direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C82, C6⋊C82, C246⋊3C2, C123⋊4C22, C3⋊(C2×C82), SmallGroup(492,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C82 |
Generators and relations for S3×C82
G = < a,b,c | a82=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C82
►On 246 points
Generators in S246
(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)
(1 172 107)(2 173 108)(3 174 109)(4 175 110)(5 176 111)(6 177 112)(7 178 113)(8 179 114)(9 180 115)(10 181 116)(11 182 117)(12 183 118)(13 184 119)(14 185 120)(15 186 121)(16 187 122)(17 188 123)(18 189 124)(19 190 125)(20 191 126)(21 192 127)(22 193 128)(23 194 129)(24 195 130)(25 196 131)(26 197 132)(27 198 133)(28 199 134)(29 200 135)(30 201 136)(31 202 137)(32 203 138)(33 204 139)(34 205 140)(35 206 141)(36 207 142)(37 208 143)(38 209 144)(39 210 145)(40 211 146)(41 212 147)(42 213 148)(43 214 149)(44 215 150)(45 216 151)(46 217 152)(47 218 153)(48 219 154)(49 220 155)(50 221 156)(51 222 157)(52 223 158)(53 224 159)(54 225 160)(55 226 161)(56 227 162)(57 228 163)(58 229 164)(59 230 83)(60 231 84)(61 232 85)(62 233 86)(63 234 87)(64 235 88)(65 236 89)(66 237 90)(67 238 91)(68 239 92)(69 240 93)(70 241 94)(71 242 95)(72 243 96)(73 244 97)(74 245 98)(75 246 99)(76 165 100)(77 166 101)(78 167 102)(79 168 103)(80 169 104)(81 170 105)(82 171 106)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(83 189)(84 190)(85 191)(86 192)(87 193)(88 194)(89 195)(90 196)(91 197)(92 198)(93 199)(94 200)(95 201)(96 202)(97 203)(98 204)(99 205)(100 206)(101 207)(102 208)(103 209)(104 210)(105 211)(106 212)(107 213)(108 214)(109 215)(110 216)(111 217)(112 218)(113 219)(114 220)(115 221)(116 222)(117 223)(118 224)(119 225)(120 226)(121 227)(122 228)(123 229)(124 230)(125 231)(126 232)(127 233)(128 234)(129 235)(130 236)(131 237)(132 238)(133 239)(134 240)(135 241)(136 242)(137 243)(138 244)(139 245)(140 246)(141 165)(142 166)(143 167)(144 168)(145 169)(146 170)(147 171)(148 172)(149 173)(150 174)(151 175)(152 176)(153 177)(154 178)(155 179)(156 180)(157 181)(158 182)(159 183)(160 184)(161 185)(162 186)(163 187)(164 188)
G:=sub<Sym(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), (1,172,107)(2,173,108)(3,174,109)(4,175,110)(5,176,111)(6,177,112)(7,178,113)(8,179,114)(9,180,115)(10,181,116)(11,182,117)(12,183,118)(13,184,119)(14,185,120)(15,186,121)(16,187,122)(17,188,123)(18,189,124)(19,190,125)(20,191,126)(21,192,127)(22,193,128)(23,194,129)(24,195,130)(25,196,131)(26,197,132)(27,198,133)(28,199,134)(29,200,135)(30,201,136)(31,202,137)(32,203,138)(33,204,139)(34,205,140)(35,206,141)(36,207,142)(37,208,143)(38,209,144)(39,210,145)(40,211,146)(41,212,147)(42,213,148)(43,214,149)(44,215,150)(45,216,151)(46,217,152)(47,218,153)(48,219,154)(49,220,155)(50,221,156)(51,222,157)(52,223,158)(53,224,159)(54,225,160)(55,226,161)(56,227,162)(57,228,163)(58,229,164)(59,230,83)(60,231,84)(61,232,85)(62,233,86)(63,234,87)(64,235,88)(65,236,89)(66,237,90)(67,238,91)(68,239,92)(69,240,93)(70,241,94)(71,242,95)(72,243,96)(73,244,97)(74,245,98)(75,246,99)(76,165,100)(77,166,101)(78,167,102)(79,168,103)(80,169,104)(81,170,105)(82,171,106), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(83,189)(84,190)(85,191)(86,192)(87,193)(88,194)(89,195)(90,196)(91,197)(92,198)(93,199)(94,200)(95,201)(96,202)(97,203)(98,204)(99,205)(100,206)(101,207)(102,208)(103,209)(104,210)(105,211)(106,212)(107,213)(108,214)(109,215)(110,216)(111,217)(112,218)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,225)(120,226)(121,227)(122,228)(123,229)(124,230)(125,231)(126,232)(127,233)(128,234)(129,235)(130,236)(131,237)(132,238)(133,239)(134,240)(135,241)(136,242)(137,243)(138,244)(139,245)(140,246)(141,165)(142,166)(143,167)(144,168)(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,177)(154,178)(155,179)(156,180)(157,181)(158,182)(159,183)(160,184)(161,185)(162,186)(163,187)(164,188)>;
