metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D155, C5⋊D31, C31⋊D5, C155⋊1C2, sometimes denoted D310 or Dih155 or Dih310, SmallGroup(310,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C155 — D155 |
Generators and relations for D155
G = < a,b | a155=b2=1, bab=a-1 >
Smallest permutation representation of D155
►On 155 points
Generators in S155
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155)
(1 155)(2 154)(3 153)(4 152)(5 151)(6 150)(7 149)(8 148)(9 147)(10 146)(11 145)(12 144)(13 143)(14 142)(15 141)(16 140)(17 139)(18 138)(19 137)(20 136)(21 135)(22 134)(23 133)(24 132)(25 131)(26 130)(27 129)(28 128)(29 127)(30 126)(31 125)(32 124)(33 123)(34 122)(35 121)(36 120)(37 119)(38 118)(39 117)(40 116)(41 115)(42 114)(43 113)(44 112)(45 111)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 100)(57 99)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)
G:=sub<Sym(155)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155), (1,155)(2,154)(3,153)(4,152)(5,151)(6,150)(7,149)(8,148)(9,147)(10,146)(11,145)(12,144)(13,143)(14,142)(15,141)(16,140)(17,139)(18,138)(19,137)(20,136)(21,135)(22,134)(23,133)(24,132)(25,131)(26,130)(27,129)(28,128)(29,127)(30,126)(31,125)(32,124)(33,123)(34,122)(35,121)(36,120)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155), (1,155)(2,154)(3,153)(4,152)(5,151)(6,150)(7,149)(8,148)(9,147)(10,146)(11,145)(12,144)(13,143)(14,142)(15,141)(16,140)(17,139)(18,138)(19,137)(20,136)(21,135)(22,134)(23,133)(24,132)(25,131)(26,130)(27,129)(28,128)(29,127)(30,126)(31,125)(32,124)(33,123)(34,122)(35,121)(36,120)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155)], [(1,155),(2,154),(3,153),(4,152),(5,151),(6,150),(7,149),(8,148),(9,147),(10,146),(11,145),(12,144),(13,143),(14,142),(15,141),(16,140),(17,139),(18,138),(19,137),(20,136),(21,135),(22,134),(23,133),(24,132),(25,131),(26,130),(27,129),(28,128),(29,127),(30,126),(31,125),(32,124),(33,123),(34,122),(35,121),(36,120),(37,119),(38,118),(39,117),(40,116),(41,115),(42,114),(43,113),(44,112),(45,111),(46,110),(47,109),(48,108),(49,107),(50,106),(51,105),(52,104),(53,103),(54,102),(55,101),(56,100),(57,99),(58,98),(59,97),(60,96),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79)]])
79 conjugacy classes
| class | 1 | 2 | 5A | 5B | 31A | ··· | 31O | 155A | ··· | 155BH |
| order | 1 | 2 | 5 | 5 | 31 | ··· | 31 | 155 | ··· | 155 |
| size | 1 | 155 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
79 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | D5 | D31 | D155 |
| kernel | D155 | C155 | C31 | C5 | C1 |
| # reps | 1 | 1 | 2 | 15 | 60 |
Matrix representation of D155 ►in GL2(𝔽311) generated by
| 143 | 33 |
| 104 | 111 |
| 77 | 128 |
| 12 | 234 |
G:=sub<GL(2,GF(311))| [143,104,33,111],[77,12,128,234] >;
D155 in GAP, Magma, Sage, TeX
D_{155} % in TeX
G:=Group("D155"); // GroupNames label
G:=SmallGroup(310,5);
// by ID
G=gap.SmallGroup(310,5);
# by ID
G:=PCGroup([3,-2,-5,-31,49,2702]);
// Polycyclic
G:=Group<a,b|a^155=b^2=1,b*a*b=a^-1>;
// generators/relations
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