direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D86, C2×D43, C86⋊C2, C43⋊C22, sometimes denoted D172 or Dih86 or Dih172, SmallGroup(172,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — D86 |
Generators and relations for D86
G = < a,b | a86=b2=1, bab=a-1 >
Smallest permutation representation of D86
►On 86 points
Generators in S86
(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 84 85 86)
(1 86)(2 85)(3 84)(4 83)(5 82)(6 81)(7 80)(8 79)(9 78)(10 77)(11 76)(12 75)(13 74)(14 73)(15 72)(16 71)(17 70)(18 69)(19 68)(20 67)(21 66)(22 65)(23 64)(24 63)(25 62)(26 61)(27 60)(28 59)(29 58)(30 57)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)
G:=sub<Sym(86)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86), (1,86)(2,85)(3,84)(4,83)(5,82)(6,81)(7,80)(8,79)(9,78)(10,77)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86), (1,86)(2,85)(3,84)(4,83)(5,82)(6,81)(7,80)(8,79)(9,78)(10,77)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86)], [(1,86),(2,85),(3,84),(4,83),(5,82),(6,81),(7,80),(8,79),(9,78),(10,77),(11,76),(12,75),(13,74),(14,73),(15,72),(16,71),(17,70),(18,69),(19,68),(20,67),(21,66),(22,65),(23,64),(24,63),(25,62),(26,61),(27,60),(28,59),(29,58),(30,57),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44)]])
D86 is a maximal subgroup of
D172 C43⋊D4
D86 is a maximal quotient of Dic86 D172 C43⋊D4
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 43A | ··· | 43U | 86A | ··· | 86U |
| order | 1 | 2 | 2 | 2 | 43 | ··· | 43 | 86 | ··· | 86 |
| size | 1 | 1 | 43 | 43 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D43 | D86 |
| kernel | D86 | D43 | C86 | C2 | C1 |
| # reps | 1 | 2 | 1 | 21 | 21 |
Matrix representation of D86 ►in GL2(𝔽173) generated by
| 168 | 113 |
| 5 | 129 |
| 154 | 148 |
| 49 | 19 |
G:=sub<GL(2,GF(173))| [168,5,113,129],[154,49,148,19] >;
D86 in GAP, Magma, Sage, TeX
D_{86} % in TeX
G:=Group("D86"); // GroupNames label
G:=SmallGroup(172,3);
// by ID
G=gap.SmallGroup(172,3);
# by ID
G:=PCGroup([3,-2,-2,-43,1514]);
// Polycyclic
G:=Group<a,b|a^86=b^2=1,b*a*b=a^-1>;
// generators/relations
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