direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C58, C58.12C23, C116.20C22, C4.4(C2×C58), (C2×C4).3C58, (C2×C116).9C2, C22.4(C2×C58), C2.2(C22×C58), (C2×C58).15C22, SmallGroup(464,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C58
G = < a,b,c | a58=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C58
►Regular action on 464 points
Generators in S464
(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)
(1 369 310 210)(2 370 311 211)(3 371 312 212)(4 372 313 213)(5 373 314 214)(6 374 315 215)(7 375 316 216)(8 376 317 217)(9 377 318 218)(10 378 319 219)(11 379 320 220)(12 380 321 221)(13 381 322 222)(14 382 323 223)(15 383 324 224)(16 384 325 225)(17 385 326 226)(18 386 327 227)(19 387 328 228)(20 388 329 229)(21 389 330 230)(22 390 331 231)(23 391 332 232)(24 392 333 175)(25 393 334 176)(26 394 335 177)(27 395 336 178)(28 396 337 179)(29 397 338 180)(30 398 339 181)(31 399 340 182)(32 400 341 183)(33 401 342 184)(34 402 343 185)(35 403 344 186)(36 404 345 187)(37 405 346 188)(38 406 347 189)(39 349 348 190)(40 350 291 191)(41 351 292 192)(42 352 293 193)(43 353 294 194)(44 354 295 195)(45 355 296 196)(46 356 297 197)(47 357 298 198)(48 358 299 199)(49 359 300 200)(50 360 301 201)(51 361 302 202)(52 362 303 203)(53 363 304 204)(54 364 305 205)(55 365 306 206)(56 366 307 207)(57 367 308 208)(58 368 309 209)(59 426 145 244)(60 427 146 245)(61 428 147 246)(62 429 148 247)(63 430 149 248)(64 431 150 249)(65 432 151 250)(66 433 152 251)(67 434 153 252)(68 435 154 253)(69 436 155 254)(70 437 156 255)(71 438 157 256)(72 439 158 257)(73 440 159 258)(74 441 160 259)(75 442 161 260)(76 443 162 261)(77 444 163 262)(78 445 164 263)(79 446 165 264)(80 447 166 265)(81 448 167 266)(82 449 168 267)(83 450 169 268)(84 451 170 269)(85 452 171 270)(86 453 172 271)(87 454 173 272)(88 455 174 273)(89 456 117 274)(90 457 118 275)(91 458 119 276)(92 459 120 277)(93 460 121 278)(94 461 122 279)(95 462 123 280)(96 463 124 281)(97 464 125 282)(98 407 126 283)(99 408 127 284)(100 409 128 285)(101 410 129 286)(102 411 130 287)(103 412 131 288)(104 413 132 289)(105 414 133 290)(106 415 134 233)(107 416 135 234)(108 417 136 235)(109 418 137 236)(110 419 138 237)(111 420 139 238)(112 421 140 239)(113 422 141 240)(114 423 142 241)(115 424 143 242)(116 425 144 243)
(1 261 310 443)(2 262 311 444)(3 263 312 445)(4 264 313 446)(5 265 314 447)(6 266 315 448)(7 267 316 449)(8 268 317 450)(9 269 318 451)(10 270 319 452)(11 271 320 453)(12 272 321 454)(13 273 322 455)(14 274 323 456)(15 275 324 457)(16 276 325 458)(17 277 326 459)(18 278 327 460)(19 279 328 461)(20 280 329 462)(21 281 330 463)(22 282 331 464)(23 283 332 407)(24 284 333 408)(25 285 334 409)(26 286 335 410)(27 287 336 411)(28 288 337 412)(29 289 338 413)(30 290 339 414)(31 233 340 415)(32 234 341 416)(33 235 342 417)(34 236 343 418)(35 237 344 419)(36 238 345 420)(37 239 346 421)(38 240 347 422)(39 241 348 423)(40 242 291 424)(41 243 292 425)(42 244 293 426)(43 245 294 427)(44 246 295 428)(45 247 296 429)(46 248 297 430)(47 249 298 431)(48 250 299 432)(49 251 300 433)(50 252 301 434)(51 253 302 435)(52 254 303 436)(53 255 304 437)(54 256 305 438)(55 257 306 439)(56 258 307 440)(57 259 308 441)(58 260 309 442)(59 352 145 193)(60 353 146 194)(61 354 147 195)(62 355 148 196)(63 356 149 197)(64 357 150 198)(65 358 151 199)(66 359 152 200)(67 360 153 201)(68 361 154 202)(69 