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G = C57⋊C8order 456 = 23·3·19

1st semidirect product of C57 and C8 acting via C8/C4=C2

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

Aliases: C571C8, C76.2S3, C4.2D57, C2.Dic57, C38.Dic3, C6.Dic19, C228.2C2, C114.1C4, C12.2D19, C19⋊(C3⋊C8), C3⋊(C19⋊C8), SmallGroup(456,5)

Series: Derived Chief Lower central Upper central

C1C57 — C57⋊C8
C1C19C57C114C228 — C57⋊C8
C57 — C57⋊C8
C1C4

Generators and relations for C57⋊C8
 G = < a,b | a57=b8=1, bab-1=a-1 >

57C8
19C3⋊C8
3C19⋊C8

Smallest permutation representation of C57⋊C8
Regular action on 456 points
Generators in S456
(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)
(1 441 205 329 86 372 153 258)(2 440 206 328 87 371 154 257)(3 439 207 327 88 370 155 256)(4 438 208 326 89 369 156 255)(5 437 209 325 90 368 157 254)(6 436 210 324 91 367 158 253)(7 435 211 323 92 366 159 252)(8 434 212 322 93 365 160 251)(9 433 213 321 94 364 161 250)(10 432 214 320 95 363 162 249)(11 431 215 319 96 362 163 248)(12 430 216 318 97 361 164 247)(13 429 217 317 98 360 165 246)(14 428 218 316 99 359 166 245)(15 427 219 315 100 358 167 244)(16 426 220 314 101 357 168 243)(17 425 221 313 102 356 169 242)(18 424 222 312 103 355 170 241)(19 423 223 311 104 354 171 240)(20 422 224 310 105 353 115 239)(21 421 225 309 106 352 116 238)(22 420 226 308 107 351 117 237)(23 419 227 307 108 350 118 236)(24 418 228 306 109 349 119 235)(25 417 172 305 110 348 120 234)(26 416 173 304 111 347 121 233)(27 415 174 303 112 346 122 232)(28 414 175 302 113 345 123 231)(29 413 176 301 114 344 124 230)(30 412 177 300 58 343 125 229)(31 411 178 299 59 399 126 285)(32 410 179 298 60 398 127 284)(33 409 180 297 61 397 128 283)(34 408 181 296 62 396 129 282)(35 407 182 295 63 395 130 281)(36 406 183 294 64 394 131 280)(37 405 184 293 65 393 132 279)(38 404 185 292 66 392 133 278)(39 403 186 291 67 391 134 277)(40 402 187 290 68 390 135 276)(41 401 188 289 69 389 136 275)(42 400 189 288 70 388 137 274)(43 456 190 287 71 387 138 273)(44 455 191 286 72 386 139 272)(45 454 192 342 73 385 140 271)(46 453 193 341 74 384 141 270)(47 452 194 340 75 383 142 269)(48 451 195 339 76 382 143 268)(49 450 196 338 77 381 144 267)(50 449 197 337 78 380 145 266)(51 448 198 336 79 379 146 265)(52 447 199 335 80 378 147 264)(53 446 200 334 81 377 148 263)(54 445 201 333 82 376 149 262)(55 444 202 332 83 375 150 261)(56 443 203 331 84 374 151 260)(57 442 204 330 85 373 152 259)

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

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

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

120 conjugacy classes

class 1  2  3 4A4B 6 8A8B8C8D12A12B19A···19I38A···38I57A···57R76A···76R114A···114R228A···228AJ
order1234468888121219···1938···3857···5776···76114···114228···228
size11211257575757222···22···22···22···22···22···2

120 irreducible representations

dim1111222222222
type+++-+-+-
imageC1C2C4C8S3Dic3C3⋊C8D19Dic19D57C19⋊C8Dic57C57⋊C8
kernelC57⋊C8C228C114C57C76C38C19C12C6C4C3C2C1
# reps11241129918181836

Matrix representation of C57⋊C8 in GL2(𝔽457) generated by

196299
354132
,
23727
116220
G:=sub<GL(2,GF(457))| [196,354,299,132],[237,116,27,220] >;

C57⋊C8 in GAP, Magma, Sage, TeX

C_{57}\rtimes C_8
% in TeX

G:=Group("C57:C8");
// GroupNames label

G:=SmallGroup(456,5);
// by ID

G=gap.SmallGroup(456,5);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-19,10,26,323,10804]);
// Polycyclic

G:=Group<a,b|a^57=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C57⋊C8 in TeX

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