metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D71, C71⋊C2, sometimes denoted D142 or Dih71 or Dih142, SmallGroup(142,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C71 — D71 |
Generators and relations for D71
G = < a,b | a71=b2=1, bab=a-1 >
Smallest permutation representation of D71
►On 71 points: primitive
Generators in S71
(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)
(1 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 60)(13 59)(14 58)(15 57)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)
G:=sub<Sym(71)| (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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)>;
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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37) );
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)], [(1,71),(2,70),(3,69),(4,68),(5,67),(6,66),(7,65),(8,64),(9,63),(10,62),(11,61),(12,60),(13,59),(14,58),(15,57),(16,56),(17,55),(18,54),(19,53),(20,52),(21,51),(22,50),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37)]])
D71 is a maximal subgroup of
D213
D71 is a maximal quotient of Dic71 D213
37 conjugacy classes
| class | 1 | 2 | 71A | ··· | 71AI |
| order | 1 | 2 | 71 | ··· | 71 |
| size | 1 | 71 | 2 | ··· | 2 |
37 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D71 |
| kernel | D71 | C71 | C1 |
| # reps | 1 | 1 | 35 |
Matrix representation of D71 ►in GL2(𝔽569) generated by
| 34 | 568 |
| 1 | 0 |
| 34 | 568 |
| 17 | 535 |
G:=sub<GL(2,GF(569))| [34,1,568,0],[34,17,568,535] >;
D71 in GAP, Magma, Sage, TeX
D_{71} % in TeX
G:=Group("D71"); // GroupNames label
G:=SmallGroup(142,1);
// by ID
G=gap.SmallGroup(142,1);
# by ID
G:=PCGroup([2,-2,-71,561]);
// Polycyclic
G:=Group<a,b|a^71=b^2=1,b*a*b=a^-1>;
// generators/relations
Export