362 155 203)(70 363 156 204)(71 364 157 205)(72 365 158 206)(73 366 159 207)(74 367 160 208)(75 368 161 209)(76 369 162 210)(77 370 163 211)(78 371 164 212)(79 372 165 213)(80 373 166 214)(81 374 167 215)(82 375 168 216)(83 376 169 217)(84 377 170 218)(85 378 171 219)(86 379 172 220)(87 380 173 221)(88 381 174 222)(89 382 117 223)(90 383 118 224)(91 384 119 225)(92 385 120 226)(93 386 121 227)(94 387 122 228)(95 388 123 229)(96 389 124 230)(97 390 125 231)(98 391 126 232)(99 392 127 175)(100 393 128 176)(101 394 129 177)(102 395 130 178)(103 396 131 179)(104 397 132 180)(105 398 133 181)(106 399 134 182)(107 400 135 183)(108 401 136 184)(109 402 137 185)(110 403 138 186)(111 404 139 187)(112 405 140 188)(113 406 141 189)(114 349 142 190)(115 350 143 191)(116 351 144 192)
G:=sub<Sym(464)| (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), (1,369,310,210)(2,370,311,211)(3,371,312,212)(4,372,313,213)(5,373,314,214)(6,374,315,215)(7,375,316,216)(8,376,317,217)(9,377,318,218)(10,378,319,219)(11,379,320,220)(12,380,321,221)(13,381,322,222)(14,382,323,223)(15,383,324,224)(16,384,325,225)(17,385,326,226)(18,386,327,227)(19,387,328,228)(20,388,329,229)(21,389,330,230)(22,390,331,231)(23,391,332,232)(24,392,333,175)(25,393,334,176)(26,394,335,177)(27,395,336,178)(28,396,337,179)(29,397,338,180)(30,398,339,181)(31,399,340,182)(32,400,341,183)(33,401,342,184)(34,402,343,185)(35,403,344,186)(36,404,345,187)(37,405,346,188)(38,406,347,189)(39,349,348,190)(40,350,291,191)(41,351,292,192)(42,352,293,193)(43,353,294,194)(44,354,295,195)(45,355,296,196)(46,356,297,197)(47,357,298,198)(48,358,299,199)(49,359,300,200)(50,360,301,201)(51,361,302,202)(52,362,303,203)(53,363,304,204)(54,364,305,205)(55,365,306,206)(56,366,307,207)(57,367,308,208)(58,368,309,209)(59,426,145,244)(60,427,146,245)(61,428,147,246)(62,429,148,247)(63,430,149,248)(64,431,150,249)(65,432,151,250)(66,433,152,251)(67,434,153,252)(68,435,154,253)(69,436,155,254)(70,437,156,255)(71,438,157,256)(72,439,158,257)(73,440,159,258)(74,441,160,259)(75,442,161,260)(76,443,162,261)(77,444,163,262)(78,445,164,263)(79,446,165,264)(80,447,166,265)(81,448,167,266)(82,449,168,267)(83,450,169,268)(84,451,170,269)(85,452,171,270)(86,453,172,271)(87,454,173,272)(88,455,174,273)(89,456,117,274)(90,457,118,275)(91,458,119,276)(92,459,120,277)(93,460,121,278)(94,461,122,279)(95,462,123,280)(96,463,124,281)(97,464,125,282)(98,407,126,283)(99,408,127,284)(100,409,128,285)(101,410,129,286)(102,411,130,287)(103,412,131,288)(104,413,132,289)(105,414,133,290)(106,415,134,233)(107,416,135,234)(108,417,136,235)(109,418,137,236)(110,419,138,237)(111,420,139,238)(112,421,140,239)(113,422,141,240)(114,423,142,241)(115,424,143,242)(116,425,144,243), (1,261,310,443)(2,262,311,444)(3,263,312,445)(4,264,313,446)(5,265,314,447)(6,266,315,448)(7,267,316,449)(8,268,317,450)(9,269,318,451)(10,270,319,452)(11,271,320,453)(12,272,321,454)(13,273,322,455)(14,274,323,456)(15,275,324,457)(16,276,325,458)(17,277,326,459)(18,278,327,460)(19,279,328,461)(20,280,329,462)(21,281,330,463)(22,282,331,464)(23,283,332,407)(24,284,333,408)(25,285,334,409)(26,286,335,410)(27,287,336,411)(28,288,337,412)(29,289,338,413)(30,290,339,414)(31,233,340,415)(32,234,341,416)(33,235,342,417)(34,236,343,418)(35,237,344,419)(36,238,345,420)(37,239,346,421)(38,240,347,422)(39,241,348,423)(40,242,291,424)(41,243,292,425)(42