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G = C608Q8order 480 = 25·3·5

1st semidirect product of C60 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C608Q8, C4.4D60, C42Dic30, C207Dic6, C127Dic10, C20.29D12, C12.29D20, C60.160D4, C42.4D15, (C4×C60).2C2, (C4×C20).2S3, C1510(C4⋊Q8), C2.4(C2×D60), (C4×C12).2D5, C52(C122Q8), (C2×C4).76D30, C32(C202Q8), C6.29(C2×D20), C30.66(C2×Q8), C605C4.6C2, (C2×C20).388D6, C30.258(C2×D4), C10.30(C2×D12), C2.4(C2×Dic30), (C2×C12).391D10, (C2×Dic30).3C2, C6.33(C2×Dic10), C10.33(C2×Dic6), (C2×C60).475C22, (C2×C30).266C23, (C2×Dic15).1C22, C22.34(C22×D15), (C2×C6).262(C22×D5), (C2×C10).261(C22×S3), SmallGroup(480,834)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C608Q8
C1C5C15C30C2×C30C2×Dic15C2×Dic30 — C608Q8
C15C2×C30 — C608Q8
C1C22C42

Generators and relations for C608Q8
 G = < a,b,c | a60=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 708 in 136 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C4 [×4], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10, C10 [×2], Dic3 [×4], C12 [×6], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×4], C20 [×6], C2×C10, Dic6 [×4], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C30, C30 [×2], C4⋊Q8, Dic10 [×4], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C4⋊Dic3 [×4], C4×C12, C2×Dic6 [×2], Dic15 [×4], C60 [×6], C2×C30, C4⋊Dic5 [×4], C4×C20, C2×Dic10 [×2], C122Q8, Dic30 [×4], C2×Dic15 [×4], C2×C60, C2×C60 [×2], C202Q8, C605C4 [×4], C4×C60, C2×Dic30 [×2], C608Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D5, D6 [×3], C2×D4, C2×Q8 [×2], D10 [×3], Dic6 [×4], D12 [×2], C22×S3, D15, C4⋊Q8, Dic10 [×4], D20 [×2], C22×D5, C2×Dic6 [×2], C2×D12, D30 [×3], C2×Dic10 [×2], C2×D20, C122Q8, Dic30 [×4], D60 [×2], C22×D15, C202Q8, C2×Dic30 [×2], C2×D60, C608Q8

