direct product, abelian, monomial, 3-elementary
Aliases: C3×C117, SmallGroup(351,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C117 |
| C1 — C3×C117 |
| C1 — C3×C117 |
Generators and relations for C3×C117
G = < a,b | a3=b117=1, ab=ba >
Smallest permutation representation of C3×C117
►Regular action on 351 points
Generators in S351
(1 248 219)(2 249 220)(3 250 221)(4 251 222)(5 252 223)(6 253 224)(7 254 225)(8 255 226)(9 256 227)(10 257 228)(11 258 229)(12 259 230)(13 260 231)(14 261 232)(15 262 233)(16 263 234)(17 264 118)(18 265 119)(19 266 120)(20 267 121)(21 268 122)(22 269 123)(23 270 124)(24 271 125)(25 272 126)(26 273 127)(27 274 128)(28 275 129)(29 276 130)(30 277 131)(31 278 132)(32 279 133)(33 280 134)(34 281 135)(35 282 136)(36 283 137)(37 284 138)(38 285 139)(39 286 140)(40 287 141)(41 288 142)(42 289 143)(43 290 144)(44 291 145)(45 292 146)(46 293 147)(47 294 148)(48 295 149)(49 296 150)(50 297 151)(51 298 152)(52 299 153)(53 300 154)(54 301 155)(55 302 156)(56 303 157)(57 304 158)(58 305 159)(59 306 160)(60 307 161)(61 308 162)(62 309 163)(63 310 164)(64 311 165)(65 312 166)(66 313 167)(67 314 168)(68 315 169)(69 316 170)(70 317 171)(71 318 172)(72 319 173)(73 320 174)(74 321 175)(75 322 176)(76 323 177)(77 324 178)(78 325 179)(79 326 180)(80 327 181)(81 328 182)(82 329 183)(83 330 184)(84 331 185)(85 332 186)(86 333 187)(87 334 188)(88 335 189)(89 336 190)(90 337 191)(91 338 192)(92 339 193)(93 340 194)(94 341 195)(95 342 196)(96 343 197)(97 344 198)(98 345 199)(99 346 200)(100 347 201)(101 348 202)(102 349 203)(103 350 204)(104 351 205)(105 235 206)(106 236 207)(107 237 208)(108 238 209)(109 239 210)(110 240 211)(111 241 212)(112 242 213)(113 243 214)(114 244 215)(115 245 216)(116 246 217)(117 247 218)
(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)(235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351)
G:=sub<Sym(351)| (1,248,219)(2,249,220)(3,250,221)(4,251,222)(5,252,223)(6,253,224)(7,254,225)(8,255,226)(9,256,227)(10,257,228)(11,258,229)(12,259,230)(13,260,231)(14,261,232)(15,262,233)(16,263,234)(17,264,118)(18,265,119)(19,266,120)(20,267,121)(21,268,122)(22,269,123)(23,270,124)(24,271,125)(25,272,126)(26,273,127)(27,274,128)(28,275,129)(29,276,130)(30,277,131)(31,278,132)(32,279,133)(33,280,134)(34,281,135)(35,282,136)(36,283,137)(37,284,138)(38,285,139)(39,286,140)(40,287,141)(41,288,142)(42,289,143)(43,290,144)(44,291,145)(45,292,146)(46,293,147)(47,294,148)(48,295,149)(49,296,150)(50,297,151)(51,298,152)(52,299,153)(53,300,154)(54,301,155)(55,302,156)(56,303,157)(57,304,158)(58,305,159)(59,306,160)(60,307,161)(61,308,162)(62,309,163)(63,310,164)(64,311,165)(65,312,166)(66,313,167)(67,314,168)(68,315,169)(69,316,170)(70,317,171)(71,318,172)(72,319,173)(73,320,174)(74,321,175)(75,322,176)(76,323,177)(77,324,178)(78,325,179)(79,326,180)(80,327,181)(81,328,182)(82,329,183)(83,330,184)(84,331,185)(85,332,186)(86,333,187)(87,334,188)(88,335,189)(89,336,190)(90,337,191)(91,338,192)(92,339,193)(93,340,194)(94,341,195)(95,342,196)(96,343,197)(97,344,198)(98,345,199)(99,346,200)(100,347,201)(101,348,202)(102,349,203)(103,350,204)(104,351,205)(105,235,206)(106,236,207)(107,237,208)(108,238,209)(109,239,210)(110,240,211)(111,241,212)(112,242,213)(113,243,214)(114,244,215)(115,245,216)(116,246,217)(117,247,218), (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)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351)>;
