direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D41, C82⋊C6, C246⋊2C2, C123⋊3C22, C41⋊(C2×C6), SmallGroup(492,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C6×D41 |
Generators and relations for C6×D41
G = < a,b,c | a6=b41=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D41
►On 246 points
Generators in S246
(1 167 108 136 73 206)(2 168 109 137 74 207)(3 169 110 138 75 208)(4 170 111 139 76 209)(5 171 112 140 77 210)(6 172 113 141 78 211)(7 173 114 142 79 212)(8 174 115 143 80 213)(9 175 116 144 81 214)(10 176 117 145 82 215)(11 177 118 146 42 216)(12 178 119 147 43 217)(13 179 120 148 44 218)(14 180 121 149 45 219)(15 181 122 150 46 220)(16 182 123 151 47 221)(17 183 83 152 48 222)(18 184 84 153 49 223)(19 185 85 154 50 224)(20 186 86 155 51 225)(21 187 87 156 52 226)(22 188 88 157 53 227)(23 189 89 158 54 228)(24 190 90 159 55 229)(25 191 91 160 56 230)(26 192 92 161 57 231)(27 193 93 162 58 232)(28 194 94 163 59 233)(29 195 95 164 60 234)(30 196 96 124 61 235)(31 197 97 125 62 236)(32 198 98 126 63 237)(33 199 99 127 64 238)(34 200 100 128 65 239)(35 201 101 129 66 240)(36 202 102 130 67 241)(37 203 103 131 68 242)(38 204 104 132 69 243)(39 205 105 133 70 244)(40 165 106 134 71 245)(41 166 107 135 72 246)
(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)(206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246)
(1 135)(2 134)(3 133)(4 132)(5 131)(6 130)(7 129)(8 128)(9 127)(10 126)(11 125)(12 124)(13 164)(14 163)(15 162)(16 161)(17 160)(18 159)(19 158)(20 157)(21 156)(22 155)(23 154)(24 153)(25 152)(26 151)(27 150)(28 149)(29 148)(30 147)(31 146)(32 145)(33 144)(34 143)(35 142)(36 141)(37 140)(38 139)(39 138)(40 137)(41 136)(42 197)(43 196)(44 195)(45 194)(46 193)(47 192)(48 191)(49 190)(50 189)(51 188)(52 187)(53 186)(54 185)(55 184)(56 183)(57 182)(58 181)(59 180)(60 179)(61 178)(62 177)(63 176)(64 175)(65 174)(66 173)(67 172)(68 171)(69 170)(70 169)(71 168)(72 167)(73 166)(74 165)(75 205)(76 204)(77 203)(78 202)(79 201)(80 200)(81 199)(82 198)(83 230)(84 229)(85 228)(86 227)(87 226)(88 225)(89 224)(90 223)(91 222)(92 221)(93 220)(94 219)(95 218)(96 217)(97 216)(98 215)(99 214)(100 213)(101 212)(102 211)(103 210)(104 209)(105 208)(106 207)(107 206)(108 246)(109 245)(110 244)(111 243)(112 242)(113 241)(114 240)(115 239)(116 238)(117 237)(118 236)(119 235)(120 234)(121 233)(122 232)(123 231)
G:=sub<Sym(246)| (1,167,108,136,73,206)(2,168,109,137,74,207)(3,169,110,138,75,208)(4,170,111,139,76,209)(5,171,112,140,77,210)(6,172,113,141,78,211)(7,173,114,142,79,212)(8,174,115,143,80,213)(9,175,116,144,81,214)(10,176,117,145,82,215)(11,177,118,146,42,216)(12,178,119,147,43,217)(13,179,120,148,44,218)(14,180,121,149,45,219)(15,181,122,150,46,220)(16,182,123,151,47,221)(17,183,83,152,48,222)(18,184,84,153,49,223)(19,185,85,154,50,224)(20,186,86,155,51,225)(21,187,87,156,52,226)(22,188,88,157,53,227)(23,189,89,158,54,228)(24,190,90,159,55,229)(25,191,91,160,56,230)(26,192,92,161,57,231)(27,193,93,162,58,232)(28,194,94,163,59,233)(29,195,95,164,60,234)(30,196,96,124,61,235)(31,197,97,125,62,236)(32,198,98,126,63,237)(33,199,99,127,64,238)(34,200,100,128,65,239)(35,201,101,129,66,240)(36,202,102,130,67,241)(