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G = C60.2Q8order 480 = 25·3·5

2nd non-split extension by C60 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.2Q8, C20.6Dic6, C4.2Dic30, C12.6Dic10, C30.25SD16, C153C88C4, C4⋊C4.2D15, C157(C4.Q8), C20.46(C4×S3), C60.76(C2×C4), (C2×C20).65D6, C4.12(C4×D15), C12.14(C4×D5), (C2×C4).33D30, C6.7(Q8⋊D5), C30.37(C4⋊C4), (C2×C30).136D4, (C2×C12).66D10, C6.7(D4.D5), C605C4.11C2, C33(C20.Q8), C54(C12.Q8), C10.7(D4.S3), C2.1(D4.D15), (C2×C60).51C22, C10.7(Q82S3), C2.1(Q82D15), C2.4(C30.4Q8), C10.18(Dic3⋊C4), C6.11(C10.D4), C22.13(C157D4), (C5×C4⋊C4).2S3, (C3×C4⋊C4).2D5, (C15×C4⋊C4).2C2, (C2×C153C8).2C2, (C2×C6).68(C5⋊D4), (C2×C10).68(C3⋊D4), SmallGroup(480,168)

Series: Derived Chief Lower central Upper central

C1C60 — C60.2Q8
C1C5C15C30C2×C30C2×C60C2×C153C8 — C60.2Q8
C15C30C60 — C60.2Q8
C1C22C2×C4C4⋊C4

Generators and relations for C60.2Q8
 G = < a,b,c | a60=b4=1, c2=a45b2, bab-1=a31, cac-1=a29, cbc-1=a15b-1 >

Subgroups: 324 in 72 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3, C12 [×2], C12, C2×C6, C15, C4⋊C4, C4⋊C4, C2×C8, Dic5, C20 [×2], C20, C2×C10, C3⋊C8 [×2], C2×Dic3, C2×C12, C2×C12, C30 [×3], C4.Q8, C52C8 [×2], C2×Dic5, C2×C20, C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, Dic15, C60 [×2], C60, C2×C30, C2×C52C8, C4⋊Dic5, C5×C4⋊C4, C12.Q8, C153C8 [×2], C2×Dic15, C2×C60, C2×C60, C20.Q8, C2×C153C8, C605C4, C15×C4⋊C4, C60.2Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, SD16 [×2], D10, Dic6, C4×S3, C3⋊D4, D15, C4.Q8, Dic10, C4×D5, C5⋊D4, Dic3⋊C4, D4.S3, Q82S3, D30, C10.D4, D4.D5, Q8⋊D5, C12.Q8, Dic30, C4×D15, C157D4, C20.Q8, C30.4Q8, D4.D15, Q82D15, C60.2Q8

