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G = C7×C70order 490 = 2·5·72

Abelian group of type [7,70]

direct product, abelian, monomial, 7-elementary

Aliases: C7×C70, SmallGroup(490,10)

Series: Derived Chief Lower central Upper central

C1 — C7×C70
C1C7C72C7×C35 — 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
(1 195 459 280 377 303 111)(2 196 460 211 378 304 112)(3 197 461 212 379 305 113)(4 198 462 213 380 306 114)(5 199 463 214 381 307 115)(6 200 464 215 382 308 116)(7 201 465 216 383 309 117)(8 202 466 217 384 310 118)(9 203 467 218 385 311 119)(10 204 468 219 386 312 120)(11 205 469 220 387 313 121)(12 206 470 221 388 314 122)(13 207 471 222 389 315 123)(14 208 472 223 390 316 124)(15 209 473 224 391 317 125)(16 210 474 225 392 318 126)(17 141 475 226 393 319 127)(18 142 476 227 394 320 128)(19 143 477 228 395 321 129)(20 144 478 229 396 322 130)(21 145 479 230 397 323 131)(22 146 480 231 398 324 132)(23 147 481 232 399 325 133)(24 148 482 233 400 326 134)(25 149 483 234 401 327 135)(26 150 484 235 402 328 136)(27 151 485 236 403 329 137)(28 152 486 237 404 330 138)(29 153 487 238 405 331 139)(30 154 488 239 406 332 140)(31 155 489 240 407 333 71)(32 156 490 241 408 334 72)(33 157 421 242 409 335 73)(34 158 422 243 410 336 74)(35 159 423 244 411 337 75)(36 160 424 245 412 338 76)(37 161 425 246 413 339 77)(38 162 426 247 414 340 78)(39 163 427 248 415 341 79)(40 164 428 249 416 342 80)(41 165 429 250 417 343 81)(42 166 430 251 418 344 82)(43 167 431 252 419 345 83)(44 168 432 253 420 346 84)(45 169 433 254 351 347 85)(46 170 434 255 352 348 86)(47 171 435 256 353 349 87)(48 172 436 257 354 350 88)(49 173 437 258 355 281 89)(50 174 438 259 356 282 90)(51 175 439 260 357 283 91)(52 176 440 261 358 284 92)(53 177 441 262 359 285 93)(54 178 442 263 360 286 94)(55 179 443 264 361 287 95)(56 180 444 265 362 288 96)(57 181 445 266 363 289 97)(58 182 446 267 364 290 98)(59 183 447 268 365 291 99)(60 184 448 269 366 292 100)(61 185 449 270 367 293 101)(62 186 450 271 368 294 102)(63 187 451 272 369 295 103)(64 188 452 273 370 296 104)(65 189 453 274 371 297 105)(66 190 454 275 372 298 106)(67 191 455 276 373 299 107)(68 192 456 277 374 300 108)(69 193 457 278 375 301 109)(70 194 458 279 376 302 110)
(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)| (1,195,459,280,377,303,111)(2,196,460,211,378,304,112)(3,197,461,212,379,305,113)(4,198,462,213,380,306,114)(5,199,463,214,381,307,115)(6,200,464,215,382,308,116)(7,201,465,216,383,309,117)(8,202,466,217,384,310,118)(9,203,467,218,385,311,119)(10,204,468,219,386,312,120)(11,205,469,220,387,313,121)(12,206,470,221,388,314,122)(13,207,471,222,389,315,123)(14,208,472,223,390,316,124)(15,209,473,224,391,317,125)(16,210,474,225,392,318,126)(17,141,475,226,393,319,127)(18,142,476,227,394,320,128)(19,143,477,228,395,321,129)(20,144,478,229,396,322,130)(21,145,479,230,397,323,131)(22,146,480,231,398,324,132)(23,147,481,232,399,325,133)(24,148,482,233,400,326,134)(25,149,483,234,401,327,135)(26,150,484,235,402,328,136)(27,151,485,236,403,329,137)(28,152,486,237,404,330,138)(29,153,487,238,405,331,139)(30,154,488,239,406,332,140)(31,155,489,240,407,333,71)(32,156,490,241,408,334,72)(33,157,421,242,409,335,73)(34,158,422,243,410,336,74)(35,159,423,244,411,337,75)(36,160,424,245,412,338,76)(37,161,425,246,413,339,77)(38,162,426,247,414,340,78)(39,163,427,248,415,341,79)(40,164,428,249,416,342,80)(41,165,429,250,417,343,81)(42,166,430,251,418,344,82)(43,167,431,252,419,345,83)(44,168,432,253,420,346,84)(45,169,433,254,351,347,85)(46,170,434,255,352,348,86)(47,171,435,256,353,349,87)(48,172,436,257,354,350,88)(49,173,437,258,355,281,89)(50,174,438,259,356,282,90)(51,175,439,260,357,283,91)(52,176,440,261,358,284,92)(53,177,441,262,359,285,93)(54,178,442,263,360,286,94)(55,179,443,264,361,287,95)(56,180,444,265,362,288,96)(57,181,445,266,363,289,97)(58,182,446,267,364,290,98)(59,183,447,268,365,291,99)(60,184,448,269,366,292,100)(61,185,449,270,367,293,101)(62,186,450,271,368,294,102)(63,187,451,272,369,295,103)(64,188,452,273,370,296,104)(65,189,453,274,371,297,105)(66,190,454,275,372,298,106)(67,191,455,276,373,299,107)(68,192,456,277,374,300,108)(69,193,457,278,375,301,109)(70,194,458,279,376,302,110), (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( (1,195,459,280,377,303,111)(2,196,460,211,378,304,112)(3,197,461,212,379,305,113)(4,198,462,213,380,306,114)(5,199,463