direct product, p-group, abelian, monomial
Aliases: C3×C81, SmallGroup(243,23)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C3×C81 |
| C1 — C3×C81 |
| C1 — C3×C81 |
| C1 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C3 — C9 — C9 — C9 — C9 — C9 — C9 — C27 — C27 — C3×C81 |
Generators and relations for C3×C81
G = < a,b | a3=b81=1, ab=ba >
Smallest permutation representation of C3×C81
►Regular action on 243 points
Generators in S243
(1 130 200)(2 131 201)(3 132 202)(4 133 203)(5 134 204)(6 135 205)(7 136 206)(8 137 207)(9 138 208)(10 139 209)(11 140 210)(12 141 211)(13 142 212)(14 143 213)(15 144 214)(16 145 215)(17 146 216)(18 147 217)(19 148 218)(20 149 219)(21 150 220)(22 151 221)(23 152 222)(24 153 223)(25 154 224)(26 155 225)(27 156 226)(28 157 227)(29 158 228)(30 159 229)(31 160 230)(32 161 231)(33 162 232)(34 82 233)(35 83 234)(36 84 235)(37 85 236)(38 86 237)(39 87 238)(40 88 239)(41 89 240)(42 90 241)(43 91 242)(44 92 243)(45 93 163)(46 94 164)(47 95 165)(48 96 166)(49 97 167)(50 98 168)(51 99 169)(52 100 170)(53 101 171)(54 102 172)(55 103 173)(56 104 174)(57 105 175)(58 106 176)(59 107 177)(60 108 178)(61 109 179)(62 110 180)(63 111 181)(64 112 182)(65 113 183)(66 114 184)(67 115 185)(68 116 186)(69 117 187)(70 118 188)(71 119 189)(72 120 190)(73 121 191)(74 122 192)(75 123 193)(76 124 194)(77 125 195)(78 126 196)(79 127 197)(80 128 198)(81 129 199)
(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)
G:=sub<Sym(243)| (1,130,200)(2,131,201)(3,132,202)(4,133,203)(5,134,204)(6,135,205)(7,136,206)(8,137,207)(9,138,208)(10,139,209)(11,140,210)(12,141,211)(13,142,212)(14,143,213)(15,144,214)(16,145,215)(17,146,216)(18,147,217)(19,148,218)(20,149,219)(21,150,220)(22,151,221)(23,152,222)(24,153,223)(25,154,224)(26,155,225)(27,156,226)(28,157,227)(29,158,228)(30,159,229)(31,160,230)(32,161,231)(33,162,232)(34,82,233)(35,83,234)(36,84,235)(37,85,236)(38,86,237)(39,87,238)(40,88,239)(41,89,240)(42,90,241)(43,91,242)(44,92,243)(45,93,163)(46,94,164)(47,95,165)(48,96,166)(49,97,167)(50,98,168)(51,99,169)(52,100,170)(53,101,171)(54,102,172)(55,103,173)(56,104,174)(57,105,175)(58,106,176)(59,107,177)(60,108,178)(61,109,179)(62,110,180)(63,111,181)(64,112,182)(65,113,183)(66,114,184)(67,115,185)(68,116,186)(69,117,187)(70,118,188)(71,119,189)(72,120,190)(73,121,191)(74,122,192)(75,123,193)(76,124,194)(77,125,195)(78,126,196)(79,127,197)(80,128,198)(81,129,199), (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)>;
G:=Group( (1,130,200)(2,131,201)(3,132,202)(4,133,203)(5,134,204)(6,135,205)(7,136,206)(8,137,207)(9,138,208)(10,139,209)(11,140,210)(12,141,211)(13,142,212)(14,143,213)(15,144,214)(16,145,215)(17,146,216)(18,147,217)(19,148,218)(20,149,219)(21,150,220)(22,151,221)(23,152,222)(24,153,223)(25,154,224)(26,155,225)(27,156,226)(28,157,227)(29,158,228)(30,159,229)(31,160,230)(32,161,231)(33,162,232)(34,82,233)(35,83,234)(36,84,235)(37,85,236)(38,86,237)(39,87,238)(40,88,239)(41,89,240)(42,90,241)(43,91,242)(44,92,243)(45,93,163)(46,94,164)(47,95,165)(48,96,166)(49,97,167)(50,98,168)(51,99,169)(52,100,170)(53,101,171)(54,102,172)(55,103,173)(56,104,174)(57,105,175)(58,106,176)(59,107,177)(60,108,178)(61,109,179)(62,110,180)(63,111,181)(64,112,182)(65,113,183)(66,114,184)(67,115,185)(68,116,186)(69,117,187)(70,118,188)(71,119,189)(72,120,190)(73,121,191)(74,122,192)(75,123,193)(76,124,194)(77,125,195)(78,126,196)(79,127,197)(80,128,198)(81,129,199), (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) );
G=PermutationGroup([[(1,130,200),(2,131,201),(3,132,202),(4,133,203),(5,134,204),(6,135,205),(7,136,206),(8,137,207),(9,138,208),(10,139,209),(11,140,210),(12,141,211),(13,142,212),(14,143,213),(15,144,214),(16,145,215),(17,146,216),(18,147,217),(19,148,218),(20,149,219),(21,150,220),(22,151,221),(23,152,222),(24,153,223),(25,154,224),(26,155,225),(27,156,226),(28,157,227),(29,158,228),(30,159,229),(31,160,230),(32,161,231),(33,162,232),(34,82,233),(35,83,234),(36,84,235),(37,85,236),(38,86,237),(39,87,238),(40,88,239),(41,89,240),(42,90,241),(43,91,242),(44,92,243),(45,93,163),(46,94,164),(47,95,165),(48,96,166),(49,97,167),(50,98,168),(51,99,169),(52,100,170),(53,101,171),(54,102,172),(55,103,173),(56,104,174),(57,105,175),(58,106,176),(59,107,177),(60,108,178),(61,109,179),(62,110,180),(63,111,181),(64,112,182),(65,113,183),(66,114,184),(67,115,185),(68,116,186),(69,117,187),(70,118,188),(71,119,189),(72,120,190),(73,121,191),(74,122,192),(75,123,193),(76,124,194),(77,125,195),(78,126,196),(79,127,197),(80,128,198),(81,129,199)], [(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)]])
C3×C81 is a maximal subgroup of
C81⋊S3
243 conjugacy classes
| class | 1 | 3A | ··· | 3H | 9A | ··· | 9R | 27A | ··· | 27BB | 81A | ··· | 81FF |
| order | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 27 | ··· | 27 | 81 | ··· | 81 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
243 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||||
| image | C1 | C3 | C3 | C9 | C9 | C27 | C27 | C81 |
| kernel | C3×C81 | C81 | C3×C27 | C27 | C3×C9 | C9 | C32 | C3 |
| # reps | 1 | 6 | 2 | 12 | 6 | 36 | 18 | 162 |
Matrix representation of C3×C81 ►in GL2(𝔽163) generated by
| 58 | 0 |
| 0 | 104 |
| 49 | 0 |
| 0 | 1 |
G:=sub<GL(2,GF(163))| [58,0,0,104],[49,0,0,1] >;
C3×C81 in GAP, Magma, Sage, TeX
C_3\times C_{81} % in TeX
G:=Group("C3xC81"); // GroupNames label
G:=SmallGroup(243,23);
// by ID
G=gap.SmallGroup(243,23);
# by ID
G:=PCGroup([5,-3,3,-3,-3,-3,45,57,78]);
// Polycyclic
G:=Group<a,b|a^3=b^81=1,a*b=b*a>;
// generators/relations
Export