metabelian, supersoluble, monomial, A-group
Aliases: C81⋊S3, C3⋊D81, C27.2D9, C9.2D27, C32.3D27, (C3×C81)⋊3C2, C27.(C3⋊S3), (C3×C9).9D9, C9.2(C9⋊S3), (C3×C27).6S3, C3.2(C27⋊S3), SmallGroup(486,60)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C81 — C81⋊S3 |
Generators and relations for C81⋊S3
G = < a,b,c | a81=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C81⋊S3
►On 243 points
Generators in S243
(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)
(1 116 174)(2 117 175)(3 118 176)(4 119 177)(5 120 178)(6 121 179)(7 122 180)(8 123 181)(9 124 182)(10 125 183)(11 126 184)(12 127 185)(13 128 186)(14 129 187)(15 130 188)(16 131 189)(17 132 190)(18 133 191)(19 134 192)(20 135 193)(21 136 194)(22 137 195)(23 138 196)(24 139 197)(25 140 198)(26 141 199)(27 142 200)(28 143 201)(29 144 202)(30 145 203)(31 146 204)(32 147 205)(33 148 206)(34 149 207)(35 150 208)(36 151 209)(37 152 210)(38 153 211)(39 154 212)(40 155 213)(41 156 214)(42 157 215)(43 158 216)(44 159 217)(45 160 218)(46 161 219)(47 162 220)(48 82 221)(49 83 222)(50 84 223)(51 85 224)(52 86 225)(53 87 226)(54 88 227)(55 89 228)(56 90 229)(57 91 230)(58 92 231)(59 93 232)(60 94 233)(61 95 234)(62 96 235)(63 97 236)(64 98 237)(65 99 238)(66 100 239)(67 101 240)(68 102 241)(69 103 242)(70 104 243)(71 105 163)(72 106 164)(73 107 165)(74 108 166)(75 109 167)(76 110 168)(77 111 169)(78 112 170)(79 113 171)(80 114 172)(81 115 173)
(2 81)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 72)(12 71)(13 70)(14 69)(15 68)(16 67)(17 66)(18 65)(19 64)(20 63)(21 62)(22 61)(23 60)(24 59)(25 58)(26 57)(27 56)(28 55)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(82 208)(83 207)(84 206)(85 205)(86 204)(87 203)(88 202)(89 201)(90 200)(91 199)(92 198)(93 197)(94 196)(95 195)(96 194)(97 193)(98 192)(99 191)(100 190)(101 189)(102 188)(103 187)(104 186)(105 185)(106 184)(107 183)(108 182)(109 181)(110 180)(111 179)(112 178)(113 177)(114 176)(115 175)(116 174)(117 173)(118 172)(119 171)(120 170)(121 169)(122 168)(123 167)(124 166)(125 165)(126 164)(127 163)(128 243)(129 242)(130 241)(131 240)(132 239)(133 238)(134 237)(135 236)(136 235)(137 234)(138 233)(139 232)(140 231)(141 230)(142 229)(143 228)(144 227)(145 226)(146 225)(147 224)(148 223)(149 222)(150 221)(151 220)(152 219)(153 218)(154 217)(155 216)(156 215)(157 214)(158 213)(159 212)(160 211)(161 210)(162 209)
G:=sub<Sym(243)| (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), (1,116,174)(2,117,175)(3,118,176)(4,119,177)(5,120,178)(6,121,179)(7,122,180)(8,123,181)(9,124,182)(10,125,183)(11,126,184)(12,127,185)(13,128,186)(14,129,187)(15,130,188)(16,131,189)(17,132,190)(18,133,191)(19,134,192)(20,135,193)(21,136,194)(22,137,195)(23,138,196)(24,139,197)(25,140,198)(26,141,199)(27,142,200)(28,143,201)(29,144,202)(30,145,203)(31,146,204)(32,147,205)(33,148,206)(34,149,207)(35,150,208)(36,151,209)(37,152,210)(38,153,211)(39,154,212)(40,155,213)(41,156,214)(42,157,215)(43,158,216)(44,159,217)(45,160,218)(46,161,219)(47,162,220)(48,82,221)(49,83,222)(50,84,223)(51,85,224)(52,86,225)(53,87,226)(54,88,227)(55,89,228)(56,90,229)(57,91,230)(58,92,231)(59,