direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D154, C2×D77, C22⋊D7, C14⋊D11, C7⋊2D22, C11⋊2D14, C154⋊1C2, C77⋊2C22, sometimes denoted D308 or Dih154 or Dih308, SmallGroup(308,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C77 — D154 |
Generators and relations for D154
G = < a,b | a154=b2=1, bab=a-1 >
Smallest permutation representation of D154
►On 154 points
Generators in S154
(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)
(1 154)(2 153)(3 152)(4 151)(5 150)(6 149)(7 148)(8 147)(9 146)(10 145)(11 144)(12 143)(13 142)(14 141)(15 140)(16 139)(17 138)(18 137)(19 136)(20 135)(21 134)(22 133)(23 132)(24 131)(25 130)(26 129)(27 128)(28 127)(29 126)(30 125)(31 124)(32 123)(33 122)(34 121)(35 120)(36 119)(37 118)(38 117)(39 116)(40 115)(41 114)(42 113)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 100)(56 99)(57 98)(58 97)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)
G:=sub<Sym(154)| (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), (1,154)(2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,145)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,134)(22,133)(23,132)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)>;
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), (1,154)(2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,145)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,134)(22,133)(23,132)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78) );
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)], [(1,154),(2,153),(3,152),(4,151),(5,150),(6,149),(7,148),(8,147),(9,146),(10,145),(11,144),(12,143),(13,142),(14,141),(15,140),(16,139),(17,138),(18,137),(19,136),(20,135),(21,134),(22,133),(23,132),(24,131),(25,130),(26,129),(27,128),(28,127),(29,126),(30,125),(31,124),(32,123),(33,122),(34,121),(35,120),(36,119),(37,118),(38,117),(39,116),(40,115),(41,114),(42,113),(43,112),(44,111),(45,110),(46,109),(47,108),(48,107),(49,106),(50,105),(51,104),(52,103),(53,102),(54,101),(55,100),(56,99),(57,98),(58,97),(59,96),(60,95),(61,94),(62,93),(63,92),(64,91),(65,90),(66,89),(67,88),(68,87),(69,86),(70,85),(71,84),(72,83),(73,82),(74,81),(75,80),(76,79),(77,78)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | 7B | 7C | 11A | ··· | 11E | 14A | 14B | 14C | 22A | ··· | 22E | 77A | ··· | 77AD | 154A | ··· | 154AD |
| order | 1 | 2 | 2 | 2 | 7 | 7 | 7 | 11 | ··· | 11 | 14 | 14 | 14 | 22 | ··· | 22 | 77 | ··· | 77 | 154 | ··· | 154 |
| size | 1 | 1 | 77 | 77 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D7 | D11 | D14 | D22 | D77 | D154 |
| kernel | D154 | D77 | C154 | C22 | C14 | C11 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 3 | 5 | 3 | 5 | 30 | 30 |
Matrix representation of D154 ►in GL3(𝔽463) generated by
| 462 | 0 | 0 |
| 0 | 231 | 236 |
| 0 | 107 | 424 |
| 1 | 0 | 0 |
| 0 | 273 | 385 |
| 0 | 445 | 190 |
G:=sub<GL(3,GF(463))| [462,0,0,0,231,107,0,236,424],[1,0,0,0,273,445,0,385,190] >;
D154 in GAP, Magma, Sage, TeX
D_{154} % in TeX
G:=Group("D154"); // GroupNames label
G:=SmallGroup(308,8);
// by ID
G=gap.SmallGroup(308,8);
# by ID
G:=PCGroup([4,-2,-2,-7,-11,290,4483]);
// Polycyclic
G:=Group<a,b|a^154=b^2=1,b*a*b=a^-1>;
// generators/relations
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