direct product, abelian, monomial
Aliases: C15×C30, SmallGroup(450,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C15×C30 |
| C1 — C15×C30 |
| C1 — C15×C30 |
Generators and relations for C15×C30
G = < a,b | a15=b30=1, ab=ba >
Subgroups: 96, all normal (8 characteristic)
C1, C2, C3, C5, C6, C32, C10, C15, C3×C6, C52, C30, C3×C15, C5×C10, C5×C15, C3×C30, C5×C30, C152, C15×C30
Quotients: C1, C2, C3, C5, C6, C32, C10, C15, C3×C6, C52, C30, C3×C15, C5×C10, C5×C15, C3×C30, C5×C30, C152, C15×C30
►Regular action on 450 points
Generators in S450
(1 393 123 329 270 334 449 234 77 201 111 38 174 273 382)(2 394 124 330 241 335 450 235 78 202 112 39 175 274 383)(3 395 125 301 242 336 421 236 79 203 113 40 176 275 384)(4 396 126 302 243 337 422 237 80 204 114 41 177 276 385)(5 397 127 303 244 338 423 238 81 205 115 42 178 277 386)(6 398 128 304 245 339 424 239 82 206 116 43 179 278 387)(7 399 129 305 246 340 425 240 83 207 117 44 180 279 388)(8 400 130 306 247 341 426 211 84 208 118 45 151 280 389)(9 401 131 307 248 342 427 212 85 209 119 46 152 281 390)(10 402 132 308 249 343 428 213 86 210 120 47 153 282 361)(11 403 133 309 250 344 429 214 87 181 91 48 154 283 362)(12 404 134 310 251 345 430 215 88 182 92 49 155 284 363)(13 405 135 311 252 346 431 216 89 183 93 50 156 285 364)(14 406 136 312 253 347 432 217 90 184 94 51 157 286 365)(15 407 137 313 254 348 433 218 61 185 95 52 158 287 366)(16 408 138 314 255 349 434 219 62 186 96 53 159 288 367)(17 409 139 315 256 350 435 220 63 187 97 54 160 289 368)(18 410 140 316 257 351 436 221 64 188 98 55 161 290 369)(19 411 141 317 258 352 437 222 65 189 99 56 162 291 370)(20 412 142 318 259 353 438 223 66 190 100 57 163 292 371)(21 413 143 319 260 354 439 224 67 191 101 58 164 293 372)(22 414 144 320 261 355 440 225 68 192 102 59 165 294 373)(23 415 145 321 262 356 441 226 69 193 103 60 166 295 374)(24 416 146 322 263 357 442 227 70 194 104 31 167 296 375)(25 417 147 323 264 358 443 228 71 195 105 32 168 297 376)(26 418 148 324 265 359 444 229 72 196 106 33 169 298 377)(27 419 149 325 266 360 445 230 73 197 107 34 170 299 378)(28 420 150 326 267 331 446 231 74 198 108 35 171 300 379)(29 391 121 327 268 332 447 232 75 199 109 36 172 271 380)(30 392 122 328 269 333 448 233 76 200 110 37 173 272 381)
(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)(391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450)
G:=sub<Sym(450)| (1,393,123,329,270,334,449,234,77,201,111,38,174,273,382)(2,394,124,330,241,335,450,235,78,202,112,39,175,274,383)(3,395,125,301,242,336,421,236,79,203,113,40,176,275,384)(4,396,126,302,243,337,422,237,80,204,114,41,177,276,385)(5,397,127,303,244,338,423,238,81,205,115,42,178,277,386)(6,398,128,304,245,339,424,239,82,206,116,43,179,278,387)(7,399,129,305,246,340,425,240,83,207,117,44,180,279,388)(8,400,130,306,247,341,426,211,84,208,118,45,151,280,389)(9,401,131,307,248,342,427,212,85,209,119,46,152,281,390)(10,402,132,308,249,343,428,213,86,210,120,47,153,282,361)(11,403,133,309,250,344,429,214,87,181,91,48,154,283,362)(12,404,134,310,251,345,430,215,88,182,92,49,155,284,363)(13,405,135,311,252,346,431,216,89,183,93,50,156,285,364)(14,406,136,312,253,347,432,217,90,184,94,51,157,286,365)(15,407,137,313,254,348,433,218,61,185,95,52,158,287,366)(16,408,138,314,255,349,434,219,62,186,96,53,159,288,367)(17,409,139,315,256,350,435,220,63,187,97,54,160,289,368)(18,410,140,316,257,351,436,221,64,188,98,55,161,290,369)(19,411,141,317,258,352,437,222,65,189,99,56,162,291,370)(20,412,142,318,259,353,438,223,66,190,100,57,163,292,371)(21,413,143,319,260,354,439,224,67,191,101,58,164,293,372)(22,414,144,320,261,355,440,225,68,192,102,59,165,294,373)(23,415,145,321,262,356,441,226,69,193,103,60,166,295,374)(24,416,146,322,263,357,442,227,70,194,104,31,167,296,375)(25,417,147,323,264,358,443,228,71,195,105,32,168,297,376)(26,418,148,324,265,359,444,229,72,196,106,33,169,298,377)(27,419,149,325,266,360,445,230,73,197,107,34,170,299,378)(28,420,150,326,267,331,446,231,74,198,108,35,171,300,379)(29,391,121,327,268,332,447,232,75,199,109,36,172,271,380)(30,392,122,328,269,333,448,233,76,200,110,37,173,272,381), (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)(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450)>;
