metabelian, supersoluble, monomial, A-group
Aliases: C13⋊D13, C132⋊2C2, SmallGroup(338,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C132 — C13⋊D13 |
Generators and relations for C13⋊D13
G = < a,b,c | a13=b13=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C13⋊D13
►On 169 points
Generators in S169
(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)(144 145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168 169)
(1 159 25 136 65 80 107 152 121 43 95 75 38)(2 160 26 137 53 81 108 153 122 44 96 76 39)(3 161 14 138 54 82 109 154 123 45 97 77 27)(4 162 15 139 55 83 110 155 124 46 98 78 28)(5 163 16 140 56 84 111 156 125 47 99 66 29)(6 164 17 141 57 85 112 144 126 48 100 67 30)(7 165 18 142 58 86 113 145 127 49 101 68 31)(8 166 19 143 59 87 114 146 128 50 102 69 32)(9 167 20 131 60 88 115 147 129 51 103 70 33)(10 168 21 132 61 89 116 148 130 52 104 71 34)(11 169 22 133 62 90 117 149 118 40 92 72 35)(12 157 23 134 63 91 105 150 119 41 93 73 36)(13 158 24 135 64 79 106 151 120 42 94 74 37)
(1 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 39)(14 93)(15 92)(16 104)(17 103)(18 102)(19 101)(20 100)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(40 139)(41 138)(42 137)(43 136)(44 135)(45 134)(46 133)(47 132)(48 131)(49 143)(50 142)(51 141)(52 140)(53 120)(54 119)(55 118)(56 130)(57 129)(58 128)(59 127)(60 126)(61 125)(62 124)(63 123)(64 122)(65 121)(66 168)(67 167)(68 166)(69 165)(70 164)(71 163)(72 162)(73 161)(74 160)(75 159)(76 158)(77 157)(78 169)(79 153)(80 152)(81 151)(82 150)(83 149)(84 148)(85 147)(86 146)(87 145)(88 144)(89 156)(90 155)(91 154)(105 109)(106 108)(110 117)(111 116)(112 115)(113 114)
G:=sub<Sym(169)| (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)(144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169), (1,159,25,136,65,80,107,152,121,43,95,75,38)(2,160,26,137,53,81,108,153,122,44,96,76,39)(3,161,14,138,54,82,109,154,123,45,97,77,27)(4,162,15,139,55,83,110,155,124,46,98,78,28)(5,163,16,140,56,84,111,156,125,47,99,66,29)(6,164,17,141,57,85,112,144,126,48,100,67,30)(7,165,18,142,58,86,113,145,127,49,101,68,31)(8,166,19,143,59,87,114,146,128,50,102,69,32)(9,167,20,131,60,88,115,147,129,51,103,70,33)(10,168,21,132,61,89,116,148,130,52,104,71,34)(11,169,22,133,62,90,117,149,118,40,92,72,35)(12,157,23,134,63,91,105,150,119,41,93,73,36)(13,158,24,135,64,79,106,151,120,42,94,74,37), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,39)(14,93)(15,92)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(40,139)(41,138)(42,137)(43,136)(44,135)(45,134)(46,133)(47,132)(48,131)(49,143)(50,142)(51,141)(52,140)(53,120)(54,119)(55,118)(56,130)(57,129)(58,128)(59,127)(60,126)(61,125)(62,124)(63,123)(64,122)(65,121)(66,168)(67,167)(68,166)(69,165)(70,164)(71,163)(72,162)(73,161)(74,160)(75,159)(76,158)(77,157)(78,169)(79,153)(80,152)(81,151)(82,150)(83,149)(84,148)(85,147)(86,146)(87,145)(88,144)(89,156)(90,155)(91,154)(105,109)(106,108)(110,117)(111,116)(112,115)(113,114)>;
