direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C10×D17, C34⋊C10, C170⋊2C2, C85⋊3C22, C17⋊(C2×C10), SmallGroup(340,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C10×D17 |
Generators and relations for C10×D17
G = < a,b,c | a10=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C10×D17
►On 170 points
Generators in S170
(1 126 71 112 67 92 48 160 21 153)(2 127 72 113 68 93 49 161 22 137)(3 128 73 114 52 94 50 162 23 138)(4 129 74 115 53 95 51 163 24 139)(5 130 75 116 54 96 35 164 25 140)(6 131 76 117 55 97 36 165 26 141)(7 132 77 118 56 98 37 166 27 142)(8 133 78 119 57 99 38 167 28 143)(9 134 79 103 58 100 39 168 29 144)(10 135 80 104 59 101 40 169 30 145)(11 136 81 105 60 102 41 170 31 146)(12 120 82 106 61 86 42 154 32 147)(13 121 83 107 62 87 43 155 33 148)(14 122 84 108 63 88 44 156 34 149)(15 123 85 109 64 89 45 157 18 150)(16 124 69 110 65 90 46 158 19 151)(17 125 70 111 66 91 47 159 20 152)
(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)(154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170)
(1 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 102)(8 101)(9 100)(10 99)(11 98)(12 97)(13 96)(14 95)(15 94)(16 93)(17 92)(18 114)(19 113)(20 112)(21 111)(22 110)(23 109)(24 108)(25 107)(26 106)(27 105)(28 104)(29 103)(30 119)(31 118)(32 117)(33 116)(34 115)(35 121)(36 120)(37 136)(38 135)(39 134)(40 133)(41 132)(42 131)(43 130)(44 129)(45 128)(46 127)(47 126)(48 125)(49 124)(50 123)(51 122)(52 150)(53 149)(54 148)(55 147)(56 146)(57 145)(58 144)(59 143)(60 142)(61 141)(62 140)(63 139)(64 138)(65 137)(66 153)(67 152)(68 151)(69 161)(70 160)(71 159)(72 158)(73 157)(74 156)(75 155)(76 154)(77 170)(78 169)(79 168)(80 167)(81 166)(82 165)(83 164)(84 163)(85 162)
G:=sub<Sym(170)| (1,126,71,112,67,92,48,160,21,153)(2,127,72,113,68,93,49,161,22,137)(3,128,73,114,52,94,50,162,23,138)(4,129,74,115,53,95,51,163,24,139)(5,130,75,116,54,96,35,164,25,140)(6,131,76,117,55,97,36,165,26,141)(7,132,77,118,56,98,37,166,27,142)(8,133,78,119,57,99,38,167,28,143)(9,134,79,103,58,100,39,168,29,144)(10,135,80,104,59,101,40,169,30,145)(11,136,81,105,60,102,41,170,31,146)(12,120,82,106,61,86,42,154,32,147)(13,121,83,107,62,87,43,155,33,148)(14,122,84,108,63,88,44,156,34,149)(15,123,85,109,64,89,45,157,18,150)(16,124,69,110,65,90,46,158,19,151)(17,125,70,111,66,91,47,159,20,152), (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)(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,102)(8,101)(9,100)(10,99)(11,98)(12,97)(13,96)(14,95)(15,94)(16,93)(17,92)(18,114)(19,113)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,103)(30,119)(31,118)(32,117)(33,116)(34,115)(35,121)(36,120)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,150)(53,149)(54,148)(55,147)(56,146)(57,145)(58,144)(59,143)(60,142)(61,141)(62,140)(63,139)(64,138)(65,137)(66,153)(67,152)(68,151)(69,161)(70,160)(71,159)(72,158)(73,157)(74,156)(75,155)(76,154)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162)>;
