap.SmallGroup(166,1);
# by ID
G:=PCGroup([2,-2,-83,657]);
// Polycyclic
G:=Group<a,b|a^83=b^2=1,b*a*b=a^-1>;
// generators/relations
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