metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D149, C149⋊C2, sometimes denoted D298 or Dih149 or Dih298, SmallGroup(298,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C149 — D149 |
Generators and relations for D149
G = < a,b | a149=b2=1, bab=a-1 >
Smallest permutation representation of D149
►On 149 points: primitive
Generators in S149
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149)
(1 149)(2 148)(3 147)(4 146)(5 145)(6 144)(7 143)(8 142)(9 141)(10 140)(11 139)(12 138)(13 137)(14 136)(15 135)(16 134)(17 133)(18 132)(19 131)(20 130)(21 129)(22 128)(23 127)(24 126)(25 125)(26 124)(27 123)(28 122)(29 121)(30 120)(31 119)(32 118)(33 117)(34 116)(35 115)(36 114)(37 113)(38 112)(39 111)(40 110)(41 109)(42 108)(43 107)(44 106)(45 105)(46 104)(47 103)(48 102)(49 101)(50 100)(51 99)(52 98)(53 97)(54 96)(55 95)(56 94)(57 93)(58 92)(59 91)(60 90)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(149)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149), (1,149)(2,148)(3,147)(4,146)(5,145)(6,144)(7,143)(8,142)(9,141)(10,140)(11,139)(12,138)(13,137)(14,136)(15,135)(16,134)(17,133)(18,132)(19,131)(20,130)(21,129)(22,128)(23,127)(24,126)(25,125)(26,124)(27,123)(28,122)(29,121)(30,120)(31,119)(32,118)(33,117)(34,116)(35,115)(36,114)(37,113)(38,112)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,95)(56,94)(57,93)(58,92)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149), (1,149)(2,148)(3,147)(4,146)(5,145)(6,144)(7,143)(8,142)(9,141)(10,140)(11,139)(12,138)(13,137)(14,136)(15,135)(16,134)(17,133)(18,132)(19,131)(20,130)(21,129)(22,128)(23,127)(24,126)(25,125)(26,124)(27,123)(28,122)(29,121)(30,120)(31,119)(32,118)(33,117)(34,116)(35,115)(36,114)(37,113)(38,112)(39,111)(40,110)(41,109)(42,108)(43,107)(44,106)(45,105)(46,104)(47,103)(48,102)(49,101)(50,100)(51,99)(52,98)(53,97)(54,96)(55,95)(56,94)(57,93)(58,92)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149)], [(1,149),(2,148),(3,147),(4,146),(5,145),(6,144),(7,143),(8,142),(9,141),(10,140),(11,139),(12,138),(13,137),(14,136),(15,135),(16,134),(17,133),(18,132),(19,131),(20,130),(21,129),(22,128),(23,127),(24,126),(25,125),(26,124),(27,123),(28,122),(29,121),(30,120),(31,119),(32,118),(33,117),(34,116),(35,115),(36,114),(37,113),(38,112),(39,111),(40,110),(41,109),(42,108),(43,107),(44,106),(45,105),(46,104),(47,103),(48,102),(49,101),(50,100),(51,99),(52,98),(53,97),(54,96),(55,95),(56,94),(57,93),(58,92),(59,91),(60,90),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76)]])
76 conjugacy classes
| class | 1 | 2 | 149A | ··· | 149BV |
| order | 1 | 2 | 149 | ··· | 149 |
| size | 1 | 149 | 2 | ··· | 2 |
76 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D149 |
| kernel | D149 | C149 | C1 |
| # reps | 1 | 1 | 74 |
Matrix representation of D149 ►in GL2(𝔽1193) generated by
| 278 | 1192 |
| 1 | 0 |
| 278 | 1192 |
| 931 | 915 |
G:=sub<GL(2,GF(1193))| [278,1,1192,0],[278,931,1192,915] >;
D149 in GAP, Magma, Sage, TeX
D_{149} % in TeX
G:=Group("D149"); // GroupNames label
G:=SmallGroup(298,1);
// by ID
G=gap.SmallGroup(298,1);
# by ID
G:=PCGroup([2,-2,-149,1185]);
// Polycyclic
G:=Group<a,b|a^149=b^2=1,b*a*b=a^-1>;
// generators/relations
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