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)(144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169), (1,159,25,136,65,80,107,152,121,43,95,75,38)(2,160,26,137,53,81,108,153,122,44,96,76,39)(3,161,14,138,54,82,109,154,123,45,97,77,27)(4,162,15,139,55,83,110,155,124,46,98,78,28)(5,163,16,140,56,84,111,156,125,47,99,66,29)(6,164,17,141,57,85,112,144,126,48,100,67,30)(7,165,18,142,58,86,113,145,127,49,101,68,31)(8,166,19,143,59,87,114,146,128,50,102,69,32)(9,167,20,131,60,88,115,147,129,51,103,70,33)(10,168,21,132,61,89,116,148,130,52,104,71,34)(11,169,22,133,62,90,117,149,118,40,92,72,35)(12,157,23,134,63,91,105,150,119,41,93,73,36)(13,158,24,135,64,79,106,151,120,42,94,74,37), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,39)(14,93)(15,92)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(40,139)(41,138)(42,137)(43,136)(44,135)(45,134)(46,133)(47,132)(48,131)(49,143)(50,142)(51,141)(52,140)(53,120)(54,119)(55,118)(56,130)(57,129)(58,128)(59,127)(60,126)(61,125)(62,124)(63,123)(64,122)(65,121)(66,168)(67,167)(68,166)(69,165)(70,164)(71,163)(72,162)(73,161)(74,160)(75,159)(76,158)(77,157)(78,169)(79,153)(80,152)(81,151)(82,150)(83,149)(84,148)(85,147)(86,146)(87,145)(88,144)(89,156)(90,155)(91,154)(105,109)(106,108)(110,117)(111,116)(112,115)(113,114) );
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),(144,145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168,169)], [(1,159,25,136,65,80,107,152,121,43,95,75,38),(2,160,26,137,53,81,108,153,122,44,96,76,39),(3,161,14,138,54,82,109,154,123,45,97,77,27),(4,162,15,139,55,83,110,155,124,46,98,78,28),(5,163,16,140,56,84,111,156,125,47,99,66,29),(6,164,17,141,57,85,112,144,126,48,100,67,30),(7,165,18,142,58,86,113,145,127,49,101,68,31),(8,166,19,143,59,87,114,146,128,50,102,69,32),(9,167,20,131,60,88,115,147,129,51,103,70,33),(10,168,21,132,61,89,116,148,130,52,104,71,34),(11,169,22,133,62,90,117,149,118,40,92,72,35),(12,157,23,134,63,91,105,150,119,41,93,73,36),(13,158,24,135,64,79,106,151,120,42,94,74,37)], [(1,38),(2,37),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,39),(14,93),(15,92),(16,104),(17,103),(18,102),(19,101),(20,100),(21,99),(22,98),(23,97),(24,96),(25,95),(26,94),(40,139),(41,138),(42,137),(43,136),(44,135),(45,134),(46,133),(47,132),(48,131),(49,143),(50,142),(51,141),(52,140),(53,120),(54,119),(55,118),(56,130),(57,129),(58,128),(59,127),(60,126),(61,125),(62,124),(63,123),(64,122),(65,121),(66,168),(67,167),(68,166),(69,165),(70,164),(71,163),(72,162),(73,161),(74,160),(75,159),(76,158),(77,157),(78,169),(79,153),(80,152),(81,151),(82,150),(83,149),(84,148),(85,147),(86,146),(87,145),(88,144),(89,156),(90,155),(91,154),(105,109),(106,108),(110,117),(111,116),(112,115),(113,114)]])
86 conjugacy classes
| class | 1 | 2 | 13A | ··· | 13CF |
| order | 1 | 2 | 13 | ··· | 13 |
| size | 1 | 169 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | D13 |
| kernel | C13⋊D13 | C132 | C13 |
| # reps | 1 | 1 | 84 |
Matrix representation of C13⋊D13 ►in GL4(𝔽53) generated by
| 22 | 16 | 0 | 0 |
| 37 | 39 | 0 | 0 |
| 0 | 0 | 1 | 47 |
| 0 | 0 | 7 | 12 |
| 0 | 1 | 0 | 0 |
| 52 | 11 | 0 | 0 |
| 0 | 0 | 42 | 1 |
| 0 | 0 | 43 | 49 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 49 | 52 |
| 0 | 0 | 15 | 4 |
G:=sub<GL(4,GF(53))| [22,37,0,0,16,39,0,0,0,0,1,7,0,0,47,12],[0,52,0,0,1,11,0,0,0,0,42,43,0,0,1,49],[0,1,0,0,1,0,0,0,0,0,49,15,0,0,52,4] >;
C13⋊D13 in GAP, Magma, Sage, TeX
C_{13}\rtimes D_{13} % in TeX
G:=Group("C13:D13"); // GroupNames label
G:=SmallGroup(338,4);
// by ID
G=gap.SmallGroup(338,4);
# by ID
G:=PCGroup([3,-2,-13,-13,145,2810]);
// Polycyclic
G:=Group<a,b,c|a^13=b^13=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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