direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D146, C2×D73, C146⋊C2, C73⋊C22, sometimes denoted D292 or Dih146 or Dih292, SmallGroup(292,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C73 — D146 |
Generators and relations for D146
G = < a,b | a146=b2=1, bab=a-1 >
Smallest permutation representation of D146
►On 146 points
Generators in S146
(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)
(1 146)(2 145)(3 144)(4 143)(5 142)(6 141)(7 140)(8 139)(9 138)(10 137)(11 136)(12 135)(13 134)(14 133)(15 132)(16 131)(17 130)(18 129)(19 128)(20 127)(21 126)(22 125)(23 124)(24 123)(25 122)(26 121)(27 120)(28 119)(29 118)(30 117)(31 116)(32 115)(33 114)(34 113)(35 112)(36 111)(37 110)(38 109)(39 108)(40 107)(41 106)(42 105)(43 104)(44 103)(45 102)(46 101)(47 100)(48 99)(49 98)(50 97)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)
G:=sub<Sym(146)| (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), (1,146)(2,145)(3,144)(4,143)(5,142)(6,141)(7,140)(8,139)(9,138)(10,137)(11,136)(12,135)(13,134)(14,133)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)>;
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), (1,146)(2,145)(3,144)(4,143)(5,142)(6,141)(7,140)(8,139)(9,138)(10,137)(11,136)(12,135)(13,134)(14,133)(15,132)(16,131)(17,130)(18,129)(19,128)(20,127)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74) );
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)], [(1,146),(2,145),(3,144),(4,143),(5,142),(6,141),(7,140),(8,139),(9,138),(10,137),(11,136),(12,135),(13,134),(14,133),(15,132),(16,131),(17,130),(18,129),(19,128),(20,127),(21,126),(22,125),(23,124),(24,123),(25,122),(26,121),(27,120),(28,119),(29,118),(30,117),(31,116),(32,115),(33,114),(34,113),(35,112),(36,111),(37,110),(38,109),(39,108),(40,107),(41,106),(42,105),(43,104),(44,103),(45,102),(46,101),(47,100),(48,99),(49,98),(50,97),(51,96),(52,95),(53,94),(54,93),(55,92),(56,91),(57,90),(58,89),(59,88),(60,87),(61,86),(62,85),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 73A | ··· | 73AJ | 146A | ··· | 146AJ |
| order | 1 | 2 | 2 | 2 | 73 | ··· | 73 | 146 | ··· | 146 |
| size | 1 | 1 | 73 | 73 | 2 | ··· | 2 | 2 | ··· | 2 |
76 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D73 | D146 |
| kernel | D146 | D73 | C146 | C2 | C1 |
| # reps | 1 | 2 | 1 | 36 | 36 |
Matrix representation of D146 ►in GL3(𝔽293) generated by
| 292 | 0 | 0 |
| 0 | 138 | 200 |
| 0 | 188 | 92 |
| 1 | 0 | 0 |
| 0 | 207 | 129 |
| 0 | 4 | 86 |
G:=sub<GL(3,GF(293))| [292,0,0,0,138,188,0,200,92],[1,0,0,0,207,4,0,129,86] >;
D146 in GAP, Magma, Sage, TeX
D_{146} % in TeX
G:=Group("D146"); // GroupNames label
G:=SmallGroup(292,4);
// by ID
G=gap.SmallGroup(292,4);
# by ID
G:=PCGroup([3,-2,-2,-73,2594]);
// Polycyclic
G:=Group<a,b|a^146=b^2=1,b*a*b=a^-1>;
// generators/relations
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