metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D113, C113⋊C2, sometimes denoted D226 or Dih113 or Dih226, SmallGroup(226,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C113 — D113 |
Generators and relations for D113
G = < a,b | a113=b2=1, bab=a-1 >
Smallest permutation representation of D113
►On 113 points: primitive
Generators in S113
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113)
(1 113)(2 112)(3 111)(4 110)(5 109)(6 108)(7 107)(8 106)(9 105)(10 104)(11 103)(12 102)(13 101)(14 100)(15 99)(16 98)(17 97)(18 96)(19 95)(20 94)(21 93)(22 92)(23 91)(24 90)(25 89)(26 88)(27 87)(28 86)(29 85)(30 84)(31 83)(32 82)(33 81)(34 80)(35 79)(36 78)(37 77)(38 76)(39 75)(40 74)(41 73)(42 72)(43 71)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)
G:=sub<Sym(113)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113), (1,113)(2,112)(3,111)(4,110)(5,109)(6,108)(7,107)(8,106)(9,105)(10,104)(11,103)(12,102)(13,101)(14,100)(15,99)(16,98)(17,97)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,90)(25,89)(26,88)(27,87)(28,86)(29,85)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,73)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113), (1,113)(2,112)(3,111)(4,110)(5,109)(6,108)(7,107)(8,106)(9,105)(10,104)(11,103)(12,102)(13,101)(14,100)(15,99)(16,98)(17,97)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,90)(25,89)(26,88)(27,87)(28,86)(29,85)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,73)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113)], [(1,113),(2,112),(3,111),(4,110),(5,109),(6,108),(7,107),(8,106),(9,105),(10,104),(11,103),(12,102),(13,101),(14,100),(15,99),(16,98),(17,97),(18,96),(19,95),(20,94),(21,93),(22,92),(23,91),(24,90),(25,89),(26,88),(27,87),(28,86),(29,85),(30,84),(31,83),(32,82),(33,81),(34,80),(35,79),(36,78),(37,77),(38,76),(39,75),(40,74),(41,73),(42,72),(43,71),(44,70),(45,69),(46,68),(47,67),(48,66),(49,65),(50,64),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58)]])
D113 is a maximal subgroup of
C113⋊C4
D113 is a maximal quotient of Dic113
58 conjugacy classes
| class | 1 | 2 | 113A | ··· | 113BD |
| order | 1 | 2 | 113 | ··· | 113 |
| size | 1 | 113 | 2 | ··· | 2 |
58 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D113 |
| kernel | D113 | C113 | C1 |
| # reps | 1 | 1 | 56 |
Matrix representation of D113 ►in GL2(𝔽227) generated by
| 156 | 226 |
| 136 | 117 |
| 63 | 155 |
| 156 | 164 |
G:=sub<GL(2,GF(227))| [156,136,226,117],[63,156,155,164] >;
D113 in GAP, Magma, Sage, TeX
D_{113} % in TeX
G:=Group("D113"); // GroupNames label
G:=SmallGroup(226,1);
// by ID
G=gap.SmallGroup(226,1);
# by ID
G:=PCGroup([2,-2,-113,897]);
// Polycyclic
G:=Group<a,b|a^113=b^2=1,b*a*b=a^-1>;
// generators/relations
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