37,203,103,131,68,242)(38,204,104,132,69,243)(39,205,105,133,70,244)(40,165,106,134,71,245)(41,166,107,135,72,246), (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)(206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,164)(14,163)(15,162)(16,161)(17,160)(18,159)(19,158)(20,157)(21,156)(22,155)(23,154)(24,153)(25,152)(26,151)(27,150)(28,149)(29,148)(30,147)(31,146)(32,145)(33,144)(34,143)(35,142)(36,141)(37,140)(38,139)(39,138)(40,137)(41,136)(42,197)(43,196)(44,195)(45,194)(46,193)(47,192)(48,191)(49,190)(50,189)(51,188)(52,187)(53,186)(54,185)(55,184)(56,183)(57,182)(58,181)(59,180)(60,179)(61,178)(62,177)(63,176)(64,175)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,168)(72,167)(73,166)(74,165)(75,205)(76,204)(77,203)(78,202)(79,201)(80,200)(81,199)(82,198)(83,230)(84,229)(85,228)(86,227)(87,226)(88,225)(89,224)(90,223)(91,222)(92,221)(93,220)(94,219)(95,218)(96,217)(97,216)(98,215)(99,214)(100,213)(101,212)(102,211)(103,210)(104,209)(105,208)(106,207)(107,206)(108,246)(109,245)(110,244)(111,243)(112,242)(113,241)(114,240)(115,239)(116,238)(117,237)(118,236)(119,235)(120,234)(121,233)(122,232)(123,231)>;
G:=Group( (1,167,108,136,73,206)(2,168,109,137,74,207)(3,169,110,138,75,208)(4,170,111,139,76,209)(5,171,112,140,77,210)(6,172,113,141,78,211)(7,173,114,142,79,212)(8,174,115,143,80,213)(9,175,116,144,81,214)(10,176,117,145,82,215)(11,177,118,146,42,216)(12,178,119,147,43,217)(13,179,120,148,44,218)(14,180,121,149,45,219)(15,181,122,150,46,220)(16,182,123,151,47,221)(17,183,83,152,48,222)(18,184,84,153,49,223)(19,185,85,154,50,224)(20,186,86,155,51,225)(21,187,87,156,52,226)(22,188,88,157,53,227)(23,189,89,158,54,228)(24,190,90,159,55,229)(25,191,91,160,56,230)(26,192,92,161,57,231)(27,193,93,162,58,232)(28,194,94,163,59,233)(29,195,95,164,60,234)(30,196,96,124,61,235)(31,197,97,125,62,236)(32,198,98,126,63,237)(33,199,99,127,64,238)(34,200,100,128,65,239)(35,201,101,129,66,240)(36,202,102,130,67,241)(37,203,103,131,68,242)(38,204,104,132,69,243)(39,205,105,133,70,244)(40,165,106,134,71,245)(41,166,107,135,72,246), (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)(206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,126)(11,125)(12,124)(13,164)(14,163)(15,162)(16,161)(17,160)(18,159)(19,158)(20,157)(21,156)(22,155)(23,154)(24,153)(25,152)(26,151)(27,150)(28,149)(29,148)(30,147)(31,146)(32,145)(33,144)(34,143)(35,142)(36,141)(37,140)(38,139)(39,138)(40,137)(41,136)(42,197)(43,196)(44,195)(45,194)(46,193)(47,192)(48,191)(49,190)(50,189)(51,188)(52,187)(53,186)(54,185)(55,184)(56,183)(57,182)(58,181)(59,180)(60,179)(61,178)(62,177)(63,176)(64,175)(65,174)(66,173)(67,172)(68,171)(69,170)(70,169)(71,168)(72,167)(73,166)(74,165)(75,205)(76,204)(77,203)(78,202)(79,201)(80,200)(81,199)(82,198)(83,230)(84,229)(85,228)(86,227)(87,226)(88,225)(89,224)(90,223)(91,222)(92,221)(93,220)(94,219)(95,218)(96,217)(97,216)(98,215)(99,214)(100,213)(101,212)(102,211)(103,210)(104,209)(105,208)(106,207)(107,206)(108,246)(109,245)(110,244)(111,243)(112,242)(113,241)(114,240)(115,239)(116,238)(117,237)(118,236)(119,235)(120,234)(121,233)(122,232)(123,231) );