,214,381,307,115)(6,200,464,215,382,308,116)(7,201,465,216,383,309,117)(8,202,466,217,384,310,118)(9,203,467,218,385,311,119)(10,204,468,219,386,312,120)(11,205,469,220,387,313,121)(12,206,470,221,388,314,122)(13,207,471,222,389,315,123)(14,208,472,223,390,316,124)(15,209,473,224,391,317,125)(16,210,474,225,392,318,126)(17,141,475,226,393,319,127)(18,142,476,227,394,320,128)(19,143,477,228,395,321,129)(20,144,478,229,396,322,130)(21,145,479,230,397,323,131)(22,146,480,231,398,324,132)(23,147,481,232,399,325,133)(24,148,482,233,400,326,134)(25,149,483,234,401,327,135)(26,150,484,235,402,328,136)(27,151,485,236,403,329,137)(28,152,486,237,404,330,138)(29,153,487,238,405,331,139)(30,154,488,239,406,332,140)(31,155,489,240,407,333,71)(32,156,490,241,408,334,72)(33,157,421,242,409,335,73)(34,158,422,243,410,336,74)(35,159,423,244,411,337,75)(36,160,424,245,412,338,76)(37,161,425,246,413,339,77)(38,162,426,247,414,340,78)(39,163,427,248,415,341,79)(40,164,428,249,416,342,80)(41,165,429,250,417,343,81)(42,166,430,251,418,344,82)(43,167,431,252,419,345,83)(44,168,432,253,420,346,84)(45,169,433,254,351,347,85)(46,170,434,255,352,348,86)(47,171,435,256,353,349,87)(48,172,436,257,354,350,88)(49,173,437,258,355,281,89)(50,174,438,259,356,282,90)(51,175,439,260,357,283,91)(52,176,440,261,358,284,92)(53,177,441,262,359,285,93)(54,178,442,263,360,286,94)(55,179,443,264,361,287,95)(56,180,444,265,362,288,96)(57,181,445,266,363,289,97)(58,182,446,267,364,290,98)(59,183,447,268,365,291,99)(60,184,448,269,366,292,100)(61,185,449,270,367,293,101)(62,186,450,271,368,294,102)(63,187,451,272,369,295,103)(64,188,452,273,370,296,104)(65,189,453,274,371,297,105)(66,190,454,275,372,298,106)(67,191,455,276,373,299,107)(68,192,456,277,374,300,108)(69,193,457,278,375,301,109)(70,194,458,279,376,302,110), (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([(1,195,459,280,377,303,111),(2,196,460,211,378,304,112),(3,197,461,212,379,305,113),(4,198,462,213,380,306,114),(5,199,463,214,381,307,115),(6,200,464,215,382,308,116),(7,201,465,216,383,309,117),(8,202,466,217,384,310,118),(9,203,467,218,385,311,119),(10,204,468,219,386,312,120),(11,205,469,220,387,313,121),(12,206,470,221,388,314,122),(13,207,471,222,389,315,123),(14,208,472,223,390,316,124),(15,209,473,224,391,317,125),(16,210,474,225,392,318,126),(17,141,475,226,393,319,127),(18,142,476,227,394,320,128),(19,143,477,228,395,321,129),(20,144,478,229,396,322,130),(21,145,479,230,397,323,131),(22,146,480,231,398,324,132),(23,147,481,232,399,325,133),(24,148,482,233,400,326,134),(25,149,483,234,401,327,135),(26,150,484,235,402,328,136),(27,151,485,236,403,329,137),(28,152,486,237,404,330,138),(29,153,487,238,405,331,139),(30,154,488,239,406,332,140),(31,155,489,240,407,333,71),(32,156,490,241,408,334,72),(33,157,421,242,409,335,73),(34,158,422,243,410,336,74),(35,159,423,244,411,337,75),(36,160,424,245,412,338,76),(37,161,425,246,413,339,77),(38,162,426,247,414,340,78),(39,163,427,248,415,341,79),(40,164,428,249,416,342,80),(41,165,429,250,417,343,81),(42,166,430,251,418,344,82),(43,167,431,252,419,345,83),(44,168,432,253,420,346,84),(45,169,433,254,351,347,85),(46,170,434,255,352,348,86),(47,171,435,256,353,349,87),(48,172,436,257,354,350,88),(49,173,437,258,355,281,89),(50,174,438,259,356,282,90),(51,175,439,260,357,283,91),(52,176,440,261,358,284,92),(53,177,441,262,359,285,93),(54,178,442,263,360,286,94),(55,179,443,264,361,287,95),(56,180,444,265,362,288,96),(57,181,445,266,363,289,97),(58,182,446,267,364,290,98),(59,183,447,268,365,291,99),(60,184,448,269,366,292,100),(61,185,449,270,367,293,101),(62,186,450,271,368,294,102),(63,187,451,272,369,295,103),(64,188,452,273,370,296,104),(65,189,453,274,371,297,105),(66,190,454,275,372,298,106),(67,191,455,276,373,299,107),(68,192,456,277,374,300,108),(69,193,457,278,375,301,109),(70,194,458,279,376,302,110)], [(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 5A5B5C5D7A···7AV10A10B10C10D14A···14AV35A···35GJ70A···70GJ
order1255557···71010101014···1435···3570···70
size1111111···111111···11···11···1

490 irreducible representations

dim11111111
type++
imageC1C2C5C7C10C14C35C70
kernelC7×C70C7×C35C7×C14C70C72C35C14C7
# reps11448448192192

Matrix representation of C7×C70 in GL2(𝔽71) generated by

200
037
,
220
050
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

Subgroup lattice of C7×C70 in TeX

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