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G = C60.5Q8order 480 = 25·3·5

5th non-split extension by C60 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.5Q8, C30.21D8, C6.12D40, C30.9Q16, C6.5Dic20, C12.4Dic10, C20.11Dic6, C3⋊C82Dic5, C31(C405C4), C156(C2.D8), C4.2(C15⋊Q8), C60.97(C2×C4), (C2×C30).34D4, (C2×C6).35D20, C52(C6.Q16), C4⋊Dic5.3S3, C30.30(C4⋊C4), C20.101(C4×S3), C10.7(D4⋊S3), (C2×C12).62D10, (C2×C20).284D6, C2.3(C3⋊D40), C6.3(C4⋊Dic5), C605C4.17C2, C4.12(S3×Dic5), C12.5(C2×Dic5), C10.3(C3⋊Q16), C2.3(C3⋊Dic20), (C2×C60).103C22, C2.4(C6.Dic10), C10.13(Dic3⋊C4), C22.17(C3⋊D20), (C5×C3⋊C8)⋊8C4, (C2×C3⋊C8).4D5, (C10×C3⋊C8).5C2, (C2×C4).89(S3×D5), (C3×C4⋊Dic5).3C2, (C2×C10).28(C3⋊D4), SmallGroup(480,66)

Series: Derived Chief Lower central Upper central

C1C60 — C60.5Q8
C1C5C15C30C2×C30C2×C60C3×C4⋊Dic5 — C60.5Q8
C15C30C60 — C60.5Q8
C1C22C2×C4

Generators and relations for C60.5Q8
 G = < a,b,c | a60=1, b4=a30, c2=a45b2, bab-1=a41, cac-1=a19, cbc-1=a30b3 >

Subgroups: 332 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, C12 [×2], C12, C2×C6, C15, C4⋊C4 [×2], C2×C8, Dic5 [×2], C20 [×2], C2×C10, C3⋊C8 [×2], C2×Dic3, C2×C12, C2×C12, C30 [×3], C2.D8, C40 [×2], C2×Dic5 [×2], C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C3×Dic5, Dic15, C60 [×2], C2×C30, C4⋊Dic5, C4⋊Dic5, C2×C40, C6.Q16, C5×C3⋊C8 [×2], C6×Dic5, C2×Dic15, C2×C60, C405C4, C3×C4⋊Dic5, C10×C3⋊C8, C605C4, C60.5Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, D8, Q16, Dic5 [×2], D10, Dic6, C4×S3, C3⋊D4, C2.D8, Dic10, D20, C2×Dic5, Dic3⋊C4, D4⋊S3, C3⋊Q16, S3×D5, D40, Dic20, C4⋊Dic5, C6.Q16, S3×Dic5, C3⋊D20, C15⋊Q8, C405C4, C3⋊D40, C3⋊Dic20, C6.Dic10, C60.5Q8

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

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

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

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

72 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D12E12F15A15B20A···20H30A···30F40A···40P60A···60H
order1222344444455666888810···10121212121212151520···2030···3040···4060···60
size1111222202060602222266662···24420202020442···24···46···64···4

72 irreducible representations

dim11111222222222222222244444444
type+++++-++++--+--++-+-+--++-
imageC1C2C2C2C4S3Q8D4D5D6D8Q16Dic5D10Dic6C4×S3C3⋊D4Dic10D20D40Dic20D4⋊S3C3⋊Q16S3×D5S3×Dic5C15⋊Q8C3⋊D20C3⋊D40C3⋊Dic20
kernelC60.5Q8C3×C4⋊Dic5C10×C3⋊C8C605C4C5×C3⋊C8C4⋊Dic5C60C2×C30C2×C3⋊C8C2×C20C30C30C3⋊C8C2×C12C20C20C2×C10C12C2×C6C6C6C10C10C2×C4C4C4C22C2C2
# reps11114111212242222448811222244

Matrix representation of C60.5Q8 in GL6(𝔽241)

44780000
1631630000
0018918900
0052100
00001240
000010
,
132080000
3390000
001000
000100
000011120
0000131130
,
1661810000
150750000
0012814400
0023611300
0000171140
000010170

G:=sub<GL(6,GF(241))| [44,163,0,0,0,0,78,163,0,0,0,0,0,0,189,52,0,0,0,0,189,1,0,0,0,0,0,0,1,1,0,0,0,0,240,0],[13,33,0,0,0,0,208,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,111,131,0,0,0,0,20,130],[166,150,0,0,0,0,181,75,0,0,0,0,0,0,128,236,0,0,0,0,144,113,0,0,0,0,0,0,171,101,0,0,0,0,140,70] >;

C60.5Q8 in GAP, Magma, Sage, TeX

C_{60}._5Q_8
% in TeX

G:=Group("C60.5Q8");
// GroupNames label

G:=SmallGroup(480,66);
// by ID

G=gap.SmallGroup(480,66);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45*b^2,b*a*b^-1=a^41,c*a*c^-1=a^19,c*b*c^-1=a^30*b^3>;
// generators/relations

׿
×
𝔽