G:=Group( (1,126,71,112,67,92,48,160,21,153)(2,127,72,113,68,93,49,161,22,137)(3,128,73,114,52,94,50,162,23,138)(4,129,74,115,53,95,51,163,24,139)(5,130,75,116,54,96,35,164,25,140)(6,131,76,117,55,97,36,165,26,141)(7,132,77,118,56,98,37,166,27,142)(8,133,78,119,57,99,38,167,28,143)(9,134,79,103,58,100,39,168,29,144)(10,135,80,104,59,101,40,169,30,145)(11,136,81,105,60,102,41,170,31,146)(12,120,82,106,61,86,42,154,32,147)(13,121,83,107,62,87,43,155,33,148)(14,122,84,108,63,88,44,156,34,149)(15,123,85,109,64,89,45,157,18,150)(16,124,69,110,65,90,46,158,19,151)(17,125,70,111,66,91,47,159,20,152), (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)(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,102)(8,101)(9,100)(10,99)(11,98)(12,97)(13,96)(14,95)(15,94)(16,93)(17,92)(18,114)(19,113)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,103)(30,119)(31,118)(32,117)(33,116)(34,115)(35,121)(36,120)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,150)(53,149)(54,148)(55,147)(56,146)(57,145)(58,144)(59,143)(60,142)(61,141)(62,140)(63,139)(64,138)(65,137)(66,153)(67,152)(68,151)(69,161)(70,160)(71,159)(72,158)(73,157)(74,156)(75,155)(76,154)(77,170)(78,169)(79,168)(80,167)(81,166)(82,165)(83,164)(84,163)(85,162) );
G=PermutationGroup([[(1,126,71,112,67,92,48,160,21,153),(2,127,72,113,68,93,49,161,22,137),(3,128,73,114,52,94,50,162,23,138),(4,129,74,115,53,95,51,163,24,139),(5,130,75,116,54,96,35,164,25,140),(6,131,76,117,55,97,36,165,26,141),(7,132,77,118,56,98,37,166,27,142),(8,133,78,119,57,99,38,167,28,143),(9,134,79,103,58,100,39,168,29,144),(10,135,80,104,59,101,40,169,30,145),(11,136,81,105,60,102,41,170,31,146),(12,120,82,106,61,86,42,154,32,147),(13,121,83,107,62,87,43,155,33,148),(14,122,84,108,63,88,44,156,34,149),(15,123,85,109,64,89,45,157,18,150),(16,124,69,110,65,90,46,158,19,151),(17,125,70,111,66,91,47,159,20,152)], [(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),(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170)], [(1,91),(2,90),(3,89),(4,88),(5,87),(6,86),(7,102),(8,101),(9,100),(10,99),(11,98),(12,97),(13,96),(14,95),(15,94),(16,93),(17,92),(18,114),(19,113),(20,112),(21,111),(22,110),(23,109),(24,108),(25,107),(26,106),(27,105),(28,104),(29,103),(30,119),(31,118),(32,117),(33,116),(34,115),(35,121),(36,120),(37,136),(38,135),(39,134),(40,133),(41,132),(42,131),(43,130),(44,129),(45,128),(46,127),(47,126),(48,125),(49,124),(50,123),(51,122),(52,150),(53,149),(54,148),(55,147),(56,146),(57,145),(58,144),(59,143),(60,142),(61,141),(62,140),(63,139),(64,138),(65,137),(66,153),(67,152),(68,151),(69,161),(70,160),(71,159),(72,158),(73,157),(74,156),(75,155),(76,154),(77,170),(78,169),(79,168),(80,167),(81,166),(82,165),(83,164),(84,163),(85,162)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 17A | ··· | 17H | 34A | ··· | 34H | 85A | ··· | 85AF | 170A | ··· | 170AF |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 17 | ··· | 17 | 34 | ··· | 34 | 85 | ··· | 85 | 170 | ··· | 170 |
| size | 1 | 1 | 17 | 17 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 17 | ··· | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D17 | D34 | C5×D17 | C10×D17 |
| kernel | C10×D17 | C5×D17 | C170 | D34 | D17 | C34 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 8 | 8 | 32 | 32 |
Matrix representation of C10×D17 ►in GL2(𝔽1021) generated by
| 345 | 0 |
| 0 | 345 |
| 0 | 1 |
| 1020 | 560 |
| 0 | 1020 |
| 1020 | 0 |
G:=sub<GL(2,GF(1021))| [345,0,0,345],[0,1020,1,560],[0,1020,1020,0] >;
C10×D17 in GAP, Magma, Sage, TeX
C_{10}\times D_{17} % in TeX
G:=Group("C10xD17"); // GroupNames label
G:=SmallGroup(340,12);
// by ID
G=gap.SmallGroup(340,12);
# by ID
G:=PCGroup([4,-2,-2,-5,-17,5123]);
// Polycyclic
G:=Group<a,b,c|a^10=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export