,244,293,426)(43,245,294,427)(44,246,295,428)(45,247,296,429)(46,248,297,430)(47,249,298,431)(48,250,299,432)(49,251,300,433)(50,252,301,434)(51,253,302,435)(52,254,303,436)(53,255,304,437)(54,256,305,438)(55,257,306,439)(56,258,307,440)(57,259,308,441)(58,260,309,442)(59,352,145,193)(60,353,146,194)(61,354,147,195)(62,355,148,196)(63,356,149,197)(64,357,150,198)(65,358,151,199)(66,359,152,200)(67,360,153,201)(68,361,154,202)(69,362,155,203)(70,363,156,204)(71,364,157,205)(72,365,158,206)(73,366,159,207)(74,367,160,208)(75,368,161,209)(76,369,162,210)(77,370,163,211)(78,371,164,212)(79,372,165,213)(80,373,166,214)(81,374,167,215)(82,375,168,216)(83,376,169,217)(84,377,170,218)(85,378,171,219)(86,379,172,220)(87,380,173,221)(88,381,174,222)(89,382,117,223)(90,383,118,224)(91,384,119,225)(92,385,120,226)(93,386,121,227)(94,387,122,228)(95,388,123,229)(96,389,124,230)(97,390,125,231)(98,391,126,232)(99,392,127,175)(100,393,128,176)(101,394,129,177)(102,395,130,178)(103,396,131,179)(104,397,132,180)(105,398,133,181)(106,399,134,182)(107,400,135,183)(108,401,136,184)(109,402,137,185)(110,403,138,186)(111,404,139,187)(112,405,140,188)(113,406,141,189)(114,349,142,190)(115,350,143,191)(116,351,144,192)>;
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), (1,369,310,210)(2,370,311,211)(3,371,312,212)(4,372,313,213)(5,373,314,214)(6,374,315,215)(7,375,316,216)(8,376,317,217)(9,377,318,218)(10,378,319,219)(11,379,320,220)(12,380,321,221)(13,381,322,222)(14,382,323,223)(15,383,324,224)(16,384,325,225)(17,385,326,226)(18,386,327,227)(19,387,328,228)(20,388,329,229)(21,389,330,230)(22,390,331,231)(23,391,332,232)(24,392,333,175)(25,393,334,176)(26,394,335,177)(27,395,336,178)(28,396,337,179)(29,397,338,180)(30,398,339,181)(31,399,340,182)(32,400,341,183)(33,401,342,184)(34,402,343,185)(35,403,344,186)(36,404,345,187)(37,405,346,188)(38,406,347,189)(39,349,348,190)(40,350,291,191)(41,351,292,192)(42,352,293,193)(43,353,294,194)(44,354,295,195)(45,355,296,196)(46,356,297,197)(47,357,298,198)(48,358,299,199)(49,359,300,200)(50,360,301,201)(51,361,302,202)(52,362,303,203)(53,363,304,204)(54,364,305,205)(55,365,306,206)(56,366,307,207)(57,367,308,208)(58,368,309,209)(59,426,145,244)(60,427,146,245)(61,428,147,246)(62,429,148,247)(63,430,149,248)(64,431,150,249)(65,432,151,250)(66,433,152,251)(67,434,153,252)(68,435,154,253)(69,436,155,254)(70,437,156,255)(71,438,157,256)(72,439,158,257)(73,440,159,258)(74,441,160,259)(75,442,161,260)(76,443,162,261)(77,444,163,262)(78,445,164,263)(79,446,165,264)(80,447,166,265)(81,448,167,266)(82,449,168,267)(83,450,169,268)(84,451,170,269)(85,452,171,270)(86,453,172,271)(87,454,173,272)(88,455,174,273)(89,456,117,274)(90,457,118,275)(91,458,119,276)(92,459,120,277)(93,460,121,278)(94,461,122,279)(95,462,123,280)(96,463,124,281)(97,464,125,282)(98,407,126,283)(99,408,127,284)(100,409,128,285)(101,410,129,286)(102,411,130,287)(103,412,131,288)(104,413,132,289)(105,414,133,290)(106,415,134,233)(107,416,135,234)(108,417,136,235)(109,418,137,236)(110,419,138,237)(111,420,139,238)(112,421,140,239)(113,422,141,240)(114,423,142,241)(115,424,143,242)(116,425,144,243), (1,261,310,443)(2,262,311,444)(3,263,312,445)(4,264,313,446)(5,265,314,447)(6,266,315,448)(7,267,316,449)(8,268,317,450)(9,269,318,451)(10,270,319,452)(11,271,320,453)(12,272,321,454)(13,273,322,455)(14,274,323,456)(15,275,324,457)(16,