direct product, abelian, monomial, 3-elementary
Aliases: C3×C87, SmallGroup(261,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C87 |
| C1 — C3×C87 |
| C1 — C3×C87 |
Generators and relations for C3×C87
G = < a,b | a3=b87=1, ab=ba >
Smallest permutation representation of C3×C87
►Regular action on 261 points
Generators in S261
(1 208 131)(2 209 132)(3 210 133)(4 211 134)(5 212 135)(6 213 136)(7 214 137)(8 215 138)(9 216 139)(10 217 140)(11 218 141)(12 219 142)(13 220 143)(14 221 144)(15 222 145)(16 223 146)(17 224 147)(18 225 148)(19 226 149)(20 227 150)(21 228 151)(22 229 152)(23 230 153)(24 231 154)(25 232 155)(26 233 156)(27 234 157)(28 235 158)(29 236 159)(30 237 160)(31 238 161)(32 239 162)(33 240 163)(34 241 164)(35 242 165)(36 243 166)(37 244 167)(38 245 168)(39 246 169)(40 247 170)(41 248 171)(42 249 172)(43 250 173)(44 251 174)(45 252 88)(46 253 89)(47 254 90)(48 255 91)(49 256 92)(50 257 93)(51 258 94)(52 259 95)(53 260 96)(54 261 97)(55 175 98)(56 176 99)(57 177 100)(58 178 101)(59 179 102)(60 180 103)(61 181 104)(62 182 105)(63 183 106)(64 184 107)(65 185 108)(66 186 109)(67 187 110)(68 188 111)(69 189 112)(70 190 113)(71 191 114)(72 192 115)(73 193 116)(74 194 117)(75 195 118)(76 196 119)(77 197 120)(78 198 121)(79 199 122)(80 200 123)(81 201 124)(82 202 125)(83 203 126)(84 204 127)(85 205 128)(86 206 129)(87 207 130)
(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)(175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261)
G:=sub<Sym(261)| (1,208,131)(2,209,132)(3,210,133)(4,211,134)(5,212,135)(6,213,136)(7,214,137)(8,215,138)(9,216,139)(10,217,140)(11,218,141)(12,219,142)(13,220,143)(14,221,144)(15,222,145)(16,223,146)(17,224,147)(18,225,148)(19,226,149)(20,227,150)(21,228,151)(22,229,152)(23,230,153)(24,231,154)(25,232,155)(26,233,156)(27,234,157)(28,235,158)(29,236,159)(30,237,160)(31,238,161)(32,239,162)(33,240,163)(34,241,164)(35,242,165)(36,243,166)(37,244,167)(38,245,168)(39,246,169)(40,247,170)(41,248,171)(42,249,172)(43,250,173)(44,251,174)(45,252,88)(46,253,89)(47,254,90)(48,255,91)(49,256,92)(50,257,93)(51,258,94)(52,259,95)(53,260,96)(54,261,97)(55,175,98)(56,176,99)(57,177,100)(58,178,101)(59,179,102)(60,180,103)(61,181,104)(62,182,105)(63,183,106)(64,184,107)(65,185,108)(66,186,109)(67,187,110)(68,188,111)(69,189,112)(70,190,113)(71,191,114)(72,192,115)(73,193,116)(74,194,117)(75,195,118)(76,196,119)(77,197,120)(78,198,121)(79,199,122)(80,200,123)(81,201,124)(82,202,125)(83,203,126)(84,204,127)(85,205,128)(86,206,129)(87,207,130), (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)(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261)>;
