metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D152, C19⋊1D8, C8⋊1D19, C152⋊1C2, D76⋊1C2, C2.4D76, C4.9D38, C38.2D4, C76.9C22, sometimes denoted D304 or Dih152 or Dih304, SmallGroup(304,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D152
G = < a,b | a152=b2=1, bab=a-1 >
Smallest permutation representation of D152
►On 152 points
Generators in S152
(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)
(1 152)(2 151)(3 150)(4 149)(5 148)(6 147)(7 146)(8 145)(9 144)(10 143)(11 142)(12 141)(13 140)(14 139)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 132)(22 131)(23 130)(24 129)(25 128)(26 127)(27 126)(28 125)(29 124)(30 123)(31 122)(32 121)(33 120)(34 119)(35 118)(36 117)(37 116)(38 115)(39 114)(40 113)(41 112)(42 111)(43 110)(44 109)(45 108)(46 107)(47 106)(48 105)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 96)(58 95)(59 94)(60 93)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)
G:=sub<Sym(152)| (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), (1,152)(2,151)(3,150)(4,149)(5,148)(6,147)(7,146)(8,145)(9,144)(10,143)(11,142)(12,141)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,132)(22,131)(23,130)(24,129)(25,128)(26,127)(27,126)(28,125)(29,124)(30,123)(31,122)(32,121)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,152)(2,151)(3,150)(4,149)(5,148)(6,147)(7,146)(8,145)(9,144)(10,143)(11,142)(12,141)(13,140)(14,139)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,132)(22,131)(23,130)(24,129)(25,128)(26,127)(27,126)(28,125)(29,124)(30,123)(31,122)(32,121)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)], [(1,152),(2,151),(3,150),(4,149),(5,148),(6,147),(7,146),(8,145),(9,144),(10,143),(11,142),(12,141),(13,140),(14,139),(15,138),(16,137),(17,136),(18,135),(19,134),(20,133),(21,132),(22,131),(23,130),(24,129),(25,128),(26,127),(27,126),(28,125),(29,124),(30,123),(31,122),(32,121),(33,120),(34,119),(35,118),(36,117),(37,116),(38,115),(39,114),(40,113),(41,112),(42,111),(43,110),(44,109),(45,108),(46,107),(47,106),(48,105),(49,104),(50,103),(51,102),(52,101),(53,100),(54,99),(55,98),(56,97),(57,96),(58,95),(59,94),(60,93),(61,92),(62,91),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76R | 152A | ··· | 152AJ |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 76 | 76 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
79 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D8 | D19 | D38 | D76 | D152 |
| kernel | D152 | C152 | D76 | C38 | C19 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 9 | 9 | 18 | 36 |
Matrix representation of D152 ►in GL2(𝔽457) generated by
| 94 | 270 |
| 187 | 85 |
| 94 | 270 |
| 145 | 363 |
G:=sub<GL(2,GF(457))| [94,187,270,85],[94,145,270,363] >;
D152 in GAP, Magma, Sage, TeX
D_{152} % in TeX
G:=Group("D152"); // GroupNames label
G:=SmallGroup(304,6);
// by ID
G=gap.SmallGroup(304,6);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,61,66,182,42,7204]);
// Polycyclic
G:=Group<a,b|a^152=b^2=1,b*a*b=a^-1>;
// generators/relations
Export