direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C12×D17, C68⋊2C6, C204⋊4C2, D34.2C6, C6.14D34, Dic17⋊2C6, C102.14C22, C51⋊8(C2×C4), C17⋊2(C2×C12), C34.2(C2×C6), C2.1(C6×D17), (C6×D17).4C2, (C3×Dic17)⋊5C2, SmallGroup(408,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C12×D17 |
Generators and relations for C12×D17
G = < a,b,c | a12=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C12×D17
►On 204 points
Generators in S204
(1 194 62 105 79 155 32 181 36 131 87 147)(2 195 63 106 80 156 33 182 37 132 88 148)(3 196 64 107 81 157 34 183 38 133 89 149)(4 197 65 108 82 158 18 184 39 134 90 150)(5 198 66 109 83 159 19 185 40 135 91 151)(6 199 67 110 84 160 20 186 41 136 92 152)(7 200 68 111 85 161 21 187 42 120 93 153)(8 201 52 112 69 162 22 171 43 121 94 137)(9 202 53 113 70 163 23 172 44 122 95 138)(10 203 54 114 71 164 24 173 45 123 96 139)(11 204 55 115 72 165 25 174 46 124 97 140)(12 188 56 116 73 166 26 175 47 125 98 141)(13 189 57 117 74 167 27 176 48 126 99 142)(14 190 58 118 75 168 28 177 49 127 100 143)(15 191 59 119 76 169 29 178 50 128 101 144)(16 192 60 103 77 170 30 179 51 129 102 145)(17 193 61 104 78 154 31 180 35 130 86 146)
(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)(137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153)(154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170)(171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187)(188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 28)(19 27)(20 26)(21 25)(22 24)(29 34)(30 33)(31 32)(35 36)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 45)(52 54)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(69 71)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(86 87)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(103 106)(104 105)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(120 124)(121 123)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(137 139)(140 153)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(154 155)(156 170)(157 169)(158 168)(159 167)(160 166)(161 165)(162 164)(171 173)(174 187)(175 186)(176 185)(177 184)(178 183)(179 182)(180 181)(188 199)(189 198)(190 197)(191 196)(192 195)(193 194)(200 204)(201 203)
G:=sub<Sym(204)| (1,194,62,105,79,155,32,181,36,131,87,147)(2,195,63,106,80,156,33,182,37,132,88,148)(3,196,64,107,81,157,34,183,38,133,89,149)(4,197,65,108,82,158,18,184,39,134,90,150)(5,198,66,109,83,159,19,185,40,135,91,151)(6,199,67,110,84,160,20,186,41,136,92,152)(7,200,68,111,85,161,21,187,42,120,93,153)(8,201,52,112,69,162,22,171,43,121,94,137)(9,202,53,113,70,163,23,172,44,122,95,138)(10,203,54,114,71,164,24,173,45,123,96,139)(11,204,55,115,72,165,25,174,46,124,97,140)(12,188,56,116,73,166,26,175,47,125,98,141)(13,189,57,117,74,167,27,176,48,126,99,142)(14,190,58,118,75,168,28,177,49,127,100,143)(15,191,59,119,76,169,29,178,50,128,101,144)(16,192,60,103,77,170,30,179,51,129,102,145)(17,193,61,104,78,154,31,180,35,130,86,146), (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)(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153)(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187)(188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)(35,36)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,45)(52,54)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(69,71)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(86,87)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(103,106)(104,105)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(120,124)(121,123)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(137,139)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(154,155)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(171,173)(174,187)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)(188,199)(189,198)(190,197)(191,196)(192,195)(193,194)(200,204)(201,203)>;