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)(165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246), (1,172,107)(2,173,108)(3,174,109)(4,175,110)(5,176,111)(6,177,112)(7,178,113)(8,179,114)(9,180,115)(10,181,116)(11,182,117)(12,183,118)(13,184,119)(14,185,120)(15,186,121)(16,187,122)(17,188,123)(18,189,124)(19,190,125)(20,191,126)(21,192,127)(22,193,128)(23,194,129)(24,195,130)(25,196,131)(26,197,132)(27,198,133)(28,199,134)(29,200,135)(30,201,136)(31,202,137)(32,203,138)(33,204,139)(34,205,140)(35,206,141)(36,207,142)(37,208,143)(38,209,144)(39,210,145)(40,211,146)(41,212,147)(42,213,148)(43,214,149)(44,215,150)(45,216,151)(46,217,152)(47,218,153)(48,219,154)(49,220,155)(50,221,156)(51,222,157)(52,223,158)(53,224,159)(54,225,160)(55,226,161)(56,227,162)(57,228,163)(58,229,164)(59,230,83)(60,231,84)(61,232,85)(62,233,86)(63,234,87)(64,235,88)(65,236,89)(66,237,90)(67,238,91)(68,239,92)(69,240,93)(70,241,94)(71,242,95)(72,243,96)(73,244,97)(74,245,98)(75,246,99)(76,165,100)(77,166,101)(78,167,102)(79,168,103)(80,169,104)(81,170,105)(82,171,106), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(83,189)(84,190)(85,191)(86,192)(87,193)(88,194)(89,195)(90,196)(91,197)(92,198)(93,199)(94,200)(95,201)(96,202)(97,203)(98,204)(99,205)(100,206)(101,207)(102,208)(103,209)(104,210)(105,211)(106,212)(107,213)(108,214)(109,215)(110,216)(111,217)(112,218)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,225)(120,226)(121,227)(122,228)(123,229)(124,230)(125,231)(126,232)(127,233)(128,234)(129,235)(130,236)(131,237)(132,238)(133,239)(134,240)(135,241)(136,242)(137,243)(138,244)(139,245)(140,246)(141,165)(142,166)(143,167)(144,168)(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,177)(154,178)(155,179)(156,180)(157,181)(158,182)(159,183)(160,184)(161,185)(162,186)(163,187)(164,188) );
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),(165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246)], [(1,172,107),(2,173,108),(3,174,109),(4,175,110),(5,176,111),(6,177,112),(7,178,113),(8,179,114),(9,180,115),(10,181,116),(11,182,117),(12,183,118),(13,184,119),(14,185,120),(15,186,121),(16,187,122),(17,188,123),(18,189,124),(19,190,125),(20,191,126),(21,192,127),(22,193,128),(23,194,129),(24,195,130),(25,196,131),(26,197,132),(27,198,133),(28,199,134),(29,200,135),(30,201,136),(31,202,137),(32,203,138),(33,204,139),(34,205,140),(35,206,141),(36,207,142),(37,208,143),(38,209,144),(39,210,145),(40,211,146),(41,212,147),(42,213,148),(43,214,149),(44,215,150),(45,216,151),(46,217,152),(47,218,153),(48,219,154),(49,220,155),(50,221,156),(51,222,157),(52,223,158),(53,224,159),(54,225,160),(55,226,161),(56,227,162),(57,228,163),(58,229,164),(59,230,83),(60,231,84),(61,232,85),(62,233,86),(63,234,87),(64,235,88),(65,236,89),(66,237,90),(67,238,91),(68,239,92),(69,240,93),(70,241,94),(71,242,95),(72,243,96),(73,244,97),(74,245,98),(75,246,99),(76,165,100),(77,166,101),(78,167,102),(79,168,103),(80,169,104),(81,170,105),(82,171,106)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82),(83,189),(84,190),(85,191),(86,192),(87,193),(88,194),(89,195),(90,196),(91,197),(92,198),(93,199),(94,200),(95,201),(96,202),(97,203),(98,204),(99,205),(100,206),(101,207),(102,208),(103,209),(104,210),(105,211),(106,212),(107,213),(108,214),(109,215),(110,216),(111,217),(112,218),(113,219),(114,220),(115,221),(116,222),(117,223),(118,224),(119,225),(120,226),(121,227),(122,228),(123,229),(124,230),(125,231),(126,232),(127,233),(128,234),(129,235),(130,236),(131,237),(132,238),(133,239),(134,240),(135,241),(136,242),(137,243),(138,244),(139,245),(140,246),(141,165),(142,166),(143,167),(144,168),(145,169),(146,170),(147,171),(148,172),(149,173),(150,174),(151,175),(152,176),(153,177),(154,178),(155,179),(156,180),(157,181),(158,182),(159,183),(160,184),(161,185),(162,186),(163,187),(164,188)]])
246 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 41A | ··· | 41AN | 82A | ··· | 82AN | 82AO | ··· | 82DP | 123A | ··· | 123AN | 246A | ··· | 246AN |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 41 | ··· | 41 | 82 | ··· | 82 | 82 | ··· | 82 | 123 | ··· | 123 | 246 | ··· | 246 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
246 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C41 | C82 | C82 | S3 | D6 | S3×C41 | S3×C82 |
| kernel | S3×C82 | S3×C41 | C246 | D6 | S3 | C6 | C82 | C41 | C2 | C1 |
| # reps | 1 | 2 | 1 | 40 | 80 | 40 | 1 | 1 | 40 | 40 |
Matrix representation of S3×C82 ►in GL3(𝔽739) generated by
| 738 | 0 | 0 |
| 0 | 438 | 0 |
| 0 | 0 | 438 |
| 1 | 0 | 0 |
| 0 | 0 | 738 |
| 0 | 1 | 738 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(739))| [738,0,0,0,438,0,0,0,438],[1,0,0,0,0,1,0,738,738],[1,0,0,0,0,1,0,1,0] >;
S3×C82 in GAP, Magma, Sage, TeX
S_3\times C_{82} % in TeX
G:=Group("S3xC82"); // GroupNames label
G:=SmallGroup(492,10);
// by ID
G=gap.SmallGroup(492,10);
# by ID
G:=PCGroup([4,-2,-2,-41,-3,5251]);
// Polycyclic
G:=Group<a,b,c|a^82=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export