metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D151, C151⋊C2, sometimes denoted D302 or Dih151 or Dih302, SmallGroup(302,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C151 — D151 |
Generators and relations for D151
G = < a,b | a151=b2=1, bab=a-1 >
Smallest permutation representation of D151
►On 151 points: primitive
Generators in S151
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151)
(1 151)(2 150)(3 149)(4 148)(5 147)(6 146)(7 145)(8 144)(9 143)(10 142)(11 141)(12 140)(13 139)(14 138)(15 137)(16 136)(17 135)(18 134)(19 133)(20 132)(21 131)(22 130)(23 129)(24 128)(25 127)(26 126)(27 125)(28 124)(29 123)(30 122)(31 121)(32 120)(33 119)(34 118)(35 117)(36 116)(37 115)(38 114)(39 113)(40 112)(41 111)(42 110)(43 109)(44 108)(45 107)(46 106)(47 105)(48 104)(49 103)(50 102)(51 101)(52 100)(53 99)(54 98)(55 97)(56 96)(57 95)(58 94)(59 93)(60 92)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)
G:=sub<Sym(151)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151), (1,151)(2,150)(3,149)(4,148)(5,147)(6,146)(7,145)(8,144)(9,143)(10,142)(11,141)(12,140)(13,139)(14,138)(15,137)(16,136)(17,135)(18,134)(19,133)(20,132)(21,131)(22,130)(23,129)(24,128)(25,127)(26,126)(27,125)(28,124)(29,123)(30,122)(31,121)(32,120)(33,119)(34,118)(35,117)(36,116)(37,115)(38,114)(39,113)(40,112)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,100)(53,99)(54,98)(55,97)(56,96)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151), (1,151)(2,150)(3,149)(4,148)(5,147)(6,146)(7,145)(8,144)(9,143)(10,142)(11,141)(12,140)(13,139)(14,138)(15,137)(16,136)(17,135)(18,134)(19,133)(20,132)(21,131)(22,130)(23,129)(24,128)(25,127)(26,126)(27,125)(28,124)(29,123)(30,122)(31,121)(32,120)(33,119)(34,118)(35,117)(36,116)(37,115)(38,114)(39,113)(40,112)(41,111)(42,110)(43,109)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,100)(53,99)(54,98)(55,97)(56,96)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151)], [(1,151),(2,150),(3,149),(4,148),(5,147),(6,146),(7,145),(8,144),(9,143),(10,142),(11,141),(12,140),(13,139),(14,138),(15,137),(16,136),(17,135),(18,134),(19,133),(20,132),(21,131),(22,130),(23,129),(24,128),(25,127),(26,126),(27,125),(28,124),(29,123),(30,122),(31,121),(32,120),(33,119),(34,118),(35,117),(36,116),(37,115),(38,114),(39,113),(40,112),(41,111),(42,110),(43,109),(44,108),(45,107),(46,106),(47,105),(48,104),(49,103),(50,102),(51,101),(52,100),(53,99),(54,98),(55,97),(56,96),(57,95),(58,94),(59,93),(60,92),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77)]])
77 conjugacy classes
| class | 1 | 2 | 151A | ··· | 151BW |
| order | 1 | 2 | 151 | ··· | 151 |
| size | 1 | 151 | 2 | ··· | 2 |
77 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D151 |
| kernel | D151 | C151 | C1 |
| # reps | 1 | 1 | 75 |
Matrix representation of D151 ►in GL2(𝔽907) generated by
| 431 | 906 |
| 1 | 0 |
| 431 | 906 |
| 732 | 476 |
G:=sub<GL(2,GF(907))| [431,1,906,0],[431,732,906,476] >;
D151 in GAP, Magma, Sage, TeX
D_{151} % in TeX
G:=Group("D151"); // GroupNames label
G:=SmallGroup(302,1);
// by ID
G=gap.SmallGroup(302,1);
# by ID
G:=PCGroup([2,-2,-151,1201]);
// Polycyclic
G:=Group<a,b|a^151=b^2=1,b*a*b=a^-1>;
// generators/relations
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