G:=Group( (1,208,131)(2,209,132)(3,210,133)(4,211,134)(5,212,135)(6,213,136)(7,214,137)(8,215,138)(9,216,139)(10,217,140)(11,218,141)(12,219,142)(13,220,143)(14,221,144)(15,222,145)(16,223,146)(17,224,147)(18,225,148)(19,226,149)(20,227,150)(21,228,151)(22,229,152)(23,230,153)(24,231,154)(25,232,155)(26,233,156)(27,234,157)(28,235,158)(29,236,159)(30,237,160)(31,238,161)(32,239,162)(33,240,163)(34,241,164)(35,242,165)(36,243,166)(37,244,167)(38,245,168)(39,246,169)(40,247,170)(41,248,171)(42,249,172)(43,250,173)(44,251,174)(45,252,88)(46,253,89)(47,254,90)(48,255,91)(49,256,92)(50,257,93)(51,258,94)(52,259,95)(53,260,96)(54,261,97)(55,175,98)(56,176,99)(57,177,100)(58,178,101)(59,179,102)(60,180,103)(61,181,104)(62,182,105)(63,183,106)(64,184,107)(65,185,108)(66,186,109)(67,187,110)(68,188,111)(69,189,112)(70,190,113)(71,191,114)(72,192,115)(73,193,116)(74,194,117)(75,195,118)(76,196,119)(77,197,120)(78,198,121)(79,199,122)(80,200,123)(81,201,124)(82,202,125)(83,203,126)(84,204,127)(85,205,128)(86,206,129)(87,207,130), (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)(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261) );
G=PermutationGroup([[(1,208,131),(2,209,132),(3,210,133),(4,211,134),(5,212,135),(6,213,136),(7,214,137),(8,215,138),(9,216,139),(10,217,140),(11,218,141),(12,219,142),(13,220,143),(14,221,144),(15,222,145),(16,223,146),(17,224,147),(18,225,148),(19,226,149),(20,227,150),(21,228,151),(22,229,152),(23,230,153),(24,231,154),(25,232,155),(26,233,156),(27,234,157),(28,235,158),(29,236,159),(30,237,160),(31,238,161),(32,239,162),(33,240,163),(34,241,164),(35,242,165),(36,243,166),(37,244,167),(38,245,168),(39,246,169),(40,247,170),(41,248,171),(42,249,172),(43,250,173),(44,251,174),(45,252,88),(46,253,89),(47,254,90),(48,255,91),(49,256,92),(50,257,93),(51,258,94),(52,259,95),(53,260,96),(54,261,97),(55,175,98),(56,176,99),(57,177,100),(58,178,101),(59,179,102),(60,180,103),(61,181,104),(62,182,105),(63,183,106),(64,184,107),(65,185,108),(66,186,109),(67,187,110),(68,188,111),(69,189,112),(70,190,113),(71,191,114),(72,192,115),(73,193,116),(74,194,117),(75,195,118),(76,196,119),(77,197,120),(78,198,121),(79,199,122),(80,200,123),(81,201,124),(82,202,125),(83,203,126),(84,204,127),(85,205,128),(86,206,129),(87,207,130)], [(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),(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261)]])
261 conjugacy classes
| class | 1 | 3A | ··· | 3H | 29A | ··· | 29AB | 87A | ··· | 87HP |
| order | 1 | 3 | ··· | 3 | 29 | ··· | 29 | 87 | ··· | 87 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
261 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C29 | C87 |
| kernel | C3×C87 | C87 | C32 | C3 |
| # reps | 1 | 8 | 28 | 224 |
Matrix representation of C3×C87 ►in GL2(𝔽349) generated by
| 226 | 0 |
| 0 | 1 |
| 274 | 0 |
| 0 | 266 |
G:=sub<GL(2,GF(349))| [226,0,0,1],[274,0,0,266] >;
C3×C87 in GAP, Magma, Sage, TeX
C_3\times C_{87} % in TeX
G:=Group("C3xC87"); // GroupNames label
G:=SmallGroup(261,2);
// by ID
G=gap.SmallGroup(261,2);
# by ID
G:=PCGroup([3,-3,-3,-29]);
// Polycyclic
G:=Group<a,b|a^3=b^87=1,a*b=b*a>;
// generators/relations
Export