direct product, abelian, monomial, 7-elementary
Aliases: C7×C70, SmallGroup(490,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C7×C70 |
| C1 — C7×C70 |
| C1 — C7×C70 |
Generators and relations for C7×C70
G = < a,b | a7=b70=1, ab=ba >
Smallest permutation representation of C7×C70
►Regular action on 490 points
Generators in S490
(1 449 329 155 376 133 246)(2 450 330 156 377 134 247)(3 451 331 157 378 135 248)(4 452 332 158 379 136 249)(5 453 333 159 380 137 250)(6 454 334 160 381 138 251)(7 455 335 161 382 139 252)(8 456 336 162 383 140 253)(9 457 337 163 384 71 254)(10 458 338 164 385 72 255)(11 459 339 165 386 73 256)(12 460 340 166 387 74 257)(13 461 341 167 388 75 258)(14 462 342 168 389 76 259)(15 463 343 169 390 77 260)(16 464 344 170 391 78 261)(17 465 345 171 392 79 262)(18 466 346 172 393 80 263)(19 467 347 173 394 81 264)(20 468 348 174 395 82 265)(21 469 349 175 396 83 266)(22 470 350 176 397 84 267)(23 471 281 177 398 85 268)(24 472 282 178 399 86 269)(25 473 283 179 400 87 270)(26 474 284 180 401 88 271)(27 475 285 181 402 89 272)(28 476 286 182 403 90 273)(29 477 287 183 404 91 274)(30 478 288 184 405 92 275)(31 479 289 185 406 93 276)(32 480 290 186 407 94 277)(33 481 291 187 408 95 278)(34 482 292 188 409 96 279)(35 483 293 189 410 97 280)(36 484 294 190 411 98 211)(37 485 295 191 412 99 212)(38 486 296 192 413 100 213)(39 487 297 193 414 101 214)(40 488 298 194 415 102 215)(41 489 299 195 416 103 216)(42 490 300 196 417 104 217)(43 421 301 197 418 105 218)(44 422 302 198 419 106 219)(45 423 303 199 420 107 220)(46 424 304 200 351 108 221)(47 425 305 201 352 109 222)(48 426 306 202 353 110 223)(49 427 307 203 354 111 224)(50 428 308 204 355 112 225)(51 429 309 205 356 113 226)(52 430 310 206 357 114 227)(53 431 311 207 358 115 228)(54 432 312 208 359 116 229)(55 433 313 209 360 117 230)(56 434 314 210 361 118 231)(57 435 315 141 362 119 232)(58 436 316 142 363 120 233)(59 437 317 143 364 121 234)(60 438 318 144 365 122 235)(61 439 319 145 366 123 236)(62 440 320 146 367 124 237)(63 441 321 147 368 125 238)(64 442 322 148 369 126 239)(65 443 323 149 370 127 240)(66 444 324 150 371 128 241)(67 445 325 151 372 129 242)(68 446 326 152 373 130 243)(69 447 327 153 374 131 244)(70 448 328 154 375 132 245)
(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)(421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490)
G:=sub<Sym(490)| (1,449,329,155,376,133,246)(2,450,330,156,377,134,247)(3,451,331,157,378,135,248)(4,452,332,158,379,136,249)(5,453,333,159,380,137,250)(6,454,334,160,381,138,251)(7,455,335,161,382,139,252)(8,456,336,162,383,140,253)(9,457,337,163,384,71,254)(10,458,338,164,385,72,255)(11,459,339,165,386,73,256)(12,460,340,166,387,74,257)(13,461,341,167,388,75,258)(14,462,342,168,389,76,259)(15,463,343,169,390,77,260)(16,464,344,170,391,78,261)(17,465,345,171,392,79,262)(18,466,346,172,393,80,263)(19,467,347,173,394,81,264)(20,468,348,174,395,82,265)(21,469,349,175,396,83,266)(22,470,350,176,397,84,267)(23,471,281,177,398,85,268)(24,472,282,178,399,86,269)(25,473,283,179,400,87,270)(26,474,284,180,401,88,271)(27,475,285,181,402,89,272)(28,476,286,182,403,90,273)(29,477,287,183,404,91,274)(