93,232)(60,94,233)(61,95,234)(62,96,235)(63,97,236)(64,98,237)(65,99,238)(66,100,239)(67,101,240)(68,102,241)(69,103,242)(70,104,243)(71,105,163)(72,106,164)(73,107,165)(74,108,166)(75,109,167)(76,110,168)(77,111,169)(78,112,170)(79,113,171)(80,114,172)(81,115,173), (2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(82,208)(83,207)(84,206)(85,205)(86,204)(87,203)(88,202)(89,201)(90,200)(91,199)(92,198)(93,197)(94,196)(95,195)(96,194)(97,193)(98,192)(99,191)(100,190)(101,189)(102,188)(103,187)(104,186)(105,185)(106,184)(107,183)(108,182)(109,181)(110,180)(111,179)(112,178)(113,177)(114,176)(115,175)(116,174)(117,173)(118,172)(119,171)(120,170)(121,169)(122,168)(123,167)(124,166)(125,165)(126,164)(127,163)(128,243)(129,242)(130,241)(131,240)(132,239)(133,238)(134,237)(135,236)(136,235)(137,234)(138,233)(139,232)(140,231)(141,230)(142,229)(143,228)(144,227)(145,226)(146,225)(147,224)(148,223)(149,222)(150,221)(151,220)(152,219)(153,218)(154,217)(155,216)(156,215)(157,214)(158,213)(159,212)(160,211)(161,210)(162,209)>;
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), (1,116,174)(2,117,175)(3,118,176)(4,119,177)(5,120,178)(6,121,179)(7,122,180)(8,123,181)(9,124,182)(10,125,183)(11,126,184)(12,127,185)(13,128,186)(14,129,187)(15,130,188)(16,131,189)(17,132,190)(18,133,191)(19,134,192)(20,135,193)(21,136,194)(22,137,195)(23,138,196)(24,139,197)(25,140,198)(26,141,199)(27,142,200)(28,143,201)(29,144,202)(30,145,203)(31,146,204)(32,147,205)(33,148,206)(34,149,207)(35,150,208)(36,151,209)(37,152,210)(38,153,211)(39,154,212)(40,155,213)(41,156,214)(42,157,215)(43,158,216)(44,159,217)(45,160,218)(46,161,219)(47,162,220)(48,82,221)(49,83,222)(50,84,223)(51,85,224)(52,86,225)(53,87,226)(54,88,227)(55,89,228)(56,90,229)(57,91,230)(58,92,231)(59,93,232)(60,94,233)(61,95,234)(62,96,235)(63,97,236)(64,98,237)(65,99,238)(66,100,239)(67,101,240)(68,102,241)(69,103,242)(70,104,243)(71,105,163)(72,106,164)(73,107,165)(74,108,166)(75,109,167)(76,110,168)(77,111,169)(78,112,170)(79,113,171)(80,114,172)(81,115,173), (2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(82,208)(83,207)(84,206)(85,205)(86,204)(87,203)(88,202)(89,201)(90,200)(91,199)(92,198)(93,197)(94,196)(95,195)(96,194)(97,193)(98,192)(99,191)(100,190)(101,189)(102,188)(103,187)(104,186)(105,185)(106,184)(107,183)(108,182)(109,181)(110,180)(111,179)(112,178)(113,177)(114,176)(115,175)(116,174)(117,173)(118,172)(119,171)(120,170)(121,169)(122,168)(123,167)(124,166)(125,165)(126,164)(127,163)(128,243)(129,242)(130,241)(131,240)(132,239)(133,238)(134,237)(135,236)(136,235)(137,234)(138,233)(139,232)(140,231)(141,230)(142,229)(143,228)(144,227)(145,226)(146,225)(147,224)(148,223)(149,222)(150,221)(151,220)(152,219)(153,218)(154,217)(155,216)(156,215)(157,214)(158,213)(159,212)(160,211)(161,210)(162,209) );
