direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C14×D17, C34⋊C14, C238⋊2C2, C119⋊3C22, C17⋊(C2×C14), SmallGroup(476,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C14×D17 |
Generators and relations for C14×D17
G = < a,b,c | a14=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C14×D17
►On 238 points
Generators in S238
(1 175 119 157 100 137 83 125 63 224 47 221 19 198)(2 176 103 158 101 138 84 126 64 225 48 205 20 199)(3 177 104 159 102 139 85 127 65 226 49 206 21 200)(4 178 105 160 86 140 69 128 66 227 50 207 22 201)(5 179 106 161 87 141 70 129 67 228 51 208 23 202)(6 180 107 162 88 142 71 130 68 229 35 209 24 203)(7 181 108 163 89 143 72 131 52 230 36 210 25 204)(8 182 109 164 90 144 73 132 53 231 37 211 26 188)(9 183 110 165 91 145 74 133 54 232 38 212 27 189)(10 184 111 166 92 146 75 134 55 233 39 213 28 190)(11 185 112 167 93 147 76 135 56 234 40 214 29 191)(12 186 113 168 94 148 77 136 57 235 41 215 30 192)(13 187 114 169 95 149 78 120 58 236 42 216 31 193)(14 171 115 170 96 150 79 121 59 237 43 217 32 194)(15 172 116 154 97 151 80 122 60 238 44 218 33 195)(16 173 117 155 98 152 81 123 61 222 45 219 34 196)(17 174 118 156 99 153 82 124 62 223 46 220 18 197)
(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)(205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221)(222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238)
(1 124)(2 123)(3 122)(4 121)(5 120)(6 136)(7 135)(8 134)(9 133)(10 132)(11 131)(12 130)(13 129)(14 128)(15 127)(16 126)(17 125)(18 137)(19 153)(20 152)(21 151)(22 150)(23 149)(24 148)(25 147)(26 146)(27 145)(28 144)(29 143)(30 142)(31 141)(32 140)(33 139)(34 138)(35 168)(36 167)(37 166)(38 165)(39 164)(40 163)(41 162)(42 161)(43 160)(44 159)(45 158)(46 157)(47 156)(48 155)(49 154)(50 170)(51 169)(52 185)(53 184)(54 183)(55 182)(56 181)(57 180)(58 179)(59 178)(60 177)(61 176)(62 175)(63 174)(64 173)(65 172)(66 171)(67 187)(68 186)(69 194)(70 193)(71 192)(72 191)(73 190)(74 189)(75 188)(76 204)(77 203)(78 202)(79 201)(80 200)(81 199)(82 198)(83 197)(84 196)(85 195)(86 217)(87 216)(88 215)(89 214)(90 213)(91 212)(92 211)(93 210)(94 209)(95 208)(96 207)(97 206)(98 205)(99 221)(100 220)(101 219)(102 218)(103 222)(104 238)(105 237)(106 236)(107 235)(108 234)(109 233)(110 232)(111 231)(112 230)(113 229)(114 228)(115 227)(116 226)(117 225)(118 224)(119 223)
G:=sub<Sym(238)| (1,175,119,157,100,137,83,125,63,224,47,221,19,198)(2,176,103,158,101,138,84,126,64,225,48,205,20,199)(3,177,104,159,102,139,85,127,65,226,49,206,21,200)(4,178,105,160,86,140,69,128,66,227,50,207,22,201)(5,179,106,161,87,141,70,129,67,228,51,208,23,202)(6,180,107,162,88,142,71,130,68,229,35,209,24,203)(7,181,108,163,89,143,72,131,52,230,36,210,25,204)(8,182,109,164,90,144,73,132,53,231,37,211,26,188)(9,183,110,165,91,145,74,133,54,232,38,212,27,189)(10,184,111,166,92,146,75,134,55,233,39,213,28,190)(11,185,112,167,93,147,76,135,56,234,40,214,29,191)(12,186,113,168,94,148,77,136,57,235,41,215,30,192)(13,187,114,169,95,149,78,120,58,236,42,216,31,193)(14,171,115,170,96,150,79,121,59,237,43,217,32,194)(15,172,116,154,97,151,80,122,60,238,44,218,33,195)(16,173,117,155,98,152,81,123,61,222,45,219,34,196)(17,174,118,156,99,153,82,124,62,223,46,220