direct product, abelian, monomial, 3-elementary
Aliases: C3×C105, SmallGroup(315,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C105 |
| C1 — C3×C105 |
| C1 — C3×C105 |
Generators and relations for C3×C105
G = < a,b | a3=b105=1, ab=ba >
Smallest permutation representation of C3×C105
►Regular action on 315 points
Generators in S315
(1 295 127)(2 296 128)(3 297 129)(4 298 130)(5 299 131)(6 300 132)(7 301 133)(8 302 134)(9 303 135)(10 304 136)(11 305 137)(12 306 138)(13 307 139)(14 308 140)(15 309 141)(16 310 142)(17 311 143)(18 312 144)(19 313 145)(20 314 146)(21 315 147)(22 211 148)(23 212 149)(24 213 150)(25 214 151)(26 215 152)(27 216 153)(28 217 154)(29 218 155)(30 219 156)(31 220 157)(32 221 158)(33 222 159)(34 223 160)(35 224 161)(36 225 162)(37 226 163)(38 227 164)(39 228 165)(40 229 166)(41 230 167)(42 231 168)(43 232 169)(44 233 170)(45 234 171)(46 235 172)(47 236 173)(48 237 174)(49 238 175)(50 239 176)(51 240 177)(52 241 178)(53 242 179)(54 243 180)(55 244 181)(56 245 182)(57 246 183)(58 247 184)(59 248 185)(60 249 186)(61 250 187)(62 251 188)(63 252 189)(64 253 190)(65 254 191)(66 255 192)(67 256 193)(68 257 194)(69 258 195)(70 259 196)(71 260 197)(72 261 198)(73 262 199)(74 263 200)(75 264 201)(76 265 202)(77 266 203)(78 267 204)(79 268 205)(80 269 206)(81 270 207)(82 271 208)(83 272 209)(84 273 210)(85 274 106)(86 275 107)(87 276 108)(88 277 109)(89 278 110)(90 279 111)(91 280 112)(92 281 113)(93 282 114)(94 283 115)(95 284 116)(96 285 117)(97 286 118)(98 287 119)(99 288 120)(100 289 121)(101 290 122)(102 291 123)(103 292 124)(104 293 125)(105 294 126)
(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)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315)
G:=sub<Sym(315)| (1,295,127)(2,296,128)(3,297,129)(4,298,130)(5,299,131)(6,300,132)(7,301,133)(8,302,134)(9,303,135)(10,304,136)(11,305,137)(12,306,138)(13,307,139)(14,308,140)(15,309,141)(16,310,142)(17,311,143)(18,312,144)(19,313,145)(20,314,146)(21,315,147)(22,211,148)(23,212,149)(24,213,150)(25,214,151)(26,215,152)(27,216,153)(28,217,154)(29,218,155)(30,219,156)(31,220,157)(32,221,158)(33,222,159)(34,223,160)(35,224,161)(36,225,162)(37,226,163)(38,227,164)(39,228,165)(40,229,166)(41,230,167)(42,231,168)(43,232,169)(44,233,170)(45,234,171)(46,235,172)(47,236,173)(48,237,174)(49,238,175)(50,239,176)(51,240,177)(52,241,178)(53,242,179)(54,243,180)(55,244,181)(56,245,182)(57,246,183)(58,247,184)(59,248,185)(60,249,186)(61,250,187)(62,251,188)(63,252,189)(64,253,190)(65,254,191)(66,255,192)(67,256,193)(68,257,194)(69,258,195)(70,259,196)(71,260,197)(72,261,198)(73,262,199)(74,263,200)(75,264,201)(76,265,202)(77,266,203)(78,267,204)(79,268,205)(80,269,206)(81,270,207)(82,271,208)(83,272,209)(84,273,210)(85,274,106)(86,275,107)(87,276,108)(88,277,109)(89,278,110)(90,279,111)(91,280,112)(92,281,113)(93,282,114)(94,283,115)(95,284,116)(96,285,117)(97,286,118)(98,287,119)(99,288,120)(100,289,121)(101,290,122)(102,291,123)(103,292,124)(104,293,125)(105,294,126), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315)>;