30,478,288,184,405,92,275)(31,479,289,185,406,93,276)(32,480,290,186,407,94,277)(33,481,291,187,408,95,278)(34,482,292,188,409,96,279)(35,483,293,189,410,97,280)(36,484,294,190,411,98,211)(37,485,295,191,412,99,212)(38,486,296,192,413,100,213)(39,487,297,193,414,101,214)(40,488,298,194,415,102,215)(41,489,299,195,416,103,216)(42,490,300,196,417,104,217)(43,421,301,197,418,105,218)(44,422,302,198,419,106,219)(45,423,303,199,420,107,220)(46,424,304,200,351,108,221)(47,425,305,201,352,109,222)(48,426,306,202,353,110,223)(49,427,307,203,354,111,224)(50,428,308,204,355,112,225)(51,429,309,205,356,113,226)(52,430,310,206,357,114,227)(53,431,311,207,358,115,228)(54,432,312,208,359,116,229)(55,433,313,209,360,117,230)(56,434,314,210,361,118,231)(57,435,315,141,362,119,232)(58,436,316,142,363,120,233)(59,437,317,143,364,121,234)(60,438,318,144,365,122,235)(61,439,319,145,366,123,236)(62,440,320,146,367,124,237)(63,441,321,147,368,125,238)(64,442,322,148,369,126,239)(65,443,323,149,370,127,240)(66,444,324,150,371,128,241)(67,445,325,151,372,129,242)(68,446,326,152,373,130,243)(69,447,327,153,374,131,244)(70,448,328,154,375,132,245), (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)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490)>;
G:=Group( (1,449,329,155,376,133,246)(2,450,330,156,377,134,247)(3,451,331,157,378,135,248)(4,452,332,158,379,136,249)(5,453,333,159,380,137,250)(6,454,334,160,381,138,251)(7,455,335,161,382,139,252)(8,456,336,162,383,140,253)(9,457,337,163,384,71,254)(10,458,338,164,385,72,255)(11,459,339,165,386,73,256)(12,460,340,166,387,74,257)(13,461,341,167,388,75,258)(14,462,342,168,389,76,259)(15,463,343,169,390,77,260)(16,464,344,170,391,78,261)(17,465,345,171,392,79,262)(18,466,346,172,393,80,263)(19,467,347,173,394,81,264)(20,468,348,174,395,82,265)(21,469,349,175,396,83,266)(22,470,350,176,397,84,267)(23,471,281,177,398,85,268)(24,472,282,178,399,86,269)(25,473,283,179,400,87,270)(26,474,284,180,401,88,271)(27,475,285,181,402,89,272)(28,476,286,182,403,90,273)(29,477,287,183,404,91,274)(30,478,288,184,405,92,275)(31,479,289,185,406,93,276)(32,480,290,186,407,94,277)(33,481,291,187,408,95,278)(34,482,292,188,409,96,279)(35,483,293,189,410,97,280)(36,484,294,190,411,98,211)(37,485,295,191,412,99,212)(38,486,296,192,413,100,213)(39,487,297,193,414,101,214)(40,488,298,194,415,102,215)(41,489,299,195,416,103,216)(42,490,300,196,417,104,217)(43,421,301,197,418,105,218)(44,422,302,198,419,106,219)(45,423,303,199,420,107,220)(46,424,304,200,351,108,221)(47,425,305,201,352,109,222)(48,426,306,202,353,110,223)(49,427,307,203,354,111,224)(50,428,308,204,355,112,225)(51,429,309,205,356,113,226)(52,430,310,206,357,114,227)(53,431,311,207,358,115,228)(54,432,312,208,359,116,229)(55,433,313,209,360,117,230)(56,434,314,210,361,118,231)(57,435,315,141,362,119,232)(58,436,316,142,363,120,233)(59,437,317,143,364,121,234)(60,438,318,144,365,122,235)(61,439,319,145,366,123,236)(62,440,320,146,367,124,237)(63,441,321,147,368,125,238)(64,442,322,148,369,126,239)(65,443,323,149,370,127,240)(66,444,324,150,371,128,241)(67,445,325,151,372,129,242)(68,446,326,152,373,130,243)(69,447,327,153,374,131,244)(70,448,328,154,375,132,245), (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)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490) );