,18,197), (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)(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221)(222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238), (1,124)(2,123)(3,122)(4,121)(5,120)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,129)(14,128)(15,127)(16,126)(17,125)(18,137)(19,153)(20,152)(21,151)(22,150)(23,149)(24,148)(25,147)(26,146)(27,145)(28,144)(29,143)(30,142)(31,141)(32,140)(33,139)(34,138)(35,168)(36,167)(37,166)(38,165)(39,164)(40,163)(41,162)(42,161)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(49,154)(50,170)(51,169)(52,185)(53,184)(54,183)(55,182)(56,181)(57,180)(58,179)(59,178)(60,177)(61,176)(62,175)(63,174)(64,173)(65,172)(66,171)(67,187)(68,186)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,204)(77,203)(78,202)(79,201)(80,200)(81,199)(82,198)(83,197)(84,196)(85,195)(86,217)(87,216)(88,215)(89,214)(90,213)(91,212)(92,211)(93,210)(94,209)(95,208)(96,207)(97,206)(98,205)(99,221)(100,220)(101,219)(102,218)(103,222)(104,238)(105,237)(106,236)(107,235)(108,234)(109,233)(110,232)(111,231)(112,230)(113,229)(114,228)(115,227)(116,226)(117,225)(118,224)(119,223)>;
G:=Group( (1,175,119,157,100,137,83,125,63,224,47,221,19,198)(2,176,103,158,101,138,84,126,64,225,48,205,20,199)(3,177,104,159,102,139,85,127,65,226,49,206,21,200)(4,178,105,160,86,140,69,128,66,227,50,207,22,201)(5,179,106,161,87,141,70,129,67,228,51,208,23,202)(6,180,107,162,88,142,71,130,68,229,35,209,24,203)(7,181,108,163,89,143,72,131,52,230,36,210,25,204)(8,182,109,164,90,144,73,132,53,231,37,211,26,188)(9,183,110,165,91,145,74,133,54,232,38,212,27,189)(10,184,111,166,92,146,75,134,55,233,39,213,28,190)(11,185,112,167,93,147,76,135,56,234,40,214,29,191)(12,186,113,168,94,148,77,136,57,235,41,215,30,192)(13,187,114,169,95,149,78,120,58,236,42,216,31,193)(14,171,115,170,96,150,79,121,59,237,43,217,32,194)(15,172,116,154,97,151,80,122,60,238,44,218,33,195)(16,173,117,155,98,152,81,123,61,222,45,219,34,196)(17,174,118,156,99,153,82,124,62,223,46,220,18,197), (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)(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221)(222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238), (1,124)(2,123)(3,122)(4,121)(5,120)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,129)(14,128)(15,127)(16,126)(17,125)(18,137)(19,153)(20,152)(21,151)(22,150)(23,149)(24,148)(25,147)(26,146)(27,145)(28,144)(29,143)(30,142)(31,141)(32,140)(33,139)(34,138)(35,168)(36,167)(37,166)(38,165)(39,164)(40,163)(41,162)(42,161)(43,160)(44,159)(45,158)(46,157)(47,156)(48,155)(49,154)(50,170)(51,169)(52,185)(53,184)(54,183)(55,182)(56,181)(57,180)(58,179)(59,178)(60,177)(61,176)(62,175)(63,174)(64,173)(65,172)(66,171)(67,187)(68,186)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,204)(77,203)(78,202)(79,201)(80,200)(81,199)(82,198)(83,197)(84,196)(85,195)(86,217)(87,216)(88,215)(89,214)(90,213)(91,212)(92,211)(93,210)(94,209)(95,208)(96,207)(97,206)(98,205)(99,221)(100,220)(101,219)(102,218)(103,222)(104,238)(105,237)(106,236)(107,235)(108,234)(109,233)(110,232)(111,231)(112,230)(113,229)(114,228)(115,227)(116,226)(117,225)(118,224)(119,223) );
