direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D150, C2×D75, C50⋊S3, C6⋊D25, C3⋊2D50, C25⋊2D6, C5.D30, C150⋊1C2, C75⋊2C22, C30.2D5, C10.2D15, C15.2D10, sometimes denoted D300 or Dih150 or Dih300, SmallGroup(300,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C75 — D150 |
Generators and relations for D150
G = < a,b | a150=b2=1, bab=a-1 >
Smallest permutation representation of D150
►On 150 points
Generators in S150
(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)
(1 150)(2 149)(3 148)(4 147)(5 146)(6 145)(7 144)(8 143)(9 142)(10 141)(11 140)(12 139)(13 138)(14 137)(15 136)(16 135)(17 134)(18 133)(19 132)(20 131)(21 130)(22 129)(23 128)(24 127)(25 126)(26 125)(27 124)(28 123)(29 122)(30 121)(31 120)(32 119)(33 118)(34 117)(35 116)(36 115)(37 114)(38 113)(39 112)(40 111)(41 110)(42 109)(43 108)(44 107)(45 106)(46 105)(47 104)(48 103)(49 102)(50 101)(51 100)(52 99)(53 98)(54 97)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)
G:=sub<Sym(150)| (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), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)], [(1,150),(2,149),(3,148),(4,147),(5,146),(6,145),(7,144),(8,143),(9,142),(10,141),(11,140),(12,139),(13,138),(14,137),(15,136),(16,135),(17,134),(18,133),(19,132),(20,131),(21,130),(22,129),(23,128),(24,127),(25,126),(26,125),(27,124),(28,123),(29,122),(30,121),(31,120),(32,119),(33,118),(34,117),(35,116),(36,115),(37,114),(38,113),(39,112),(40,111),(41,110),(42,109),(43,108),(44,107),(45,106),(46,105),(47,104),(48,103),(49,102),(50,101),(51,100),(52,99),(53,98),(54,97),(55,96),(56,95),(57,94),(58,93),(59,92),(60,91),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 6 | 10A | 10B | 15A | 15B | 15C | 15D | 25A | ··· | 25J | 30A | 30B | 30C | 30D | 50A | ··· | 50J | 75A | ··· | 75T | 150A | ··· | 150T |
| order | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 6 | 10 | 10 | 15 | 15 | 15 | 15 | 25 | ··· | 25 | 30 | 30 | 30 | 30 | 50 | ··· | 50 | 75 | ··· | 75 | 150 | ··· | 150 |
| size | 1 | 1 | 75 | 75 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D5 | D6 | D10 | D15 | D25 | D30 | D50 | D75 | D150 |
| kernel | D150 | D75 | C150 | C50 | C30 | C25 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 10 | 4 | 10 | 20 | 20 |
Matrix representation of D150 ►in GL3(𝔽151) generated by
| 150 | 0 | 0 |
| 0 | 32 | 39 |
| 0 | 112 | 28 |
| 1 | 0 | 0 |
| 0 | 32 | 39 |
| 0 | 148 | 119 |
G:=sub<GL(3,GF(151))| [150,0,0,0,32,112,0,39,28],[1,0,0,0,32,148,0,39,119] >;
D150 in GAP, Magma, Sage, TeX
D_{150} % in TeX
G:=Group("D150"); // GroupNames label
G:=SmallGroup(300,11);
// by ID
G=gap.SmallGroup(300,11);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,122,2163,418,6004]);
// Polycyclic
G:=Group<a,b|a^150=b^2=1,b*a*b=a^-1>;
// generators/relations
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