metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D73, C73⋊C2, sometimes denoted D146 or Dih73 or Dih146, SmallGroup(146,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C73 — D73 |
Generators and relations for D73
G = < a,b | a73=b2=1, bab=a-1 >
Smallest permutation representation of D73
►On 73 points: primitive
Generators in S73
(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)
(1 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)
G:=sub<Sym(73)| (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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)>;
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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38) );
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)], [(1,73),(2,72),(3,71),(4,70),(5,69),(6,68),(7,67),(8,66),(9,65),(10,64),(11,63),(12,62),(13,61),(14,60),(15,59),(16,58),(17,57),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(25,49),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38)]])
D73 is a maximal subgroup of
C73⋊C4 C73⋊C6 D219
D73 is a maximal quotient of Dic73 D219
38 conjugacy classes
| class | 1 | 2 | 73A | ··· | 73AJ |
| order | 1 | 2 | 73 | ··· | 73 |
| size | 1 | 73 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D73 |
| kernel | D73 | C73 | C1 |
| # reps | 1 | 1 | 36 |
Matrix representation of D73 ►in GL2(𝔽293) generated by
| 137 | 292 |
| 59 | 222 |
| 21 | 40 |
| 282 | 272 |
G:=sub<GL(2,GF(293))| [137,59,292,222],[21,282,40,272] >;
D73 in GAP, Magma, Sage, TeX
D_{73} % in TeX
G:=Group("D73"); // GroupNames label
G:=SmallGroup(146,1);
// by ID
G=gap.SmallGroup(146,1);
# by ID
G:=PCGroup([2,-2,-73,577]);
// Polycyclic
G:=Group<a,b|a^73=b^2=1,b*a*b=a^-1>;
// generators/relations
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