metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D81, C81⋊C2, C27.S3, C3.D27, C9.1D9, sometimes denoted D162 or Dih81 or Dih162, SmallGroup(162,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C81 — D81 |
Generators and relations for D81
G = < a,b | a81=b2=1, bab=a-1 >
Smallest permutation representation of D81
►On 81 points
Generators in S81
(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)
(2 81)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 72)(12 71)(13 70)(14 69)(15 68)(16 67)(17 66)(18 65)(19 64)(20 63)(21 62)(22 61)(23 60)(24 59)(25 58)(26 57)(27 56)(28 55)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)
G:=sub<Sym(81)| (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), (2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)>;
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), (2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42) );
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)], [(2,81),(3,80),(4,79),(5,78),(6,77),(7,76),(8,75),(9,74),(10,73),(11,72),(12,71),(13,70),(14,69),(15,68),(16,67),(17,66),(18,65),(19,64),(20,63),(21,62),(22,61),(23,60),(24,59),(25,58),(26,57),(27,56),(28,55),(29,54),(30,53),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42)]])
D81 is a maximal subgroup of
D243 C81⋊C6 C81⋊S3
D81 is a maximal quotient of Dic81 D243 C81⋊S3
42 conjugacy classes
| class | 1 | 2 | 3 | 9A | 9B | 9C | 27A | ··· | 27I | 81A | ··· | 81AA |
| order | 1 | 2 | 3 | 9 | 9 | 9 | 27 | ··· | 27 | 81 | ··· | 81 |
| size | 1 | 81 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + |
| image | C1 | C2 | S3 | D9 | D27 | D81 |
| kernel | D81 | C81 | C27 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 3 | 9 | 27 |
Matrix representation of D81 ►in GL2(𝔽163) generated by
| 16 | 53 |
| 110 | 69 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(163))| [16,110,53,69],[0,1,1,0] >;
D81 in GAP, Magma, Sage, TeX
D_{81} % in TeX
G:=Group("D81"); // GroupNames label
G:=SmallGroup(162,1);
// by ID
G=gap.SmallGroup(162,1);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,101,156,452,237,1803,138,2704]);
// Polycyclic
G:=Group<a,b|a^81=b^2=1,b*a*b=a^-1>;
// generators/relations
Export