G:=Group( (1,393,123,329,270,334,449,234,77,201,111,38,174,273,382)(2,394,124,330,241,335,450,235,78,202,112,39,175,274,383)(3,395,125,301,242,336,421,236,79,203,113,40,176,275,384)(4,396,126,302,243,337,422,237,80,204,114,41,177,276,385)(5,397,127,303,244,338,423,238,81,205,115,42,178,277,386)(6,398,128,304,245,339,424,239,82,206,116,43,179,278,387)(7,399,129,305,246,340,425,240,83,207,117,44,180,279,388)(8,400,130,306,247,341,426,211,84,208,118,45,151,280,389)(9,401,131,307,248,342,427,212,85,209,119,46,152,281,390)(10,402,132,308,249,343,428,213,86,210,120,47,153,282,361)(11,403,133,309,250,344,429,214,87,181,91,48,154,283,362)(12,404,134,310,251,345,430,215,88,182,92,49,155,284,363)(13,405,135,311,252,346,431,216,89,183,93,50,156,285,364)(14,406,136,312,253,347,432,217,90,184,94,51,157,286,365)(15,407,137,313,254,348,433,218,61,185,95,52,158,287,366)(16,408,138,314,255,349,434,219,62,186,96,53,159,288,367)(17,409,139,315,256,350,435,220,63,187,97,54,160,289,368)(18,410,140,316,257,351,436,221,64,188,98,55,161,290,369)(19,411,141,317,258,352,437,222,65,189,99,56,162,291,370)(20,412,142,318,259,353,438,223,66,190,100,57,163,292,371)(21,413,143,319,260,354,439,224,67,191,101,58,164,293,372)(22,414,144,320,261,355,440,225,68,192,102,59,165,294,373)(23,415,145,321,262,356,441,226,69,193,103,60,166,295,374)(24,416,146,322,263,357,442,227,70,194,104,31,167,296,375)(25,417,147,323,264,358,443,228,71,195,105,32,168,297,376)(26,418,148,324,265,359,444,229,72,196,106,33,169,298,377)(27,419,149,325,266,360,445,230,73,197,107,34,170,299,378)(28,420,150,326,267,331,446,231,74,198,108,35,171,300,379)(29,391,121,327,268,332,447,232,75,199,109,36,172,271,380)(30,392,122,328,269,333,448,233,76,200,110,37,173,272,381), (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)(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450) );
G=PermutationGroup([[(1,393,123,329,270,334,449,234,77,201,111,38,174,273,382),(2,394,124,330,241,335,450,235,78,202,112,39,175,274,383),(3,395,125,301,242,336,421,236,79,203,113,40,176,275,384),(4,396,126,302,243,337,422,237,80,204,114,41,177,276,385),(5,397,127,303,244,338,423,238,81,205,115,42,178,277,386),(6,398,128,304,245,339,424,239,82,206,116,43,179,278,387),(7,399,129,305,246,340,425,240,83,207,117,44,180,279,388),(8,400,130,306,247,341,426,211,84,208,118,45,151,280,389),(9,401,131,307,248,342,427,212,85,209,119,46,152,281,390),(10,402,132,308,249,343,428,213,86,210,120,47,153,282,361),(11,403,133,309,250,344,429,214,87,181,91,48,154,283,362),(12,404,134,310,251,345,430,215,88,182,92,49,155,284,363),(13,405,135,311,252,346,431,216,89,183,93,50,156,285,364),(14,406,136,312,253,347,432,217,90,184,94,51,157,286,365),(15,407,137,313,254,348,433,218,61,185,95,52,158,287,366),(16,408,138,314,255,349,434,219,62,186,96,53,159,288,367),(17,409,139,315,256,350,435,220,63,187,97,54,160,289,368),(18,410,140,316,257,351,436,221,64,188,98,55,161,290,369),(19,411,141,317,258,352,437,222,65,189,99,56,162,291,370),(20,412,142,318,259,353,438,223,66,190,100,57,163,292,371),(21,413,143,319,260,354,439,224,67,191,101,58,164,293,372),(22,414,144,320,261,355,440,225,68,192,102,59,165,294,373),(23,415,145,321,262,356,441,226,69,193,103,60,166,295,374),(24,416,146,322,263,357,442,227,70,194,104,31,167,296,375),(25,417,147,323,264,358,443,228,71,195,105,32,168,297,376),(26,418,148,324,265,359,444,229,72,196,106,33,169,298,377),(27,419,149,325,266,360,445,230,73,197,107,34,170,299,378),(28,420,150,326,267,331,446,231,74,198,108,35,171,300,379),(29,391,121,327,268,332,447,232,75,199,109,36,172,271,380),(30,392,122,328,269,333,448,233,76,200,110,37,173,272,381)], [(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),(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420),(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450)]])
450 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 5A | ··· | 5X | 6A | ··· | 6H | 10A | ··· | 10X | 15A | ··· | 15GJ | 30A | ··· | 30GJ |
| order | 1 | 2 | 3 | ··· | 3 | 5 | ··· | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
450 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 |
| kernel | C15×C30 | C152 | C5×C30 | C3×C30 | C5×C15 | C3×C15 | C30 | C15 |
| # reps | 1 | 1 | 8 | 24 | 8 | 24 | 192 | 192 |
Matrix representation of C15×C30 ►in GL2(𝔽31) generated by
| 14 | 0 |
| 0 | 9 |
| 23 | 0 |
| 0 | 5 |
G:=sub<GL(2,GF(31))| [14,0,0,9],[23,0,0,5] >;
C15×C30 in GAP, Magma, Sage, TeX
C_{15}\times C_{30} % in TeX
G:=Group("C15xC30"); // GroupNames label
G:=SmallGroup(450,34);
// by ID
G=gap.SmallGroup(450,34);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,-5]);
// Polycyclic
G:=Group<a,b|a^15=b^30=1,a*b=b*a>;
// generators/relations