G:=Group( (1,194,62,105,79,155,32,181,36,131,87,147)(2,195,63,106,80,156,33,182,37,132,88,148)(3,196,64,107,81,157,34,183,38,133,89,149)(4,197,65,108,82,158,18,184,39,134,90,150)(5,198,66,109,83,159,19,185,40,135,91,151)(6,199,67,110,84,160,20,186,41,136,92,152)(7,200,68,111,85,161,21,187,42,120,93,153)(8,201,52,112,69,162,22,171,43,121,94,137)(9,202,53,113,70,163,23,172,44,122,95,138)(10,203,54,114,71,164,24,173,45,123,96,139)(11,204,55,115,72,165,25,174,46,124,97,140)(12,188,56,116,73,166,26,175,47,125,98,141)(13,189,57,117,74,167,27,176,48,126,99,142)(14,190,58,118,75,168,28,177,49,127,100,143)(15,191,59,119,76,169,29,178,50,128,101,144)(16,192,60,103,77,170,30,179,51,129,102,145)(17,193,61,104,78,154,31,180,35,130,86,146), (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)(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153)(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187)(188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)(35,36)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,45)(52,54)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(69,71)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(86,87)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(103,106)(104,105)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(120,124)(121,123)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(137,139)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)(154,155)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(171,173)(174,187)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)(188,199)(189,198)(190,197)(191,196)(192,195)(193,194)(200,204)(201,203) );
G=PermutationGroup([[(1,194,62,105,79,155,32,181,36,131,87,147),(2,195,63,106,80,156,33,182,37,132,88,148),(3,196,64,107,81,157,34,183,38,133,89,149),(4,197,65,108,82,158,18,184,39,134,90,150),(5,198,66,109,83,159,19,185,40,135,91,151),(6,199,67,110,84,160,20,186,41,136,92,152),(7,200,68,111,85,161,21,187,42,120,93,153),(8,201,52,112,69,162,22,171,43,121,94,137),(9,202,53,113,70,163,23,172,44,122,95,138),(10,203,54,114,71,164,24,173,45,123,96,139),(11,204,55,115,72,165,25,174,46,124,97,140),(12,188,56,116,73,166,26,175,47,125,98,141),(13,189,57,117,74,167,27,176,48,126,99,142),(14,190,58,118,75,168,28,177,49,127,100,143),(15,191,59,119,76,169,29,178,50,128,101,144),(16,192,60,103,77,170,30,179,51,129,102,145),(17,193,61,104,78,154,31,180,35,130,86,146)], [(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),(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153),(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170),(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187),(188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,28),(19,27),(20,26),(21,25),(22,24),(29,34),(30,33),(31,32),(35,36),(37,51),(38,50),(39,49),(40,48),(41,47),(42,46),(43,45),(52,54),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(61,62),(69,71),(72,85),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(86,87),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(103,106),(104,105),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114),(120,124),(121,123),(125,136),(126,135),(127,134),(128,133),(129,132),(130,131),(137,139),(140,153),(141,152),(142,151),(143,150),(144,149),(145,148),(146,147),(154,155),(156,170),(157,169),(158,168),(159,167),(160,166),(161,165),(162,164),(171,173),(174,187),(175,186),(176,185),(177,184),(178,183),(179,182),(180,181),(188,199),(189,198),(190,197),(191,196),(192,195),(193,194),(200,204),(201,203)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51P | 68A | ··· | 68P | 102A | ··· | 102P | 204A | ··· | 204AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 | 204 | ··· | 204 |
| size | 1 | 1 | 17 | 17 | 1 | 1 | 1 | 1 | 17 | 17 | 1 | 1 | 17 | 17 | 17 | 17 | 1 | 1 | 1 | 1 | 17 | 17 | 17 | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D17 | D34 | C3×D17 | C4×D17 | C6×D17 | C12×D17 |
| kernel | C12×D17 | C3×Dic17 | C204 | C6×D17 | C4×D17 | C3×D17 | Dic17 | C68 | D34 | D17 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 8 | 8 | 16 | 16 | 16 | 32 |
Matrix representation of C12×D17 ►in GL3(𝔽409) generated by
| 143 | 0 | 0 |
| 0 | 356 | 0 |
| 0 | 0 | 356 |
| 1 | 0 | 0 |
| 0 | 304 | 119 |
| 0 | 408 | 266 |
| 408 | 0 | 0 |
| 0 | 134 | 382 |
| 0 | 256 | 275 |
G:=sub<GL(3,GF(409))| [143,0,0,0,356,0,0,0,356],[1,0,0,0,304,408,0,119,266],[408,0,0,0,134,256,0,382,275] >;
C12×D17 in GAP, Magma, Sage, TeX
C_{12}\times D_{17} % in TeX
G:=Group("C12xD17"); // GroupNames label
G:=SmallGroup(408,16);
// by ID
G=gap.SmallGroup(408,16);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-17,66,9604]);
// Polycyclic
G:=Group<a,b,c|a^12=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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