276,325,458)(17,277,326,459)(18,278,327,460)(19,279,328,461)(20,280,329,462)(21,281,330,463)(22,282,331,464)(23,283,332,407)(24,284,333,408)(25,285,334,409)(26,286,335,410)(27,287,336,411)(28,288,337,412)(29,289,338,413)(30,290,339,414)(31,233,340,415)(32,234,341,416)(33,235,342,417)(34,236,343,418)(35,237,344,419)(36,238,345,420)(37,239,346,421)(38,240,347,422)(39,241,348,423)(40,242,291,424)(41,243,292,425)(42,244,293,426)(43,245,294,427)(44,246,295,428)(45,247,296,429)(46,248,297,430)(47,249,298,431)(48,250,299,432)(49,251,300,433)(50,252,301,434)(51,253,302,435)(52,254,303,436)(53,255,304,437)(54,256,305,438)(55,257,306,439)(56,258,307,440)(57,259,308,441)(58,260,309,442)(59,352,145,193)(60,353,146,194)(61,354,147,195)(62,355,148,196)(63,356,149,197)(64,357,150,198)(65,358,151,199)(66,359,152,200)(67,360,153,201)(68,361,154,202)(69,362,155,203)(70,363,156,204)(71,364,157,205)(72,365,158,206)(73,366,159,207)(74,367,160,208)(75,368,161,209)(76,369,162,210)(77,370,163,211)(78,371,164,212)(79,372,165,213)(80,373,166,214)(81,374,167,215)(82,375,168,216)(83,376,169,217)(84,377,170,218)(85,378,171,219)(86,379,172,220)(87,380,173,221)(88,381,174,222)(89,382,117,223)(90,383,118,224)(91,384,119,225)(92,385,120,226)(93,386,121,227)(94,387,122,228)(95,388,123,229)(96,389,124,230)(97,390,125,231)(98,391,126,232)(99,392,127,175)(100,393,128,176)(101,394,129,177)(102,395,130,178)(103,396,131,179)(104,397,132,180)(105,398,133,181)(106,399,134,182)(107,400,135,183)(108,401,136,184)(109,402,137,185)(110,403,138,186)(111,404,139,187)(112,405,140,188)(113,406,141,189)(114,349,142,190)(115,350,143,191)(116,351,144,192) );
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)], [(1,369,310,210),(2,370,311,211),(3,371,312,212),(4,372,313,213),(5,373,314,214),(6,374,315,215),(7,375,316,216),(8,376,317,217),(9,377,318,218),(10,378,319,219),(11,379,320,220),(12,380,321,221),(13,381,322,222),(14,382,323,223),(15,383,324,224),(16,384,325,225),(17,385,326,226),(18,386,327,227),(19,387,328,228),(20,388,329,229),(21,389,330,230),(22,390,331,231),(23,391,332,232),(24,392,333,175),(25,393,334,176),(26,394,335,177),(27,395,336,178),(28,396,337,179),(29,397,338,180),(30,398,339,181),(31,399,340,182),(32,400,341,183),(33,401,342,184),(34,402,343,185),(35,403,344,186),(36,404,345,187),(37,405,346,188),(38,406,347,189),(39,349,348,190),(40,350,291,191),(41,351,292,192),(42,352,293,193),(43,353,294,194),(44,354,295,195),(45,355,296,196),(46,356,297,197),(47,357,298,198),(48,358,299,199),(49,359,300,200),(50,360,301,201),(51,361,302,202),(52,362,303,203),(53,363,304,204),(54,364,305,205),(55,365,306,206),(56,366,307,207),(57,367,308,208),(58,368,309,209),(59,426,145,244),(60,427,146,245),(61,428,147,246),(62,429,148,247),(63,430,149,248),(64,431,150,249),(65,432,151,250),(66,433,152,251),(67,434,153,252),(68,435,154,253),(69,436,155,254),(70,437,156,255),(71,438,157,256),(72,439,158,257),(73,440,159,258),(74,441,160,259),(75,442,161,260),(76,443,162,261),(77,444,163,262),(78,445,164,263),(79,446,165,264),(80,447,166,265),(81,448,167,266),(82,449,168,267),(83,450,169,268),(84,451,170,269),(85,452,171,270),(86,453,172,271),(87,454,173,272),(88,455,174,273),(89,456,117,274),(90,457,118,275),(91,458,119,276),(92,459,120,277),(93,460,121,278),(94,461,122,279),(95,462,123,280),(96,463,124,281),(97,464,125,282),(98,407,126,283),(99,408,127,284),(100,409