direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D142, C2×D71, C142⋊C2, C71⋊C22, sometimes denoted D284 or Dih142 or Dih284, SmallGroup(284,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C71 — D142 |
Generators and relations for D142
G = < a,b | a142=b2=1, bab=a-1 >
Smallest permutation representation of D142
►On 142 points
Generators in S142
(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)
(1 142)(2 141)(3 140)(4 139)(5 138)(6 137)(7 136)(8 135)(9 134)(10 133)(11 132)(12 131)(13 130)(14 129)(15 128)(16 127)(17 126)(18 125)(19 124)(20 123)(21 122)(22 121)(23 120)(24 119)(25 118)(26 117)(27 116)(28 115)(29 114)(30 113)(31 112)(32 111)(33 110)(34 109)(35 108)(36 107)(37 106)(38 105)(39 104)(40 103)(41 102)(42 101)(43 100)(44 99)(45 98)(46 97)(47 96)(48 95)(49 94)(50 93)(51 92)(52 91)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)
G:=sub<Sym(142)| (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), (1,142)(2,141)(3,140)(4,139)(5,138)(6,137)(7,136)(8,135)(9,134)(10,133)(11,132)(12,131)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)>;
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), (1,142)(2,141)(3,140)(4,139)(5,138)(6,137)(7,136)(8,135)(9,134)(10,133)(11,132)(12,131)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72) );
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)], [(1,142),(2,141),(3,140),(4,139),(5,138),(6,137),(7,136),(8,135),(9,134),(10,133),(11,132),(12,131),(13,130),(14,129),(15,128),(16,127),(17,126),(18,125),(19,124),(20,123),(21,122),(22,121),(23,120),(24,119),(25,118),(26,117),(27,116),(28,115),(29,114),(30,113),(31,112),(32,111),(33,110),(34,109),(35,108),(36,107),(37,106),(38,105),(39,104),(40,103),(41,102),(42,101),(43,100),(44,99),(45,98),(46,97),(47,96),(48,95),(49,94),(50,93),(51,92),(52,91),(53,90),(54,89),(55,88),(56,87),(57,86),(58,85),(59,84),(60,83),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 71A | ··· | 71AI | 142A | ··· | 142AI |
| order | 1 | 2 | 2 | 2 | 71 | ··· | 71 | 142 | ··· | 142 |
| size | 1 | 1 | 71 | 71 | 2 | ··· | 2 | 2 | ··· | 2 |
74 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D71 | D142 |
| kernel | D142 | D71 | C142 | C2 | C1 |
| # reps | 1 | 2 | 1 | 35 | 35 |
Matrix representation of D142 ►in GL3(𝔽569) generated by
| 568 | 0 | 0 |
| 0 | 236 | 176 |
| 0 | 236 | 0 |
| 1 | 0 | 0 |
| 0 | 116 | 412 |
| 0 | 430 | 453 |
G:=sub<GL(3,GF(569))| [568,0,0,0,236,236,0,176,0],[1,0,0,0,116,430,0,412,453] >;
D142 in GAP, Magma, Sage, TeX
D_{142} % in TeX
G:=Group("D142"); // GroupNames label
G:=SmallGroup(284,3);
// by ID
G=gap.SmallGroup(284,3);
# by ID
G:=PCGroup([3,-2,-2,-71,2522]);
// Polycyclic
G:=Group<a,b|a^142=b^2=1,b*a*b=a^-1>;
// generators/relations
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