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G = C61⋊C8order 488 = 23·61

The semidirect product of C61 and C8 acting via C8/C2=C4

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C61⋊C8, C122.C4, Dic61.2C2, C2.(C61⋊C4), SmallGroup(488,3)

Series: Derived Chief Lower central Upper central

C1C61 — C61⋊C8
C1C61C122Dic61 — C61⋊C8
C61 — C61⋊C8
C1C2

Generators and relations for C61⋊C8
 G = < a,b | a61=b8=1, bab-1=a50 >

61C4
61C8

Smallest permutation representation of C61⋊C8
Regular action on 488 points
Generators in S488
(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)
(1 462 184 308 75 369 123 300)(2 473 244 358 76 380 183 289)(3 484 243 347 77 391 182 278)(4 434 242 336 78 402 181 267)(5 445 241 325 79 413 180 256)(6 456 240 314 80 424 179 245)(7 467 239 364 81 374 178 295)(8 478 238 353 82 385 177 284)(9 428 237 342 83 396 176 273)(10 439 236 331 84 407 175 262)(11 450 235 320 85 418 174 251)(12 461 234 309 86 368 173 301)(13 472 233 359 87 379 172 290)(14 483 232 348 88 390 171 279)(15 433 231 337 89 401 170 268)(16 444 230 326 90 412 169 257)(17 455 229 315 91 423 168 246)(18 466 228 365 92 373 167 296)(19 477 227 354 93 384 166 285)(20 488 226 343 94 395 165 274)(21 438 225 332 95 406 164 263)(22 449 224 321 96 417 163 252)(23 460 223 310 97 367 162 302)(24 471 222 360 98 378 161 291)(25 482 221 349 99 389 160 280)(26 432 220 338 100 400 159 269)(27 443 219 327 101 411 158 258)(28 454 218 316 102 422 157 247)(29 465 217 366 103 372 156 297)(30 476 216 355 104 383 155 286)(31 487 215 344 105 394 154 275)(32 437 214 333 106 405 153 264)(33 448 213 322 107 416 152 253)(34 459 212 311 108 427 151 303)(35 470 211 361 109 377 150 292)(36 481 210 350 110 388 149 281)(37 431 209 339 111 399 148 270)(38 442 208 328 112 410 147 259)(39 453 207 317 113 421 146 248)(40 464 206 306 114 371 145 298)(41 475 205 356 115 382 144 287)(42 486 204 345 116 393 143 276)(43 436 203 334 117 404 142 265)(44 447 202 323 118 415 141 254)(45 458 201 312 119 426 140 304)(46 469 200 362 120 376 139 293)(47 480 199 351 121 387 138 282)(48 430 198 340 122 398 137 271)(49 441 197 329 62 409 136 260)(50 452 196 318 63 420 135 249)(51 463 195 307 64 370 134 299)(52 474 194 357 65 381 133 288)(53 485 193 346 66 392 132 277)(54 435 192 335 67 403 131 266)(55 446 191 324 68 414 130 255)(56 457 190 313 69 425 129 305)(57 468 189 363 70 375 128 294)(58 479 188 352 71 386 127 283)(59 429 187 341 72 397 126 272)(60 440 186 330 73 408 125 261)(61 451 185 319 74 419 124 250)

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

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

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

38 conjugacy classes

class 1  2 4A4B8A8B8C8D61A···61O122A···122O
order1244888861···61122···122
size116161616161614···44···4

38 irreducible representations

dim111144
type+++-
imageC1C2C4C8C61⋊C4C61⋊C8
kernelC61⋊C8Dic61C122C61C2C1
# reps11241515

Matrix representation of C61⋊C8 in GL4(𝔽977) generated by

0100
0010
0001
976583192583
,
168849356214
879741918793
861509486724
313355729559
G:=sub<GL(4,GF(977))| [0,0,0,976,1,0,0,583,0,1,0,192,0,0,1,583],[168,879,861,313,849,741,509,355,356,918,486,729,214,793,724,559] >;

C61⋊C8 in GAP, Magma, Sage, TeX

C_{61}\rtimes C_8
% in TeX

G:=Group("C61:C8");
// GroupNames label

G:=SmallGroup(488,3);
// by ID

G=gap.SmallGroup(488,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-61,8,21,1411,3847]);
// Polycyclic

G:=Group<a,b|a^61=b^8=1,b*a*b^-1=a^50>;
// generators/relations

Export

Subgroup lattice of C61⋊C8 in TeX

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