direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C83, C3⋊C166, C249⋊3C2, SmallGroup(498,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C83 |
Generators and relations for S3×C83
G = < a,b,c | a83=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C83
►On 249 points
Generators in S249
(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)(167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249)
(1 155 182)(2 156 183)(3 157 184)(4 158 185)(5 159 186)(6 160 187)(7 161 188)(8 162 189)(9 163 190)(10 164 191)(11 165 192)(12 166 193)(13 84 194)(14 85 195)(15 86 196)(16 87 197)(17 88 198)(18 89 199)(19 90 200)(20 91 201)(21 92 202)(22 93 203)(23 94 204)(24 95 205)(25 96 206)(26 97 207)(27 98 208)(28 99 209)(29 100 210)(30 101 211)(31 102 212)(32 103 213)(33 104 214)(34 105 215)(35 106 216)(36 107 217)(37 108 218)(38 109 219)(39 110 220)(40 111 221)(41 112 222)(42 113 223)(43 114 224)(44 115 225)(45 116 226)(46 117 227)(47 118 228)(48 119 229)(49 120 230)(50 121 231)(51 122 232)(52 123 233)(53 124 234)(54 125 235)(55 126 236)(56 127 237)(57 128 238)(58 129 239)(59 130 240)(60 131 241)(61 132 242)(62 133 243)(63 134 244)(64 135 245)(65 136 246)(66 137 247)(67 138 248)(68 139 249)(69 140 167)(70 141 168)(71 142 169)(72 143 170)(73 144 171)(74 145 172)(75 146 173)(76 147 174)(77 148 175)(78 149 176)(79 150 177)(80 151 178)(81 152 179)(82 153 180)(83 154 181)
(84 194)(85 195)(86 196)(87 197)(88 198)(89 199)(90 200)(91 201)(92 202)(93 203)(94 204)(95 205)(96 206)(97 207)(98 208)(99 209)(100 210)(101 211)(102 212)(103 213)(104 214)(105 215)(106 216)(107 217)(108 218)(109 219)(110 220)(111 221)(112 222)(113 223)(114 224)(115 225)(116 226)(117 227)(118 228)(119 229)(120 230)(121 231)(122 232)(123 233)(124 234)(125 235)(126 236)(127 237)(128 238)(129 239)(130 240)(131 241)(132 242)(133 243)(134 244)(135 245)(136 246)(137 247)(138 248)(139 249)(140 167)(141 168)(142 169)(143 170)(144 171)(145 172)(146 173)(147 174)(148 175)(149 176)(150 177)(151 178)(152 179)(153 180)(154 181)(155 182)(156 183)(157 184)(158 185)(159 186)(160 187)(161 188)(162 189)(163 190)(164 191)(165 192)(166 193)
G:=sub<Sym(249)| (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)(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249), (1,155,182)(2,156,183)(3,157,184)(4,158,185)(5,159,186)(6,160,187)(7,161,188)(8,162,189)(9,163,190)(10,164,191)(11,165,192)(12,166,193)(13,84,194)(14,85,195)(15,86,196)(16,87,197)(17,88,198)(18,89,199)(19,90,200)(20,91,201)(21,92,202)(22,93,203)(23,94,204)(24,95,205)(25,96,206)(26,97,207)(27,98,208)(28,99,209)(29,100,210)(30,101,211)(31,102,212)(32,103,213)(33,104,214)(34,105,215)(35,106,216)(36,107,217)(37,108,218)(38,109,219)(39,110,220)(40,111,221)(41,112,222)(42,113,223)(43,114,224)(44,115,225)(45,116,226)(46,117,227)(47,118,228)(48,119,229)(49,120,230)(50,121,231)(51,122,232)(52,123,233)(53,124,234)(54,125,235)(55,126,236)(56,127,237)(57,128,238)(58,129,239)(59,130,240)(60,131,241)(61,132,242)(62,133,243)(63,134,244)(64,135,245)(65,136,246)(66,137,247)(67,138,248)(68,139,249)(69,140,167)(70,141,168)(71,142,169)(72,143,170)(73,144,171)(74,145,172)(75,146,173)(76,147,174)(77,148,175)(78,149,176)(79,150,177)(80,151,178)(81,152,179)(82,153,180)(83,154,181), (84,194)(85,195)(86,196)(87,197)(88,198)(89,199)(90,200)(91,201)(92,202)(93,203)(94,204)(95,205)(96,206)(97,207)(98,208)(99,209)(100,210)(101,211)(102,212)(103,213)(104,214)(105,215)(106,216)(107,217)(108,218)(109,219)(110,220)(111,221)(112,222)(113,223)(114,224)(115,225)(116,226)(117,227)(118,228)(119,229)(120,230)(121,231)(122,232)(123,233)(124,234)(125,235)(126,236)(127,237)(128,238)(129,239)(130,240)(131,241)(132,242)(133,243)(134,244)(135,245)(136,246)(137,247)(138,248)(139,249)(140,167)(141,168)(142,169)(143,170)(144,171)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,181)(155,182)(156,183)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,193)>;
