metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D63, C9⋊D7, C7⋊D9, C63⋊1C2, C3.D21, C21.1S3, sometimes denoted D126 or Dih63 or Dih126, SmallGroup(126,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C63 — D63 |
Generators and relations for D63
G = < a,b | a63=b2=1, bab=a-1 >
Smallest permutation representation of D63
►On 63 points
Generators in S63
(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)
(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)
G:=sub<Sym(63)| (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), (2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)>;
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), (2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33) );
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)], [(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,54),(12,53),(13,52),(14,51),(15,50),(16,49),(17,48),(18,47),(19,46),(20,45),(21,44),(22,43),(23,42),(24,41),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33)]])
D63 is a maximal subgroup of
D7×D9 D189 C9⋊F7 C9⋊2F7 C9⋊5F7 D63⋊C3 C3⋊D63
D63 is a maximal quotient of Dic63 D189 C3⋊D63
33 conjugacy classes
| class | 1 | 2 | 3 | 7A | 7B | 7C | 9A | 9B | 9C | 21A | ··· | 21F | 63A | ··· | 63R |
| order | 1 | 2 | 3 | 7 | 7 | 7 | 9 | 9 | 9 | 21 | ··· | 21 | 63 | ··· | 63 |
| size | 1 | 63 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
33 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | D7 | D9 | D21 | D63 |
| kernel | D63 | C63 | C21 | C9 | C7 | C3 | C1 |
| # reps | 1 | 1 | 1 | 3 | 3 | 6 | 18 |
Matrix representation of D63 ►in GL2(𝔽127) generated by
| 66 | 27 |
| 100 | 39 |
| 63 | 23 |
| 87 | 64 |
G:=sub<GL(2,GF(127))| [66,100,27,39],[63,87,23,64] >;
D63 in GAP, Magma, Sage, TeX
D_{63} % in TeX
G:=Group("D63"); // GroupNames label
G:=SmallGroup(126,5);
// by ID
G=gap.SmallGroup(126,5);
# by ID
G:=PCGroup([4,-2,-3,-7,-3,369,341,434,1347]);
// Polycyclic
G:=Group<a,b|a^63=b^2=1,b*a*b=a^-1>;
// generators/relations
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