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G = Q8×C59order 472 = 23·59

Direct product of C59 and Q8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C59, C4.C118, C236.3C2, C118.7C22, C2.2(C2×C118), SmallGroup(472,10)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C59
C1C2C118C236 — Q8×C59
C1C2 — Q8×C59
C1C118 — Q8×C59

Generators and relations for Q8×C59
 G = < a,b,c | a59=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C59
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)
(1 199 287 402)(2 200 288 403)(3 201 289 404)(4 202 290 405)(5 203 291 406)(6 204 292 407)(7 205 293 408)(8 206 294 409)(9 207 295 410)(10 208 237 411)(11 209 238 412)(12 210 239 413)(13 211 240 355)(14 212 241 356)(15 213 242 357)(16 214 243 358)(17 215 244 359)(18 216 245 360)(19 217 246 361)(20 218 247 362)(21 219 248 363)(22 220 249 364)(23 221 250 365)(24 222 251 366)(25 223 252 367)(26 224 253 368)(27 225 254 369)(28 226 255 370)(29 227 256 371)(30 228 257 372)(31 229 258 373)(32 230 259 374)(33 231 260 375)(34 232 261 376)(35 233 262 377)(36 234 263 378)(37 235 264 379)(38 236 265 380)(39 178 266 381)(40 179 267 382)(41 180 268 383)(42 181 269 384)(43 182 270 385)(44 183 271 386)(45 184 272 387)(46 185 273 388)(47 186 274 389)(48 187 275 390)(49 188 276 391)(50 189 277 392)(51 190 278 393)(52 191 279 394)(53 192 280 395)(54 193 281 396)(55 194 282 397)(56 195 283 398)(57 196 284 399)(58 197 285 400)(59 198 286 401)(60 448 330 171)(61 449 331 172)(62 450 332 173)(63 451 333 174)(64 452 334 175)(65 453 335 176)(66 454 336 177)(67 455 337 119)(68 456 338 120)(69 457 339 121)(70 458 340 122)(71 459 341 123)(72 460 342 124)(73 461 343 125)(74 462 344 126)(75 463 345 127)(76 464 346 128)(77 465 347 129)(78 466 348 130)(79 467 349 131)(80 468 350 132)(81 469 351 133)(82 470 352 134)(83 471 353 135)(84 472 354 136)(85 414 296 137)(86 415 297 138)(87 416 298 139)(88 417 299 140)(89 418 300 141)(90 419 301 142)(91 420 302 143)(92 421 303 144)(93 422 304 145)(94 423 305 146)(95 424 306 147)(96 425 307 148)(97 426 308 149)(98 427 309 150)(99 428 310 151)(100 429 311 152)(101 430 312 153)(102 431 313 154)(103 432 314 155)(104 433 315 156)(105 434 316 157)(106 435 317 158)(107 436 318 159)(108 437 319 160)(109 438 320 161)(110 439 321 162)(111 440 322 163)(112 441 323 164)(113 442 324 165)(114 443 325 166)(115 444 326 167)(116 445 327 168)(117 446 328 169)(118 447 329 170)
(1 333 287 63)(2 334 288 64)(3 335 289 65)(4 336 290 66)(5 337 291 67)(6 338 292 68)(7 339 293 69)(8 340 294 70)(9 341 295 71)(10 342 237 72)(11 343 238 73)(12 344 239 74)(13 345 240 75)(14 346 241 76)(15 347 242 77)(16 348 243 78)(17 349 244 79)(18 350 245 80)(19 351 246 81)(20 352 247 82)(21 353 248 83)(22 354 249 84)(23 296 250 85)(24 297 251 86)(25 298 252 87)(26 299 253 88)(27 300 254 89)(28 301 255 90)(29 302 256 91)(30 303 257 92)(31 304 258 93)(32 305 259 94)(33 306 260 95)(34 307 261 96)(35 308 262 97)(36 309 263 98)(37 310 264 99)(38 311 265 100)(39 312 266 101)(40 313 267 102)(41 314 268 103)(42 315 269 104)(43 316 270 105)(44 317 271 106)(45 318 272 107)(46 319 273 108)(47 320 274 109)(48 321 275 110)(49 322 276 111)(50 323 277 112)(51 324 278 113)(52 325 279 114)(53 326 280 115)(54 327 281 116)(55 328 282 117)(56 329 283 118)(57 330 284 60)(58 331 285 61)(59 332 286 62)(119 203 455 406)(120 204 456 407)(121 205 457 408)(122 206 458 409)(123 207 459 410)(124 208 460 411)(125 209 461 412)(126 210 462 413)(127 211 463 355)(128 212 464 356)(129 213 465 357)(130 214 466 358)(131 215 467 359)(132 216 468 360)(133 217 469 361)(134 218 470 362)(135 219 471 363)(136 220 472 364)(137 221 414 365)(138 222 415 366)(139 223 416 367)(140 224 417 368)(141 225 418 369)(142 226 419 370)(143 227 420 371)(144 228 421 372)(145 229 422 373)(146 230 423 374)(147 231 424 375)(148 232 425 376)(149 233 426 377)(150 234 427 378)(151 235 428 379)(152 236 429 380)(153 178 430 381)(154 179 431 382)(155 180 432 383)(156 181 433 384)(157 182 434 385)(158 183 435 386)(159 184 436 387)(160 185 437 388)(161 186 438 389)(162 187 439 390)(163 188 440 391)(164 189 441 392)(165 190 442 393)(166 191 443 394)(167 192 444 395)(168 193 445 396)(169 194 446 397)(170 195 447 398)(171 196 448 399)(172 197 449 400)(173 198 450 401)(174 199 451 402)(175 200 452 403)(176 201 453 404)(177 202 454 405)

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

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

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

295 conjugacy classes

class 1  2 4A4B4C59A···59BF118A···118BF236A···236FR
order1244459···59118···118236···236
size112221···11···12···2

295 irreducible representations

dim111122
type++-
imageC1C2C59C118Q8Q8×C59
kernelQ8×C59C236Q8C4C59C1
# reps1358174158

Matrix representation of Q8×C59 in GL2(𝔽709) generated by

200
020
,
1707
1708
,
417489
307292
G:=sub<GL(2,GF(709))| [20,0,0,20],[1,1,707,708],[417,307,489,292] >;

Q8×C59 in GAP, Magma, Sage, TeX

Q_8\times C_{59}
% in TeX

G:=Group("Q8xC59");
// GroupNames label

G:=SmallGroup(472,10);
// by ID

G=gap.SmallGroup(472,10);
# by ID

G:=PCGroup([4,-2,-2,-59,-2,944,1905,949]);
// Polycyclic

G:=Group<a,b,c|a^59=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Q8×C59 in TeX

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