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)(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249), (1,155,182)(2,156,183)(3,157,184)(4,158,185)(5,159,186)(6,160,187)(7,161,188)(8,162,189)(9,163,190)(10,164,191)(11,165,192)(12,166,193)(13,84,194)(14,85,195)(15,86,196)(16,87,197)(17,88,198)(18,89,199)(19,90,200)(20,91,201)(21,92,202)(22,93,203)(23,94,204)(24,95,205)(25,96,206)(26,97,207)(27,98,208)(28,99,209)(29,100,210)(30,101,211)(31,102,212)(32,103,213)(33,104,214)(34,105,215)(35,106,216)(36,107,217)(37,108,218)(38,109,219)(39,110,220)(40,111,221)(41,112,222)(42,113,223)(43,114,224)(44,115,225)(45,116,226)(46,117,227)(47,118,228)(48,119,229)(49,120,230)(50,121,231)(51,122,232)(52,123,233)(53,124,234)(54,125,235)(55,126,236)(56,127,237)(57,128,238)(58,129,239)(59,130,240)(60,131,241)(61,132,242)(62,133,243)(63,134,244)(64,135,245)(65,136,246)(66,137,247)(67,138,248)(68,139,249)(69,140,167)(70,141,168)(71,142,169)(72,143,170)(73,144,171)(74,145,172)(75,146,173)(76,147,174)(77,148,175)(78,149,176)(79,150,177)(80,151,178)(81,152,179)(82,153,180)(83,154,181), (84,194)(85,195)(86,196)(87,197)(88,198)(89,199)(90,200)(91,201)(92,202)(93,203)(94,204)(95,205)(96,206)(97,207)(98,208)(99,209)(100,210)(101,211)(102,212)(103,213)(104,214)(105,215)(106,216)(107,217)(108,218)(109,219)(110,220)(111,221)(112,222)(113,223)(114,224)(115,225)(116,226)(117,227)(118,228)(119,229)(120,230)(121,231)(122,232)(123,233)(124,234)(125,235)(126,236)(127,237)(128,238)(129,239)(130,240)(131,241)(132,242)(133,243)(134,244)(135,245)(136,246)(137,247)(138,248)(139,249)(140,167)(141,168)(142,169)(143,170)(144,171)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,181)(155,182)(156,183)(157,184)(158,185)(159,186)(160,187)(161,188)(162,189)(163,190)(164,191)(165,192)(166,193) );
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),(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249)], [(1,155,182),(2,156,183),(3,157,184),(4,158,185),(5,159,186),(6,160,187),(7,161,188),(8,162,189),(9,163,190),(10,164,191),(11,165,192),(12,166,193),(13,84,194),(14,85,195),(15,86,196),(16,87,197),(17,88,198),(18,89,199),(19,90,200),(20,91,201),(21,92,202),(22,93,203),(23,94,204),(24,95,205),(25,96,206),(26,97,207),(27,98,208),(28,99,209),(29,100,210),(30,101,211),(31,102,212),(32,103,213),(33,104,214),(34,105,215),(35,106,216),(36,107,217),(37,108,218),(38,109,219),(39,110,220),(40,111,221),(41,112,222),(42,113,223),(43,114,224),(44,115,225),(45,116,226),(46,117,227),(47,118,228),(48,119,229),(49,120,230),(50,121,231),(51,122,232),(52,123,233),(53,124,234),(54,125,235),(55,126,236),(56,127,237),(57,128,238),(58,129,239),(59,130,240),(60,131,241),(61,132,242),(62,133,243),(63,134,244),(64,135,245),(65,136,246),(66,137,247),(67,138,248),(68,139,249),(69,140,167),(70,141,168),(71,142,169),(72,143,170),(73,144,171),(74,145,172),(75,146,173),(76,147,174),(77,148,175),(78,149,176),(79,150,177),(80,151,178),(81,152,179),(82,153,180),(83,154,181)], [(84,194),(85,195),(86,196),(87,197),(88,198),(89,199),(90,200),(91,201),(92,202),(93,203),(94,204),(95,205),(96,206),(97,207),(98,208),(99,209),(100,210),(101,211),(102,212),(103,213),(104,214),(105,215),(106,216),(107,217),(108,218),(109,219),(110,220),(111,221),(112,222),(113,223),(114,224),(115,225),(116,226),(117,227),(118,228),(119,229),(120,230),(121,231),(122,232),(123,233),(124,234),(125,235),(126,236),(127,237),(128,238),(129,239),(130,240),(131,241),(132,242),(133,243),(134,244),(135,245),(136,246),(137,247),(138,248),(139,249),(140,167),(141,168),(142,169),(143,170),(144,171),(145,172),(146,173),(147,174),(148,175),(149,176),(150,177),(151,178),(152,179),(153,180),(154,181),(155,182),(156,183),(157,184),(158,185),(159,186),(160,187),(161,188),(162,189),(163,190),(164,191),(165,192),(166,193)]])
249 conjugacy classes
| class | 1 | 2 | 3 | 83A | ··· | 83CD | 166A | ··· | 166CD | 249A | ··· | 249CD |
| order | 1 | 2 | 3 | 83 | ··· | 83 | 166 | ··· | 166 | 249 | ··· | 249 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
249 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C83 | C166 | S3 | S3×C83 |
| kernel | S3×C83 | C249 | S3 | C3 | C83 | C1 |
| # reps | 1 | 1 | 82 | 82 | 1 | 82 |
Matrix representation of S3×C83 ►in GL2(𝔽499) generated by
| 268 | 0 |
| 0 | 268 |
| 498 | 498 |
| 1 | 0 |
| 1 | 0 |
| 498 | 498 |
G:=sub<GL(2,GF(499))| [268,0,0,268],[498,1,498,0],[1,498,0,498] >;
S3×C83 in GAP, Magma, Sage, TeX
S_3\times C_{83} % in TeX
G:=Group("S3xC83"); // GroupNames label
G:=SmallGroup(498,1);
// by ID
G=gap.SmallGroup(498,1);
# by ID
G:=PCGroup([3,-2,-83,-3,2990]);
// Polycyclic
G:=Group<a,b,c|a^83=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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