G:=Group( (1,295,127)(2,296,128)(3,297,129)(4,298,130)(5,299,131)(6,300,132)(7,301,133)(8,302,134)(9,303,135)(10,304,136)(11,305,137)(12,306,138)(13,307,139)(14,308,140)(15,309,141)(16,310,142)(17,311,143)(18,312,144)(19,313,145)(20,314,146)(21,315,147)(22,211,148)(23,212,149)(24,213,150)(25,214,151)(26,215,152)(27,216,153)(28,217,154)(29,218,155)(30,219,156)(31,220,157)(32,221,158)(33,222,159)(34,223,160)(35,224,161)(36,225,162)(37,226,163)(38,227,164)(39,228,165)(40,229,166)(41,230,167)(42,231,168)(43,232,169)(44,233,170)(45,234,171)(46,235,172)(47,236,173)(48,237,174)(49,238,175)(50,239,176)(51,240,177)(52,241,178)(53,242,179)(54,243,180)(55,244,181)(56,245,182)(57,246,183)(58,247,184)(59,248,185)(60,249,186)(61,250,187)(62,251,188)(63,252,189)(64,253,190)(65,254,191)(66,255,192)(67,256,193)(68,257,194)(69,258,195)(70,259,196)(71,260,197)(72,261,198)(73,262,199)(74,263,200)(75,264,201)(76,265,202)(77,266,203)(78,267,204)(79,268,205)(80,269,206)(81,270,207)(82,271,208)(83,272,209)(84,273,210)(85,274,106)(86,275,107)(87,276,108)(88,277,109)(89,278,110)(90,279,111)(91,280,112)(92,281,113)(93,282,114)(94,283,115)(95,284,116)(96,285,117)(97,286,118)(98,287,119)(99,288,120)(100,289,121)(101,290,122)(102,291,123)(103,292,124)(104,293,125)(105,294,126), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315) );
G=PermutationGroup([[(1,295,127),(2,296,128),(3,297,129),(4,298,130),(5,299,131),(6,300,132),(7,301,133),(8,302,134),(9,303,135),(10,304,136),(11,305,137),(12,306,138),(13,307,139),(14,308,140),(15,309,141),(16,310,142),(17,311,143),(18,312,144),(19,313,145),(20,314,146),(21,315,147),(22,211,148),(23,212,149),(24,213,150),(25,214,151),(26,215,152),(27,216,153),(28,217,154),(29,218,155),(30,219,156),(31,220,157),(32,221,158),(33,222,159),(34,223,160),(35,224,161),(36,225,162),(37,226,163),(38,227,164),(39,228,165),(40,229,166),(41,230,167),(42,231,168),(43,232,169),(44,233,170),(45,234,171),(46,235,172),(47,236,173),(48,237,174),(49,238,175),(50,239,176),(51,240,177),(52,241,178),(53,242,179),(54,243,180),(55,244,181),(56,245,182),(57,246,183),(58,247,184),(59,248,185),(60,249,186),(61,250,187),(62,251,188),(63,252,189),(64,253,190),(65,254,191),(66,255,192),(67,256,193),(68,257,194),(69,258,195),(70,259,196),(71,260,197),(72,261,198),(73,262,199),(74,263,200),(75,264,201),(76,265,202),(77,266,203),(78,267,204),(79,268,205),(80,269,206),(81,270,207),(82,271,208),(83,272,209),(84,273,210),(85,274,106),(86,275,107),(87,276,108),(88,277,109),(89,278,110),(90,279,111),(91,280,112),(92,281,113),(93,282,114),(94,283,115),(95,284,116),(96,285,117),(97,286,118),(98,287,119),(99,288,120),(100,289,121),(101,290,122),(102,291,123),(103,292,124),(104,293,125),(105,294,126)], [(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),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315)]])
315 conjugacy classes
| class | 1 | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 7A | ··· | 7F | 15A | ··· | 15AF | 21A | ··· | 21AV | 35A | ··· | 35X | 105A | ··· | 105GJ |
| order | 1 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 15 | ··· | 15 | 21 | ··· | 21 | 35 | ··· | 35 | 105 | ··· | 105 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
315 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||||
| image | C1 | C3 | C5 | C7 | C15 | C21 | C35 | C105 |
| kernel | C3×C105 | C105 | C3×C21 | C3×C15 | C21 | C15 | C32 | C3 |
| # reps | 1 | 8 | 4 | 6 | 32 | 48 | 24 | 192 |
Matrix representation of C3×C105 ►in GL2(𝔽211) generated by
| 196 | 0 |
| 0 | 1 |
| 45 | 0 |
| 0 | 93 |
G:=sub<GL(2,GF(211))| [196,0,0,1],[45,0,0,93] >;
C3×C105 in GAP, Magma, Sage, TeX
C_3\times C_{105} % in TeX
G:=Group("C3xC105"); // GroupNames label
G:=SmallGroup(315,4);
// by ID
G=gap.SmallGroup(315,4);
# by ID
G:=PCGroup([4,-3,-3,-5,-7]);
// Polycyclic
G:=Group<a,b|a^3=b^105=1,a*b=b*a>;
// generators/relations
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