,128,285),(101,410,129,286),(102,411,130,287),(103,412,131,288),(104,413,132,289),(105,414,133,290),(106,415,134,233),(107,416,135,234),(108,417,136,235),(109,418,137,236),(110,419,138,237),(111,420,139,238),(112,421,140,239),(113,422,141,240),(114,423,142,241),(115,424,143,242),(116,425,144,243)], [(1,261,310,443),(2,262,311,444),(3,263,312,445),(4,264,313,446),(5,265,314,447),(6,266,315,448),(7,267,316,449),(8,268,317,450),(9,269,318,451),(10,270,319,452),(11,271,320,453),(12,272,321,454),(13,273,322,455),(14,274,323,456),(15,275,324,457),(16,276,325,458),(17,277,326,459),(18,278,327,460),(19,279,328,461),(20,280,329,462),(21,281,330,463),(22,282,331,464),(23,283,332,407),(24,284,333,408),(25,285,334,409),(26,286,335,410),(27,287,336,411),(28,288,337,412),(29,289,338,413),(30,290,339,414),(31,233,340,415),(32,234,341,416),(33,235,342,417),(34,236,343,418),(35,237,344,419),(36,238,345,420),(37,239,346,421),(38,240,347,422),(39,241,348,423),(40,242,291,424),(41,243,292,425),(42,244,293,426),(43,245,294,427),(44,246,295,428),(45,247,296,429),(46,248,297,430),(47,249,298,431),(48,250,299,432),(49,251,300,433),(50,252,301,434),(51,253,302,435),(52,254,303,436),(53,255,304,437),(54,256,305,438),(55,257,306,439),(56,258,307,440),(57,259,308,441),(58,260,309,442),(59,352,145,193),(60,353,146,194),(61,354,147,195),(62,355,148,196),(63,356,149,197),(64,357,150,198),(65,358,151,199),(66,359,152,200),(67,360,153,201),(68,361,154,202),(69,362,155,203),(70,363,156,204),(71,364,157,205),(72,365,158,206),(73,366,159,207),(74,367,160,208),(75,368,161,209),(76,369,162,210),(77,370,163,211),(78,371,164,212),(79,372,165,213),(80,373,166,214),(81,374,167,215),(82,375,168,216),(83,376,169,217),(84,377,170,218),(85,378,171,219),(86,379,172,220),(87,380,173,221),(88,381,174,222),(89,382,117,223),(90,383,118,224),(91,384,119,225),(92,385,120,226),(93,386,121,227),(94,387,122,228),(95,388,123,229),(96,389,124,230),(97,390,125,231),(98,391,126,232),(99,392,127,175),(100,393,128,176),(101,394,129,177),(102,395,130,178),(103,396,131,179),(104,397,132,180),(105,398,133,181),(106,399,134,182),(107,400,135,183),(108,401,136,184),(109,402,137,185),(110,403,138,186),(111,404,139,187),(112,405,140,188),(113,406,141,189),(114,349,142,190),(115,350,143,191),(116,351,144,192)]])
290 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 29A | ··· | 29AB | 58A | ··· | 58CF | 116A | ··· | 116FL |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
290 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | ||||
| image | C1 | C2 | C2 | C29 | C58 | C58 | Q8 | Q8×C29 |
| kernel | Q8×C58 | C2×C116 | Q8×C29 | C2×Q8 | C2×C4 | Q8 | C58 | C2 |
| # reps | 1 | 3 | 4 | 28 | 84 | 112 | 2 | 56 |
Matrix representation of Q8×C58 ►in GL3(𝔽233) generated by
| 232 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 46 |
| 232 | 0 | 0 |
| 0 | 149 | 2 |
| 0 | 83 | 84 |
| 232 | 0 | 0 |
| 0 | 160 | 100 |
| 0 | 203 | 73 |
G:=sub<GL(3,GF(233))| [232,0,0,0,46,0,0,0,46],[232,0,0,0,149,83,0,2,84],[232,0,0,0,160,203,0,100,73] >;
Q8×C58 in GAP, Magma, Sage, TeX
Q_8\times C_{58} % in TeX
G:=Group("Q8xC58"); // GroupNames label
G:=SmallGroup(464,47);
// by ID
G=gap.SmallGroup(464,47);
# by ID
G:=PCGroup([5,-2,-2,-2,-29,-2,1160,2341,1166]);
// Polycyclic
G:=Group<a,b,c|a^58=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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