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G = C59⋊C8order 472 = 23·59

The semidirect product of C59 and C8 acting via C8/C4=C2

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

Aliases: C59⋊C8, C118.C4, C4.2D59, C2.Dic59, C236.2C2, SmallGroup(472,1)

Series: Derived Chief Lower central Upper central

C1C59 — C59⋊C8
C1C59C118C236 — C59⋊C8
C59 — C59⋊C8
C1C4

Generators and relations for C59⋊C8
 G = < a,b | a59=b8=1, bab-1=a-1 >

59C8

Smallest permutation representation of C59⋊C8
Regular action on 472 points
Generators in S472
(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)
(1 414 214 296 79 355 124 237)(2 472 215 354 80 413 125 295)(3 471 216 353 81 412 126 294)(4 470 217 352 82 411 127 293)(5 469 218 351 83 410 128 292)(6 468 219 350 84 409 129 291)(7 467 220 349 85 408 130 290)(8 466 221 348 86 407 131 289)(9 465 222 347 87 406 132 288)(10 464 223 346 88 405 133 287)(11 463 224 345 89 404 134 286)(12 462 225 344 90 403 135 285)(13 461 226 343 91 402 136 284)(14 460 227 342 92 401 137 283)(15 459 228 341 93 400 138 282)(16 458 229 340 94 399 139 281)(17 457 230 339 95 398 140 280)(18 456 231 338 96 397 141 279)(19 455 232 337 97 396 142 278)(20 454 233 336 98 395 143 277)(21 453 234 335 99 394 144 276)(22 452 235 334 100 393 145 275)(23 451 236 333 101 392 146 274)(24 450 178 332 102 391 147 273)(25 449 179 331 103 390 148 272)(26 448 180 330 104 389 149 271)(27 447 181 329 105 388 150 270)(28 446 182 328 106 387 151 269)(29 445 183 327 107 386 152 268)(30 444 184 326 108 385 153 267)(31 443 185 325 109 384 154 266)(32 442 186 324 110 383 155 265)(33 441 187 323 111 382 156 264)(34 440 188 322 112 381 157 263)(35 439 189 321 113 380 158 262)(36 438 190 320 114 379 159 261)(37 437 191 319 115 378 160 260)(38 436 192 318 116 377 161 259)(39 435 193 317 117 376 162 258)(40 434 194 316 118 375 163 257)(41 433 195 315 60 374 164 256)(42 432 196 314 61 373 165 255)(43 431 197 313 62 372 166 254)(44 430 198 312 63 371 167 253)(45 429 199 311 64 370 168 252)(46 428 200 310 65 369 169 251)(47 427 201 309 66 368 170 250)(48 426 202 308 67 367 171 249)(49 425 203 307 68 366 172 248)(50 424 204 306 69 365 173 247)(51 423 205 305 70 364 174 246)(52 422 206 304 71 363 175 245)(53 421 207 303 72 362 176 244)(54 420 208 302 73 361 177 243)(55 419 209 301 74 360 119 242)(56 418 210 300 75 359 120 241)(57 417 211 299 76 358 121 240)(58 416 212 298 77 357 122 239)(59 415 213 297 78 356 123 238)

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

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

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

124 conjugacy classes

class 1  2 4A4B8A8B8C8D59A···59AC118A···118AC236A···236BF
order1244888859···59118···118236···236
size1111595959592···22···22···2

124 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D59Dic59C59⋊C8
kernelC59⋊C8C236C118C59C4C2C1
# reps1124292958

Matrix representation of C59⋊C8 in GL3(𝔽1889) generated by

100
001
01888957
,
180400
0503603
02791386
G:=sub<GL(3,GF(1889))| [1,0,0,0,0,1888,0,1,957],[1804,0,0,0,503,279,0,603,1386] >;

C59⋊C8 in GAP, Magma, Sage, TeX

C_{59}\rtimes C_8
% in TeX

G:=Group("C59:C8");
// GroupNames label

G:=SmallGroup(472,1);
// by ID

G=gap.SmallGroup(472,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-59,8,21,7427]);
// Polycyclic

G:=Group<a,b|a^59=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C59⋊C8 in TeX

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