Smallest permutation representation of C608Q8
Regular action on 480 points
Generators in S480
(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)
(1 206 477 132)(2 207 478 133)(3 208 479 134)(4 209 480 135)(5 210 421 136)(6 211 422 137)(7 212 423 138)(8 213 424 139)(9 214 425 140)(10 215 426 141)(11 216 427 142)(12 217 428 143)(13 218 429 144)(14 219 430 145)(15 220 431 146)(16 221 432 147)(17 222 433 148)(18 223 434 149)(19 224 435 150)(20 225 436 151)(21 226 437 152)(22 227 438 153)(23 228 439 154)(24 229 440 155)(25 230 441 156)(26 231 442 157)(27 232 443 158)(28 233 444 159)(29 234 445 160)(30 235 446 161)(31 236 447 162)(32 237 448 163)(33 238 449 164)(34 239 450 165)(35 240 451 166)(36 181 452 167)(37 182 453 168)(38 183 454 169)(39 184 455 170)(40 185 456 171)(41 186 457 172)(42 187 458 173)(43 188 459 174)(44 189 460 175)(45 190 461 176)(46 191 462 177)(47 192 463 178)(48 193 464 179)(49 194 465 180)(50 195 466 121)(51 196 467 122)(52 197 468 123)(53 198 469 124)(54 199 470 125)(55 200 471 126)(56 201 472 127)(57 202 473 128)(58 203 474 129)(59 204 475 130)(60 205 476 131)(61 333 290 392)(62 334 291 393)(63 335 292 394)(64 336 293 395)(65 337 294 396)(66 338 295 397)(67 339 296 398)(68 340 297 399)(69 341 298 400)(70 342 299 401)(71 343 300 402)(72 344 241 403)(73 345 242 404)(74 346 243 405)(75 347 244 406)(76 348 245 407)(77 349 246 408)(78 350 247 409)(79 351 248 410)(80 352 249 411)(81 353 250 412)(82 354 251 413)(83 355 252 414)(84 356 253 415)(85 357 254 416)(86 358 255 417)(87 359 256 418)(88 360 257 419)(89 301 258 420)(90 302 259 361)(91 303 260 362)(92 304 261 363)(93 305 262 364)(94 306 263 365)(95 307 264 366)(96 308 265 367)(97 309 266 368)(98 310 267 369)(99 311 268 370)(100 312 269 371)(101 313 270 372)(102 314 271 373)(103 315 272 374)(104 316 273 375)(105 317 274 376)(106 318 275 377)(107 319 276 378)(108 320 277 379)(109 321 278 380)(110 322 279 381)(111 323 280 382)(112 324 281 383)(113 325 282 384)(114 326 283 385)(115 327 284 386)(116 328 285 387)(117 329 286 388)(118 330 287 389)(119 331 288 390)(120 332 289 391)
(1 266 477 97)(2 265 478 96)(3 264 479 95)(4 263 480 94)(5 262 421 93)(6 261 422 92)(7 260 423 91)(8 259 424 90)(9 258 425 89)(10 257 426 88)(11 256 427 87)(12 255 428 86)(13 254 429 85)(14 253 430 84)(15 252 431 83)(16 251 432 82)(17 250 433 81)(18 249 434 80)(19 248 435 79)(20 247 436 78)(21 246 437 77)(22 245 438 76)(23 244 439 75)(24 243 440 74)(25 242 441 73)(26 241 442 72)(27 300 443 71)(28 299 444 70)(29 298 445 69)(30 297 446 68)(31 296 447 67)(32 295 448 66)(33 294 449 65)(34 293 450 64)(35 292 451 63)(36 291 452 62)(37 290 453 61)(38 289 454 120)(39 288 455 119)(40 287 456 118)(41 286 457 117)(42 285 458 116)(43 284 459 115)(44 283 460 114)(45 282 461 113)(46 281 462 112)(47 280 463 111)(48 279 464 110)(49 278 465 109)(50 277 466 108)(51 276 467 107)(52 275 468 106)(53 274 469 105)(54 273 470 104)(55 272 471 103)(56 271 472 102)(57 270 473 101)(58 269 474 100)(59 268 475 99)(60 267 476 98)(121 379 195 320)(122 378 196 319)(123 377 197 318)(124 376 198 317)(125 375 199 316)(126 374 200 315)(127 373 201 314)(128 372 202 313)(129 371 203 312)(130 370 204 311)(131 369 205 310)(132 368 206 309)(133 367 207 308)(134 366 208 307)(135 365 209 306)(136 364 210 305)(137 363 211 304)(138 362 212 303)(139 361 213 302)(140 420 214 301)(141 419 215 360)(142 418 216 359)(143 417 217 358)(144 416 218 357)(145 415 219 356)(146 414 220 355)(147 413 221 354)(148 412 222 353)(149 411 223 352)(150 410 224 351)(151 409 225 350)(152 408 226 349)(153 407 227 348)(154 406 228 347)(155 405 229 346)(156 404 230 345)(157 403 231 344)(158 402 232 343)(159 401 233 342)(160 400 234 341)(161 399 235 340)(162 398 236 339)(163 397 237 338)(164 396 238 337)(165 395 239 336)(166 394 240 335)(167 393 181 334)(168 392 182 333)(169 391 183 332)(170 390 184 331)(171 389 185 330)(172 388 186 329)(173 387 187 328)(174 386 188 327)(175 385 189 326)(176 384 190 325)(177 383 191 324)(178 382 192 323)(179 381 193 322)(180 380 194 321)

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

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

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

126 conjugacy classes

class 1 2A2B2C 3 4A···4F4G4H4I4J5A5B6A6B6C10A···10F12A···12L15A15B15C15D20A···20X30A···30L60A···60AV
order122234···444445566610···1012···121515151520···2030···3060···60
size111122···260606060222222···22···222222···22···22···2

126 irreducible representations

dim111122222222222222
type++++++-+++-++-++-+
imageC1C2C2C2S3D4Q8D5D6D10Dic6D12D15Dic10D20D30Dic30D60
kernelC608Q8C605C4C4×C60C2×Dic30C4×C20C60C60C4×C12C2×C20C2×C12C20C20C42C12C12C2×C4C4C4
# reps1412124236844168123216

Matrix representation of C608Q8 in GL4(𝔽61) generated by

301400
472400
005553
00841
,
32700
542900
0010
0001
,
245300
343700
004955
003412
G:=sub<GL(4,GF(61))| [30,47,0,0,14,24,0,0,0,0,55,8,0,0,53,41],[32,54,0,0,7,29,0,0,0,0,1,0,0,0,0,1],[24,34,0,0,53,37,0,0,0,0,49,34,0,0,55,12] >;

C608Q8 in GAP, Magma, Sage, TeX

C_{60}\rtimes_8Q_8
% in TeX

G:=Group("C60:8Q8");
// GroupNames label

G:=SmallGroup(480,834);
// by ID

G=gap.SmallGroup(480,834);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,120,254,58,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