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)], [(1,116,174),(2,117,175),(3,118,176),(4,119,177),(5,120,178),(6,121,179),(7,122,180),(8,123,181),(9,124,182),(10,125,183),(11,126,184),(12,127,185),(13,128,186),(14,129,187),(15,130,188),(16,131,189),(17,132,190),(18,133,191),(19,134,192),(20,135,193),(21,136,194),(22,137,195),(23,138,196),(24,139,197),(25,140,198),(26,141,199),(27,142,200),(28,143,201),(29,144,202),(30,145,203),(31,146,204),(32,147,205),(33,148,206),(34,149,207),(35,150,208),(36,151,209),(37,152,210),(38,153,211),(39,154,212),(40,155,213),(41,156,214),(42,157,215),(43,158,216),(44,159,217),(45,160,218),(46,161,219),(47,162,220),(48,82,221),(49,83,222),(50,84,223),(51,85,224),(52,86,225),(53,87,226),(54,88,227),(55,89,228),(56,90,229),(57,91,230),(58,92,231),(59,93,232),(60,94,233),(61,95,234),(62,96,235),(63,97,236),(64,98,237),(65,99,238),(66,100,239),(67,101,240),(68,102,241),(69,103,242),(70,104,243),(71,105,163),(72,106,164),(73,107,165),(74,108,166),(75,109,167),(76,110,168),(77,111,169),(78,112,170),(79,113,171),(80,114,172),(81,115,173)], [(2,81),(3,80),(4,79),(5,78),(6,77),(7,76),(8,75),(9,74),(10,73),(11,72),(12,71),(13,70),(14,69),(15,68),(16,67),(17,66),(18,65),(19,64),(20,63),(21,62),(22,61),(23,60),(24,59),(25,58),(26,57),(27,56),(28,55),(29,54),(30,53),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(82,208),(83,207),(84,206),(85,205),(86,204),(87,203),(88,202),(89,201),(90,200),(91,199),(92,198),(93,197),(94,196),(95,195),(96,194),(97,193),(98,192),(99,191),(100,190),(101,189),(102,188),(103,187),(104,186),(105,185),(106,184),(107,183),(108,182),(109,181),(110,180),(111,179),(112,178),(113,177),(114,176),(115,175),(116,174),(117,173),(118,172),(119,171),(120,170),(121,169),(122,168),(123,167),(124,166),(125,165),(126,164),(127,163),(128,243),(129,242),(130,241),(131,240),(132,239),(133,238),(134,237),(135,236),(136,235),(137,234),(138,233),(139,232),(140,231),(141,230),(142,229),(143,228),(144,227),(145,226),(146,225),(147,224),(148,223),(149,222),(150,221),(151,220),(152,219),(153,218),(154,217),(155,216),(156,215),(157,214),(158,213),(159,212),(160,211),(161,210),(162,209)]])
123 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 9A | ··· | 9I | 27A | ··· | 27AA | 81A | ··· | 81CC |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 27 | ··· | 27 | 81 | ··· | 81 |
| size | 1 | 243 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
123 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | S3 | D9 | D9 | D27 | D27 | D81 |
| kernel | C81⋊S3 | C3×C81 | C81 | C3×C27 | C27 | C3×C9 | C9 | C32 | C3 |
| # reps | 1 | 1 | 3 | 1 | 6 | 3 | 18 | 9 | 81 |
Matrix representation of C81⋊S3 ►in GL4(𝔽163) generated by
| 25 | 79 | 0 | 0 |
| 84 | 109 | 0 | 0 |
| 0 | 0 | 95 | 148 |
| 0 | 0 | 15 | 80 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 162 | 1 |
| 0 | 0 | 162 | 0 |
| 1 | 0 | 0 | 0 |
| 162 | 162 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(163))| [25,84,0,0,79,109,0,0,0,0,95,15,0,0,148,80],[1,0,0,0,0,1,0,0,0,0,162,162,0,0,1,0],[1,162,0,0,0,162,0,0,0,0,0,1,0,0,1,0] >;
C81⋊S3 in GAP, Magma, Sage, TeX
C_{81}\rtimes S_3 % in TeX
G:=Group("C81:S3"); // GroupNames label
G:=SmallGroup(486,60);
// by ID
G=gap.SmallGroup(486,60);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,265,547,218,2163,381,8104,208,11669]);
// Polycyclic
G:=Group<a,b,c|a^81=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export