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G = C60.7Q8order 480 = 25·3·5

7th non-split extension by C60 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.7Q8, C20.1Dic6, C30.13SD16, C12.10Dic10, C52(C8⋊Dic3), C154(C4.Q8), C4.4(C15⋊Q8), C12.68(C4×D5), (C2×C20).58D6, (C2×C30).29D4, C52C81Dic3, C6.3(Q8⋊D5), C4⋊Dic3.2D5, C30.25(C4⋊C4), C60.111(C2×C4), (C2×C10).34D12, C6.3(D4.D5), C605C4.22C2, C4.11(D5×Dic3), C31(C20.Q8), C10.6(C24⋊C2), (C2×C12).287D10, C10.9(C4⋊Dic3), C20.25(C2×Dic3), (C2×C60).131C22, C2.3(Dic6⋊D5), C2.3(D12.D5), C6.3(C10.D4), C2.3(C30.Q8), C22.16(C5⋊D12), (C3×C52C8)⋊1C4, (C6×C52C8).4C2, (C2×C52C8).3S3, (C2×C4).139(S3×D5), (C5×C4⋊Dic3).2C2, (C2×C6).28(C5⋊D4), SmallGroup(480,61)

Series: Derived Chief Lower central Upper central

C1C60 — C60.7Q8
C1C5C15C30C2×C30C2×C60C6×C52C8 — C60.7Q8
C15C30C60 — C60.7Q8
C1C22C2×C4

Generators and relations for C60.7Q8
 G = < a,b,c | a60=b4=1, c2=a45b2, bab-1=a11, cac-1=a49, cbc-1=a15b-1 >

Subgroups: 316 in 72 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3 [×2], C12 [×2], C2×C6, C15, C4⋊C4 [×2], C2×C8, Dic5, C20 [×2], C20, C2×C10, C24 [×2], C2×Dic3 [×2], C2×C12, C30 [×3], C4.Q8, C52C8 [×2], C2×Dic5, C2×C20, C2×C20, C4⋊Dic3, C4⋊Dic3, C2×C24, C5×Dic3, Dic15, C60 [×2], C2×C30, C2×C52C8, C4⋊Dic5, C5×C4⋊C4, C8⋊Dic3, C3×C52C8 [×2], C10×Dic3, C2×Dic15, C2×C60, C20.Q8, C6×C52C8, C5×C4⋊Dic3, C605C4, C60.7Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, Dic3 [×2], D6, C4⋊C4, SD16 [×2], D10, Dic6, D12, C2×Dic3, C4.Q8, Dic10, C4×D5, C5⋊D4, C24⋊C2 [×2], C4⋊Dic3, S3×D5, C10.D4, D4.D5, Q8⋊D5, C8⋊Dic3, D5×Dic3, C5⋊D12, C15⋊Q8, C20.Q8, D12.D5, Dic6⋊D5, C30.Q8, C60.7Q8

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

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

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

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

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L24A···24H30A···30F60A···60H
order1222344444455666888810···101212121215152020202020···2024···2430···3060···60
size11112221212606022222101010102···2222244444412···1210···104···44···4

66 irreducible representations

dim111112222222222222244444444
type+++++-++-++-+-+-+--+-+
imageC1C2C2C2C4S3Q8D4D5Dic3D6SD16D10Dic6D12Dic10C4×D5C5⋊D4C24⋊C2S3×D5D4.D5Q8⋊D5D5×Dic3C15⋊Q8C5⋊D12D12.D5Dic6⋊D5
kernelC60.7Q8C6×C52C8C5×C4⋊Dic3C605C4C3×C52C8C2×C52C8C60C2×C30C4⋊Dic3C52C8C2×C20C30C2×C12C20C2×C10C12C12C2×C6C10C2×C4C6C6C4C4C22C2C2
# reps111141112214222444822222244

Matrix representation of C60.7Q8 in GL4(𝔽241) generated by

1989900
1429900
00240190
0051190
,
237800
5521800
001973
0023844
,
669400
14721300
00183128
0019458
G:=sub<GL(4,GF(241))| [198,142,0,0,99,99,0,0,0,0,240,51,0,0,190,190],[23,55,0,0,78,218,0,0,0,0,197,238,0,0,3,44],[66,147,0,0,94,213,0,0,0,0,183,194,0,0,128,58] >;

C60.7Q8 in GAP, Magma, Sage, TeX

C_{60}._7Q_8
% in TeX

G:=Group("C60.7Q8");
// GroupNames label

G:=SmallGroup(480,61);
// by ID

G=gap.SmallGroup(480,61);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,36,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=1,c^2=a^45*b^2,b*a*b^-1=a^11,c*a*c^-1=a^49,c*b*c^-1=a^15*b^-1>;
// generators/relations

׿
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