G=PermutationGroup([[(1,167,108,136,73,206),(2,168,109,137,74,207),(3,169,110,138,75,208),(4,170,111,139,76,209),(5,171,112,140,77,210),(6,172,113,141,78,211),(7,173,114,142,79,212),(8,174,115,143,80,213),(9,175,116,144,81,214),(10,176,117,145,82,215),(11,177,118,146,42,216),(12,178,119,147,43,217),(13,179,120,148,44,218),(14,180,121,149,45,219),(15,181,122,150,46,220),(16,182,123,151,47,221),(17,183,83,152,48,222),(18,184,84,153,49,223),(19,185,85,154,50,224),(20,186,86,155,51,225),(21,187,87,156,52,226),(22,188,88,157,53,227),(23,189,89,158,54,228),(24,190,90,159,55,229),(25,191,91,160,56,230),(26,192,92,161,57,231),(27,193,93,162,58,232),(28,194,94,163,59,233),(29,195,95,164,60,234),(30,196,96,124,61,235),(31,197,97,125,62,236),(32,198,98,126,63,237),(33,199,99,127,64,238),(34,200,100,128,65,239),(35,201,101,129,66,240),(36,202,102,130,67,241),(37,203,103,131,68,242),(38,204,104,132,69,243),(39,205,105,133,70,244),(40,165,106,134,71,245),(41,166,107,135,72,246)], [(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),(206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246)], [(1,135),(2,134),(3,133),(4,132),(5,131),(6,130),(7,129),(8,128),(9,127),(10,126),(11,125),(12,124),(13,164),(14,163),(15,162),(16,161),(17,160),(18,159),(19,158),(20,157),(21,156),(22,155),(23,154),(24,153),(25,152),(26,151),(27,150),(28,149),(29,148),(30,147),(31,146),(32,145),(33,144),(34,143),(35,142),(36,141),(37,140),(38,139),(39,138),(40,137),(41,136),(42,197),(43,196),(44,195),(45,194),(46,193),(47,192),(48,191),(49,190),(50,189),(51,188),(52,187),(53,186),(54,185),(55,184),(56,183),(57,182),(58,181),(59,180),(60,179),(61,178),(62,177),(63,176),(64,175),(65,174),(66,173),(67,172),(68,171),(69,170),(70,169),(71,168),(72,167),(73,166),(74,165),(75,205),(76,204),(77,203),(78,202),(79,201),(80,200),(81,199),(82,198),(83,230),(84,229),(85,228),(86,227),(87,226),(88,225),(89,224),(90,223),(91,222),(92,221),(93,220),(94,219),(95,218),(96,217),(97,216),(98,215),(99,214),(100,213),(101,212),(102,211),(103,210),(104,209),(105,208),(106,207),(107,206),(108,246),(109,245),(110,244),(111,243),(112,242),(113,241),(114,240),(115,239),(116,238),(117,237),(118,236),(119,235),(120,234),(121,233),(122,232),(123,231)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 41A | ··· | 41T | 82A | ··· | 82T | 123A | ··· | 123AN | 246A | ··· | 246AN |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 41 | ··· | 41 | 82 | ··· | 82 | 123 | ··· | 123 | 246 | ··· | 246 |
| size | 1 | 1 | 41 | 41 | 1 | 1 | 1 | 1 | 41 | 41 | 41 | 41 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D41 | D82 | C3×D41 | C6×D41 |
| kernel | C6×D41 | C3×D41 | C246 | D82 | D41 | C82 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 20 | 20 | 40 | 40 |
Matrix representation of C6×D41 ►in GL2(𝔽739) generated by
| 419 | 0 |
| 0 | 419 |
| 166 | 1 |
| 720 | 267 |
| 668 | 445 |
| 545 | 71 |
G:=sub<GL(2,GF(739))| [419,0,0,419],[166,720,1,267],[668,545,445,71] >;
C6×D41 in GAP, Magma, Sage, TeX
C_6\times D_{41} % in TeX
G:=Group("C6xD41"); // GroupNames label
G:=SmallGroup(492,9);
// by ID
G=gap.SmallGroup(492,9);
# by ID
G:=PCGroup([4,-2,-2,-3,-41,7683]);
// Polycyclic
G:=Group<a,b,c|a^6=b^41=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export