metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D103, C103⋊C2, sometimes denoted D206 or Dih103 or Dih206, SmallGroup(206,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C103 — D103 |
Generators and relations for D103
G = < a,b | a103=b2=1, bab=a-1 >
Smallest permutation representation of D103
►On 103 points: primitive
Generators in S103
(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 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103)
(1 103)(2 102)(3 101)(4 100)(5 99)(6 98)(7 97)(8 96)(9 95)(10 94)(11 93)(12 92)(13 91)(14 90)(15 89)(16 88)(17 87)(18 86)(19 85)(20 84)(21 83)(22 82)(23 81)(24 80)(25 79)(26 78)(27 77)(28 76)(29 75)(30 74)(31 73)(32 72)(33 71)(34 70)(35 69)(36 68)(37 67)(38 66)(39 65)(40 64)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)
G:=sub<Sym(103)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103), (1,103)(2,102)(3,101)(4,100)(5,99)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,89)(16,88)(17,87)(18,86)(19,85)(20,84)(21,83)(22,82)(23,81)(24,80)(25,79)(26,78)(27,77)(28,76)(29,75)(30,74)(31,73)(32,72)(33,71)(34,70)(35,69)(36,68)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103), (1,103)(2,102)(3,101)(4,100)(5,99)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,89)(16,88)(17,87)(18,86)(19,85)(20,84)(21,83)(22,82)(23,81)(24,80)(25,79)(26,78)(27,77)(28,76)(29,75)(30,74)(31,73)(32,72)(33,71)(34,70)(35,69)(36,68)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103)], [(1,103),(2,102),(3,101),(4,100),(5,99),(6,98),(7,97),(8,96),(9,95),(10,94),(11,93),(12,92),(13,91),(14,90),(15,89),(16,88),(17,87),(18,86),(19,85),(20,84),(21,83),(22,82),(23,81),(24,80),(25,79),(26,78),(27,77),(28,76),(29,75),(30,74),(31,73),(32,72),(33,71),(34,70),(35,69),(36,68),(37,67),(38,66),(39,65),(40,64),(41,63),(42,62),(43,61),(44,60),(45,59),(46,58),(47,57),(48,56),(49,55),(50,54),(51,53)]])
D103 is a maximal quotient of Dic103
53 conjugacy classes
| class | 1 | 2 | 103A | ··· | 103AY |
| order | 1 | 2 | 103 | ··· | 103 |
| size | 1 | 103 | 2 | ··· | 2 |
53 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D103 |
| kernel | D103 | C103 | C1 |
| # reps | 1 | 1 | 51 |
Matrix representation of D103 ►in GL2(𝔽619) generated by
| 459 | 618 |
| 60 | 306 |
| 144 | 350 |
| 321 | 475 |
G:=sub<GL(2,GF(619))| [459,60,618,306],[144,321,350,475] >;
D103 in GAP, Magma, Sage, TeX
D_{103} % in TeX
G:=Group("D103"); // GroupNames label
G:=SmallGroup(206,1);
// by ID
G=gap.SmallGroup(206,1);
# by ID
G:=PCGroup([2,-2,-103,817]);
// Polycyclic
G:=Group<a,b|a^103=b^2=1,b*a*b=a^-1>;
// generators/relations
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