Smallest permutation representation of C60.2Q8
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)
(1 220 404 440)(2 191 405 471)(3 222 406 442)(4 193 407 473)(5 224 408 444)(6 195 409 475)(7 226 410 446)(8 197 411 477)(9 228 412 448)(10 199 413 479)(11 230 414 450)(12 201 415 421)(13 232 416 452)(14 203 417 423)(15 234 418 454)(16 205 419 425)(17 236 420 456)(18 207 361 427)(19 238 362 458)(20 209 363 429)(21 240 364 460)(22 211 365 431)(23 182 366 462)(24 213 367 433)(25 184 368 464)(26 215 369 435)(27 186 370 466)(28 217 371 437)(29 188 372 468)(30 219 373 439)(31 190 374 470)(32 221 375 441)(33 192 376 472)(34 223 377 443)(35 194 378 474)(36 225 379 445)(37 196 380 476)(38 227 381 447)(39 198 382 478)(40 229 383 449)(41 200 384 480)(42 231 385 451)(43 202 386 422)(44 233 387 453)(45 204 388 424)(46 235 389 455)(47 206 390 426)(48 237 391 457)(49 208 392 428)(50 239 393 459)(51 210 394 430)(52 181 395 461)(53 212 396 432)(54 183 397 463)(55 214 398 434)(56 185 399 465)(57 216 400 436)(58 187 401 467)(59 218 402 438)(60 189 403 469)(61 158 289 319)(62 129 290 350)(63 160 291 321)(64 131 292 352)(65 162 293 323)(66 133 294 354)(67 164 295 325)(68 135 296 356)(69 166 297 327)(70 137 298 358)(71 168 299 329)(72 139 300 360)(73 170 241 331)(74 141 242 302)(75 172 243 333)(76 143 244 304)(77 174 245 335)(78 145 246 306)(79 176 247 337)(80 147 248 308)(81 178 249 339)(82 149 250 310)(83 180 251 341)(84 151 252 312)(85 122 253 343)(86 153 254 314)(87 124 255 345)(88 155 256 316)(89 126 257 347)(90 157 258 318)(91 128 259 349)(92 159 260 320)(93 130 261 351)(94 161 262 322)(95 132 263 353)(96 163 264 324)(97 134 265 355)(98 165 266 326)(99 136 267 357)(100 167 268 328)(101 138 269 359)(102 169 270 330)(103 140 271 301)(104 171 272 332)(105 142 273 303)(106 173 274 334)(107 144 275 305)(108 175 276 336)(109 146 277 307)(110 177 278 338)(111 148 279 309)(112 179 280 340)(113 150 281 311)(114 121 282 342)(115 152 283 313)(116 123 284 344)(117 154 285 315)(118 125 286 346)(119 156 287 317)(120 127 288 348)
(1 167 389 313 31 137 419 343)(2 136 390 342 32 166 420 312)(3 165 391 311 33 135 361 341)(4 134 392 340 34 164 362 310)(5 163 393 309 35 133 363 339)(6 132 394 338 36 162 364 308)(7 161 395 307 37 131 365 337)(8 130 396 336 38 160 366 306)(9 159 397 305 39 129 367 335)(10 128 398 334 40 158 368 304)(11 157 399 303 41 127 369 333)(12 126 400 332 42 156 370 302)(13 155 401 301 43 125 371 331)(14 124 402 330 44 154 372 360)(15 153 403 359 45 123 373 329)(16 122 404 328 46 152 374 358)(17 151 405 357 47 121 375 327)(18 180 406 326 48 150 376 356)(19 149 407 355 49 179 377 325)(20 178 408 324 50 148 378 354)(21 147 409 353 51 177 379 323)(22 176 410 322 52 146 380 352)(23 145 411 351 53 175 381 321)(24 174 412 320 54 144 382 350)(25 143 413 349 55 173 383 319)(26 172 414 318 56 142 384 348)(27 141 415 347 57 171 385 317)(28 170 416 316 58 140 386 346)(29 139 417 345 59 169 387 315)(30 168 418 314 60 138 388 344)(61 479 274 184 91 449 244 214)(62 448 275 213 92 478 245 183)(63 477 276 182 93 447 246 212)(64 446 277 211 94 476 247 181)(65 475 278 240 95 445 248 210)(66 444 279 209 96 474 249 239)(67 473 280 238 97 443 250 208)(68 442 281 207 98 472 251 237)(69 471 282 236 99 441 252 206)(70 440 283 205 100 470 253 235)(71 469 284 234 101 439 254 204)(72 438 285 203 102 468 255 233)(73 467 286 232 103 437 256 202)(74 436 287 201 104 466 257 231)(75 465 288 230 105 435 258 200)(76 434 289 199 106 464 259 229)(77 463 290 228 107 433 260 198)(78 432 291 197 108 462 261 227)(79 461 292 226 109 431 262 196)(80 430 293 195 110 460 263 225)(81 459 294 224 111 429 264 194)(82 428 295 193 112 458 265 223)(83 457 296 222 113 427 266 192)(84 426 297 191 114 456 267 221)(85 455 298 220 115 425 268 190)(86 424 299 189 116 454 269 219)(87 453 300 218 117 423 270 188)(88 422 241 187 118 452 271 217)(89 451 242 216 119 421 272 186)(90 480 243 185 120 450 273 215)

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222344444455666888810···1012···121515151520···2030···3060···60
size111122244606022222303030302···24···422224···42···24···4

84 irreducible representations

dim11111222222222222222222444444
type+++++-++++-+-+--+-+-+
imageC1C2C2C2C4S3Q8D4D5D6SD16D10Dic6C4×S3C3⋊D4D15Dic10C4×D5C5⋊D4D30Dic30C4×D15C157D4D4.S3Q82S3D4.D5Q8⋊D5D4.D15Q82D15
kernelC60.2Q8C2×C153C8C605C4C15×C4⋊C4C153C8C5×C4⋊C4C60C2×C30C3×C4⋊C4C2×C20C30C2×C12C20C20C2×C10C4⋊C4C12C12C2×C6C2×C4C4C4C22C10C10C6C6C2C2
# reps11114111214222244444888112244

Matrix representation of C60.2Q8 in GL4(𝔽241) generated by

1617300
6817800
00154
00116240
,
418500
15620000
0021213
0023229
,
6316300
2017800
00062
0035203
G:=sub<GL(4,GF(241))| [16,68,0,0,173,178,0,0,0,0,1,116,0,0,54,240],[41,156,0,0,85,200,0,0,0,0,212,232,0,0,13,29],[63,20,0,0,163,178,0,0,0,0,0,35,0,0,62,203] >;

C60.2Q8 in GAP, Magma, Sage, TeX

C_{60}._2Q_8
% in TeX

G:=Group("C60.2Q8");
// GroupNames label

G:=SmallGroup(480,168);
// by ID

G=gap.SmallGroup(480,168);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,36,346,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=1,c^2=a^45*b^2,b*a*b^-1=a^31,c*a*c^-1=a^29,c*b*c^-1=a^15*b^-1>;
// generators/relations

׿
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