G=PermutationGroup([[(1,175,119,157,100,137,83,125,63,224,47,221,19,198),(2,176,103,158,101,138,84,126,64,225,48,205,20,199),(3,177,104,159,102,139,85,127,65,226,49,206,21,200),(4,178,105,160,86,140,69,128,66,227,50,207,22,201),(5,179,106,161,87,141,70,129,67,228,51,208,23,202),(6,180,107,162,88,142,71,130,68,229,35,209,24,203),(7,181,108,163,89,143,72,131,52,230,36,210,25,204),(8,182,109,164,90,144,73,132,53,231,37,211,26,188),(9,183,110,165,91,145,74,133,54,232,38,212,27,189),(10,184,111,166,92,146,75,134,55,233,39,213,28,190),(11,185,112,167,93,147,76,135,56,234,40,214,29,191),(12,186,113,168,94,148,77,136,57,235,41,215,30,192),(13,187,114,169,95,149,78,120,58,236,42,216,31,193),(14,171,115,170,96,150,79,121,59,237,43,217,32,194),(15,172,116,154,97,151,80,122,60,238,44,218,33,195),(16,173,117,155,98,152,81,123,61,222,45,219,34,196),(17,174,118,156,99,153,82,124,62,223,46,220,18,197)], [(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),(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221),(222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238)], [(1,124),(2,123),(3,122),(4,121),(5,120),(6,136),(7,135),(8,134),(9,133),(10,132),(11,131),(12,130),(13,129),(14,128),(15,127),(16,126),(17,125),(18,137),(19,153),(20,152),(21,151),(22,150),(23,149),(24,148),(25,147),(26,146),(27,145),(28,144),(29,143),(30,142),(31,141),(32,140),(33,139),(34,138),(35,168),(36,167),(37,166),(38,165),(39,164),(40,163),(41,162),(42,161),(43,160),(44,159),(45,158),(46,157),(47,156),(48,155),(49,154),(50,170),(51,169),(52,185),(53,184),(54,183),(55,182),(56,181),(57,180),(58,179),(59,178),(60,177),(61,176),(62,175),(63,174),(64,173),(65,172),(66,171),(67,187),(68,186),(69,194),(70,193),(71,192),(72,191),(73,190),(74,189),(75,188),(76,204),(77,203),(78,202),(79,201),(80,200),(81,199),(82,198),(83,197),(84,196),(85,195),(86,217),(87,216),(88,215),(89,214),(90,213),(91,212),(92,211),(93,210),(94,209),(95,208),(96,207),(97,206),(98,205),(99,221),(100,220),(101,219),(102,218),(103,222),(104,238),(105,237),(106,236),(107,235),(108,234),(109,233),(110,232),(111,231),(112,230),(113,229),(114,228),(115,227),(116,226),(117,225),(118,224),(119,223)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | ··· | 7F | 14A | ··· | 14F | 14G | ··· | 14R | 17A | ··· | 17H | 34A | ··· | 34H | 119A | ··· | 119AV | 238A | ··· | 238AV |
| order | 1 | 2 | 2 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 17 | ··· | 17 | 34 | ··· | 34 | 119 | ··· | 119 | 238 | ··· | 238 |
| size | 1 | 1 | 17 | 17 | 1 | ··· | 1 | 1 | ··· | 1 | 17 | ··· | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D17 | D34 | C7×D17 | C14×D17 |
| kernel | C14×D17 | C7×D17 | C238 | D34 | D17 | C34 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 6 | 12 | 6 | 8 | 8 | 48 | 48 |
Matrix representation of C14×D17 ►in GL2(𝔽239) generated by
| 141 | 0 |
| 0 | 141 |
| 96 | 1 |
| 11 | 30 |
| 15 | 61 |
| 12 | 224 |
G:=sub<GL(2,GF(239))| [141,0,0,141],[96,11,1,30],[15,12,61,224] >;
C14×D17 in GAP, Magma, Sage, TeX
C_{14}\times D_{17} % in TeX
G:=Group("C14xD17"); // GroupNames label
G:=SmallGroup(476,8);
// by ID
G=gap.SmallGroup(476,8);
# by ID
G:=PCGroup([4,-2,-2,-7,-17,7171]);
// Polycyclic
G:=Group<a,b,c|a^14=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export