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G = C60.26Q8order 480 = 25·3·5

5th non-split extension by C60 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.26Q8, Dic153C8, C60.208D4, C4.8Dic30, C20.23Dic6, C12.23Dic10, C30.25M4(2), C1513(C4⋊C8), C6.9(C8×D5), (C2×C40).3S3, C2.4(C8×D15), (C2×C24).3D5, (C2×C8).1D15, C10.18(S3×C8), (C2×C120).5C2, C30.46(C2×C8), C55(Dic3⋊C8), (C2×C4).92D30, C30.38(C4⋊C4), (C2×C20).406D6, C6.4(C8⋊D5), C33(C20.8Q8), C22.9(C4×D15), C2.1(C40⋊S3), C10.9(C8⋊S3), (C2×C12).410D10, C4.26(C157D4), (C4×Dic15).5C2, C20.105(C3⋊D4), C12.105(C5⋊D4), (C2×C60).492C22, (C2×Dic15).12C4, C2.1(C30.4Q8), C10.19(Dic3⋊C4), C6.12(C10.D4), (C2×C6).27(C4×D5), (C2×C153C8).9C2, (C2×C10).52(C4×S3), (C2×C30).134(C2×C4), SmallGroup(480,174)

Series: Derived Chief Lower central Upper central

C1C30 — C60.26Q8
C1C5C15C30C60C2×C60C4×Dic15 — C60.26Q8
C15C30 — C60.26Q8
C1C2×C4C2×C8

Generators and relations for C60.26Q8
 G = < a,b,c | a60=1, b4=a30, c2=a15b2, ab=ba, cac-1=a29, cbc-1=a15b3 >

Subgroups: 308 in 76 conjugacy classes, 43 normal (41 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3 [×3], C12 [×2], C2×C6, C15, C42, C2×C8, C2×C8, Dic5 [×3], C20 [×2], C2×C10, C3⋊C8, C24, C2×Dic3 [×2], C2×C12, C30 [×3], C4⋊C8, C52C8, C40, C2×Dic5 [×2], C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, Dic15 [×2], Dic15, C60 [×2], C2×C30, C2×C52C8, C4×Dic5, C2×C40, Dic3⋊C8, C153C8, C120, C2×Dic15 [×2], C2×C60, C20.8Q8, C2×C153C8, C4×Dic15, C2×C120, C60.26Q8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4, Q8, D5, D6, C4⋊C4, C2×C8, M4(2), D10, Dic6, C4×S3, C3⋊D4, D15, C4⋊C8, Dic10, C4×D5, C5⋊D4, S3×C8, C8⋊S3, Dic3⋊C4, D30, C8×D5, C8⋊D5, C10.D4, Dic3⋊C8, Dic30, C4×D15, C157D4, C20.8Q8, C8×D15, C40⋊S3, C30.4Q8, C60.26Q8

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

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

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

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

132 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B6C8A8B8C8D8E8F8G8H10A···10F12A12B12C12D15A15B15C15D20A···20H24A···24H30A···30L40A···40P60A···60P120A···120AF
order1222344444444556668888888810···10121212121515151520···2024···2430···3040···4060···60120···120
size11112111130303030222222222303030302···2222222222···22···22···22···22···22···2

132 irreducible representations

dim111111222222222222222222222222
type++++++-+++-+-+-
imageC1C2C2C2C4C8S3D4Q8D5D6M4(2)D10Dic6C3⋊D4C4×S3D15Dic10C5⋊D4C4×D5S3×C8C8⋊S3D30C8×D5C8⋊D5Dic30C157D4C4×D15C8×D15C40⋊S3
kernelC60.26Q8C2×C153C8C4×Dic15C2×C120C2×Dic15Dic15C2×C40C60C60C2×C24C2×C20C30C2×C12C20C20C2×C10C2×C8C12C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps11114811121222224444444888881616

Matrix representation of C60.26Q8 in GL3(𝔽241) generated by

6400
014181
0601
,
23300
025140
0101216
,
100
0352
0110206
G:=sub<GL(3,GF(241))| [64,0,0,0,14,60,0,181,1],[233,0,0,0,25,101,0,140,216],[1,0,0,0,35,110,0,2,206] >;

C60.26Q8 in GAP, Magma, Sage, TeX

C_{60}._{26}Q_8
% in TeX

G:=Group("C60.26Q8");
// GroupNames label

G:=SmallGroup(480,174);
// by ID

G=gap.SmallGroup(480,174);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,36,100,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^15*b^2,a*b=b*a,c*a*c^-1=a^29,c*b*c^-1=a^15*b^3>;
// generators/relations

׿
×
𝔽