direct product, abelian, monomial, 3-elementary
Aliases: C9×C45, SmallGroup(405,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C9×C45 |
| C1 — C9×C45 |
| C1 — C9×C45 |
Generators and relations for C9×C45
G = < a,b | a9=b45=1, ab=ba >
Smallest permutation representation of C9×C45
►Regular action on 405 points
Generators in S405
(1 314 138 201 228 348 382 119 74)(2 315 139 202 229 349 383 120 75)(3 271 140 203 230 350 384 121 76)(4 272 141 204 231 351 385 122 77)(5 273 142 205 232 352 386 123 78)(6 274 143 206 233 353 387 124 79)(7 275 144 207 234 354 388 125 80)(8 276 145 208 235 355 389 126 81)(9 277 146 209 236 356 390 127 82)(10 278 147 210 237 357 391 128 83)(11 279 148 211 238 358 392 129 84)(12 280 149 212 239 359 393 130 85)(13 281 150 213 240 360 394 131 86)(14 282 151 214 241 316 395 132 87)(15 283 152 215 242 317 396 133 88)(16 284 153 216 243 318 397 134 89)(17 285 154 217 244 319 398 135 90)(18 286 155 218 245 320 399 91 46)(19 287 156 219 246 321 400 92 47)(20 288 157 220 247 322 401 93 48)(21 289 158 221 248 323 402 94 49)(22 290 159 222 249 324 403 95 50)(23 291 160 223 250 325 404 96 51)(24 292 161 224 251 326 405 97 52)(25 293 162 225 252 327 361 98 53)(26 294 163 181 253 328 362 99 54)(27 295 164 182 254 329 363 100 55)(28 296 165 183 255 330 364 101 56)(29 297 166 184 256 331 365 102 57)(30 298 167 185 257 332 366 103 58)(31 299 168 186 258 333 367 104 59)(32 300 169 187 259 334 368 105 60)(33 301 170 188 260 335 369 106 61)(34 302 171 189 261 336 370 107 62)(35 303 172 190 262 337 371 108 63)(36 304 173 191 263 338 372 109 64)(37 305 174 192 264 339 373 110 65)(38 306 175 193 265 340 374 111 66)(39 307 176 194 266 341 375 112 67)(40 308 177 195 267 342 376 113 68)(41 309 178 196 268 343 377 114 69)(42 310 179 197 269 344 378 115 70)(43 311 180 198 270 345 379 116 71)(44 312 136 199 226 346 380 117 72)(45 313 137 200 227 347 381 118 73)
(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)(316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405)
G:=sub<Sym(405)| (1,314,138,201,228,348,382,119,74)(2,315,139,202,229,349,383,120,75)(3,271,140,203,230,350,384,121,76)(4,272,141,204,231,351,385,122,77)(5,273,142,205,232,352,386,123,78)(6,274,143,206,233,353,387,124,79)(7,275,144,207,234,354,388,125,80)(8,276,145,208,235,355,389,126,81)(9,277,146,209,236,356,390,127,82)(10,278,147,210,237,357,391,128,83)(11,279,148,211,238,358,392,129,84)(12,280,149,212,239,359,393,130,85)(13,281,150,213,240,360,394,131,86)(14,282,151,214,241,316,395,132,87)(15,283,152,215,242,317,396,133,88)(16,284,153,216,243,318,397,134,89)(17,285,154,217,244,319,398,135,90)(18,286,155,218,245,320,399,91,46)(19,287,156,219,246,321,400,92,47)(20,288,157,220,247,322,401,93,48)(21,289,158,221,248,323,402,94,49)(22,290,159,222,249,324,403,95,50)(23,291,160,223,250,325,404,96,51)(24,292,161,224,251,326,405,97,52)(25,293,162,225,252,327,361,98,53)(26,294,163,181,253,328,362,99,54)(27,295,164,182,254,329,363,100,55)(28,296,165,183,255,330,364,101,56)(29,297,166,184,256,331,365,102,57)(30,298,167,185,257,332,366,103,58)(31,299,168,186,258,333,367,104,59)(32,300,169,187,259,334,368,105,60)(33,301,170,188,260,335,369,106,61)(34,302,171,189,261,336,370,107,62)(35,303,172,190,262,337,371,108,63)(36,304,173,191,263,338,372,109,64)(37,305,174,192,264,339,373,110,65)(38,306,175,193,265,340,374,111,66)(39,307,176,194,266,341,375,112,67)(40,308,177,195,267,342,376,113,68)(41,309,178,196,268,343,377,114,69)(42,310,179,197,269,344,378,115,70)(43,311,180,198,270,345,379,116,71)(44,312,136,199,226,346,380,117,72)(45,313,137,200,227,347,381,118,73), (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)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405)>;