G:=Group( (1,248,219)(2,249,220)(3,250,221)(4,251,222)(5,252,223)(6,253,224)(7,254,225)(8,255,226)(9,256,227)(10,257,228)(11,258,229)(12,259,230)(13,260,231)(14,261,232)(15,262,233)(16,263,234)(17,264,118)(18,265,119)(19,266,120)(20,267,121)(21,268,122)(22,269,123)(23,270,124)(24,271,125)(25,272,126)(26,273,127)(27,274,128)(28,275,129)(29,276,130)(30,277,131)(31,278,132)(32,279,133)(33,280,134)(34,281,135)(35,282,136)(36,283,137)(37,284,138)(38,285,139)(39,286,140)(40,287,141)(41,288,142)(42,289,143)(43,290,144)(44,291,145)(45,292,146)(46,293,147)(47,294,148)(48,295,149)(49,296,150)(50,297,151)(51,298,152)(52,299,153)(53,300,154)(54,301,155)(55,302,156)(56,303,157)(57,304,158)(58,305,159)(59,306,160)(60,307,161)(61,308,162)(62,309,163)(63,310,164)(64,311,165)(65,312,166)(66,313,167)(67,314,168)(68,315,169)(69,316,170)(70,317,171)(71,318,172)(72,319,173)(73,320,174)(74,321,175)(75,322,176)(76,323,177)(77,324,178)(78,325,179)(79,326,180)(80,327,181)(81,328,182)(82,329,183)(83,330,184)(84,331,185)(85,332,186)(86,333,187)(87,334,188)(88,335,189)(89,336,190)(90,337,191)(91,338,192)(92,339,193)(93,340,194)(94,341,195)(95,342,196)(96,343,197)(97,344,198)(98,345,199)(99,346,200)(100,347,201)(101,348,202)(102,349,203)(103,350,204)(104,351,205)(105,235,206)(106,236,207)(107,237,208)(108,238,209)(109,239,210)(110,240,211)(111,241,212)(112,242,213)(113,243,214)(114,244,215)(115,245,216)(116,246,217)(117,247,218), (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)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351) );
G=PermutationGroup([[(1,248,219),(2,249,220),(3,250,221),(4,251,222),(5,252,223),(6,253,224),(7,254,225),(8,255,226),(9,256,227),(10,257,228),(11,258,229),(12,259,230),(13,260,231),(14,261,232),(15,262,233),(16,263,234),(17,264,118),(18,265,119),(19,266,120),(20,267,121),(21,268,122),(22,269,123),(23,270,124),(24,271,125),(25,272,126),(26,273,127),(27,274,128),(28,275,129),(29,276,130),(30,277,131),(31,278,132),(32,279,133),(33,280,134),(34,281,135),(35,282,136),(36,283,137),(37,284,138),(38,285,139),(39,286,140),(40,287,141),(41,288,142),(42,289,143),(43,290,144),(44,291,145),(45,292,146),(46,293,147),(47,294,148),(48,295,149),(49,296,150),(50,297,151),(51,298,152),(52,299,153),(53,300,154),(54,301,155),(55,302,156),(56,303,157),(57,304,158),(58,305,159),(59,306,160),(60,307,161),(61,308,162),(62,309,163),(63,310,164),(64,311,165),(65,312,166),(66,313,167),(67,314,168),(68,315,169),(69,316,170),(70,317,171),(71,318,172),(72,319,173),(73,320,174),(74,321,175),(75,322,176),(76,323,177),(77,324,178),(78,325,179),(79,326,180),(80,327,181),(81,328,182),(82,329,183),(83,330,184),(84,331,185),(85,332,186),(86,333,187),(87,334,188),(88,335,189),(89,336,190),(90,337,191),(91,338,192),(92,339,193),(93,340,194),(94,341,195),(95,342,196),(96,343,197),(97,344,198),(98,345,199),(99,346,200),(100,347,201),(101,348,202),(102,349,203),(103,350,204),(104,351,205),(105,235,206),(106,236,207),(107,237,208),(108,238,209),(109,239,210),(110,240,211),(111,241,212),(112,242,213),(113,243,214),(114,244,215),(115,245,216),(116,246,217),(117,247,218)], [(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),(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351)]])
351 conjugacy classes
| class | 1 | 3A | ··· | 3H | 9A | ··· | 9R | 13A | ··· | 13L | 39A | ··· | 39CR | 117A | ··· | 117HH |
| order | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 13 | ··· | 13 | 39 | ··· | 39 | 117 | ··· | 117 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
351 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||||
| image | C1 | C3 | C3 | C9 | C13 | C39 | C39 | C117 |
| kernel | C3×C117 | C117 | C3×C39 | C39 | C3×C9 | C9 | C32 | C3 |
| # reps | 1 | 6 | 2 | 18 | 12 | 72 | 24 | 216 |
Matrix representation of C3×C117 ►in GL2(𝔽937) generated by
| 614 | 0 |
| 0 | 614 |
| 505 | 0 |
| 0 | 676 |
G:=sub<GL(2,GF(937))| [614,0,0,614],[505,0,0,676] >;
C3×C117 in GAP, Magma, Sage, TeX
C_3\times C_{117} % in TeX
G:=Group("C3xC117"); // GroupNames label
G:=SmallGroup(351,9);
// by ID
G=gap.SmallGroup(351,9);
# by ID
G:=PCGroup([4,-3,-3,-13,-3,468]);
// Polycyclic
G:=Group<a,b|a^3=b^117=1,a*b=b*a>;
// generators/relations
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