G=PermutationGroup([[(1,449,329,155,376,133,246),(2,450,330,156,377,134,247),(3,451,331,157,378,135,248),(4,452,332,158,379,136,249),(5,453,333,159,380,137,250),(6,454,334,160,381,138,251),(7,455,335,161,382,139,252),(8,456,336,162,383,140,253),(9,457,337,163,384,71,254),(10,458,338,164,385,72,255),(11,459,339,165,386,73,256),(12,460,340,166,387,74,257),(13,461,341,167,388,75,258),(14,462,342,168,389,76,259),(15,463,343,169,390,77,260),(16,464,344,170,391,78,261),(17,465,345,171,392,79,262),(18,466,346,172,393,80,263),(19,467,347,173,394,81,264),(20,468,348,174,395,82,265),(21,469,349,175,396,83,266),(22,470,350,176,397,84,267),(23,471,281,177,398,85,268),(24,472,282,178,399,86,269),(25,473,283,179,400,87,270),(26,474,284,180,401,88,271),(27,475,285,181,402,89,272),(28,476,286,182,403,90,273),(29,477,287,183,404,91,274),(30,478,288,184,405,92,275),(31,479,289,185,406,93,276),(32,480,290,186,407,94,277),(33,481,291,187,408,95,278),(34,482,292,188,409,96,279),(35,483,293,189,410,97,280),(36,484,294,190,411,98,211),(37,485,295,191,412,99,212),(38,486,296,192,413,100,213),(39,487,297,193,414,101,214),(40,488,298,194,415,102,215),(41,489,299,195,416,103,216),(42,490,300,196,417,104,217),(43,421,301,197,418,105,218),(44,422,302,198,419,106,219),(45,423,303,199,420,107,220),(46,424,304,200,351,108,221),(47,425,305,201,352,109,222),(48,426,306,202,353,110,223),(49,427,307,203,354,111,224),(50,428,308,204,355,112,225),(51,429,309,205,356,113,226),(52,430,310,206,357,114,227),(53,431,311,207,358,115,228),(54,432,312,208,359,116,229),(55,433,313,209,360,117,230),(56,434,314,210,361,118,231),(57,435,315,141,362,119,232),(58,436,316,142,363,120,233),(59,437,317,143,364,121,234),(60,438,318,144,365,122,235),(61,439,319,145,366,123,236),(62,440,320,146,367,124,237),(63,441,321,147,368,125,238),(64,442,322,148,369,126,239),(65,443,323,149,370,127,240),(66,444,324,150,371,128,241),(67,445,325,151,372,129,242),(68,446,326,152,373,130,243),(69,447,327,153,374,131,244),(70,448,328,154,375,132,245)], [(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),(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490)]])
490 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 7A | ··· | 7AV | 10A | 10B | 10C | 10D | 14A | ··· | 14AV | 35A | ··· | 35GJ | 70A | ··· | 70GJ |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 10 | 10 | 10 | 10 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
490 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C5 | C7 | C10 | C14 | C35 | C70 |
| kernel | C7×C70 | C7×C35 | C7×C14 | C70 | C72 | C35 | C14 | C7 |
| # reps | 1 | 1 | 4 | 48 | 4 | 48 | 192 | 192 |
Matrix representation of C7×C70 ►in GL2(𝔽71) generated by
| 20 | 0 |
| 0 | 37 |
| 22 | 0 |
| 0 | 50 |
G:=sub<GL(2,GF(71))| [20,0,0,37],[22,0,0,50] >;
C7×C70 in GAP, Magma, Sage, TeX
C_7\times C_{70} % in TeX
G:=Group("C7xC70"); // GroupNames label
G:=SmallGroup(490,10);
// by ID
G=gap.SmallGroup(490,10);
# by ID
G:=PCGroup([4,-2,-5,-7,-7]);
// Polycyclic
G:=Group<a,b|a^7=b^70=1,a*b=b*a>;
// generators/relations
Export