G:=Group( (1,314,138,201,228,348,382,119,74)(2,315,139,202,229,349,383,120,75)(3,271,140,203,230,350,384,121,76)(4,272,141,204,231,351,385,122,77)(5,273,142,205,232,352,386,123,78)(6,274,143,206,233,353,387,124,79)(7,275,144,207,234,354,388,125,80)(8,276,145,208,235,355,389,126,81)(9,277,146,209,236,356,390,127,82)(10,278,147,210,237,357,391,128,83)(11,279,148,211,238,358,392,129,84)(12,280,149,212,239,359,393,130,85)(13,281,150,213,240,360,394,131,86)(14,282,151,214,241,316,395,132,87)(15,283,152,215,242,317,396,133,88)(16,284,153,216,243,318,397,134,89)(17,285,154,217,244,319,398,135,90)(18,286,155,218,245,320,399,91,46)(19,287,156,219,246,321,400,92,47)(20,288,157,220,247,322,401,93,48)(21,289,158,221,248,323,402,94,49)(22,290,159,222,249,324,403,95,50)(23,291,160,223,250,325,404,96,51)(24,292,161,224,251,326,405,97,52)(25,293,162,225,252,327,361,98,53)(26,294,163,181,253,328,362,99,54)(27,295,164,182,254,329,363,100,55)(28,296,165,183,255,330,364,101,56)(29,297,166,184,256,331,365,102,57)(30,298,167,185,257,332,366,103,58)(31,299,168,186,258,333,367,104,59)(32,300,169,187,259,334,368,105,60)(33,301,170,188,260,335,369,106,61)(34,302,171,189,261,336,370,107,62)(35,303,172,190,262,337,371,108,63)(36,304,173,191,263,338,372,109,64)(37,305,174,192,264,339,373,110,65)(38,306,175,193,265,340,374,111,66)(39,307,176,194,266,341,375,112,67)(40,308,177,195,267,342,376,113,68)(41,309,178,196,268,343,377,114,69)(42,310,179,197,269,344,378,115,70)(43,311,180,198,270,345,379,116,71)(44,312,136,199,226,346,380,117,72)(45,313,137,200,227,347,381,118,73), (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)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405) );
G=PermutationGroup([[(1,314,138,201,228,348,382,119,74),(2,315,139,202,229,349,383,120,75),(3,271,140,203,230,350,384,121,76),(4,272,141,204,231,351,385,122,77),(5,273,142,205,232,352,386,123,78),(6,274,143,206,233,353,387,124,79),(7,275,144,207,234,354,388,125,80),(8,276,145,208,235,355,389,126,81),(9,277,146,209,236,356,390,127,82),(10,278,147,210,237,357,391,128,83),(11,279,148,211,238,358,392,129,84),(12,280,149,212,239,359,393,130,85),(13,281,150,213,240,360,394,131,86),(14,282,151,214,241,316,395,132,87),(15,283,152,215,242,317,396,133,88),(16,284,153,216,243,318,397,134,89),(17,285,154,217,244,319,398,135,90),(18,286,155,218,245,320,399,91,46),(19,287,156,219,246,321,400,92,47),(20,288,157,220,247,322,401,93,48),(21,289,158,221,248,323,402,94,49),(22,290,159,222,249,324,403,95,50),(23,291,160,223,250,325,404,96,51),(24,292,161,224,251,326,405,97,52),(25,293,162,225,252,327,361,98,53),(26,294,163,181,253,328,362,99,54),(27,295,164,182,254,329,363,100,55),(28,296,165,183,255,330,364,101,56),(29,297,166,184,256,331,365,102,57),(30,298,167,185,257,332,366,103,58),(31,299,168,186,258,333,367,104,59),(32,300,169,187,259,334,368,105,60),(33,301,170,188,260,335,369,106,61),(34,302,171,189,261,336,370,107,62),(35,303,172,190,262,337,371,108,63),(36,304,173,191,263,338,372,109,64),(37,305,174,192,264,339,373,110,65),(38,306,175,193,265,340,374,111,66),(39,307,176,194,266,341,375,112,67),(40,308,177,195,267,342,376,113,68),(41,309,178,196,268,343,377,114,69),(42,310,179,197,269,344,378,115,70),(43,311,180,198,270,345,379,116,71),(44,312,136,199,226,346,380,117,72),(45,313,137,200,227,347,381,118,73)], [(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),(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360),(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405)]])
405 conjugacy classes
| class | 1 | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 9A | ··· | 9BT | 15A | ··· | 15AF | 45A | ··· | 45KB |
| order | 1 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
405 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||
| image | C1 | C3 | C5 | C9 | C15 | C45 |
| kernel | C9×C45 | C3×C45 | C92 | C45 | C3×C9 | C9 |
| # reps | 1 | 8 | 4 | 72 | 32 | 288 |
Matrix representation of C9×C45 ►in GL2(𝔽181) generated by
| 1 | 0 |
| 0 | 80 |
| 161 | 0 |
| 0 | 43 |
G:=sub<GL(2,GF(181))| [1,0,0,80],[161,0,0,43] >;
C9×C45 in GAP, Magma, Sage, TeX
C_9\times C_{45} % in TeX
G:=Group("C9xC45"); // GroupNames label
G:=SmallGroup(405,2);
// by ID
G=gap.SmallGroup(405,2);
# by ID
G:=PCGroup([5,-3,-3,-5,-3,-3,225,456]);
// Polycyclic
G:=Group<a,b|a^9=b^45=1,a*b=b*a>;
// generators/relations
Export