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G = C605C8order 480 = 25·3·5

1st semidirect product of C60 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C605C8, C4.16D60, C60.25Q8, C20.34D12, C12.34D20, C60.165D4, C4.7Dic30, C42.2D15, C20.22Dic6, C12.22Dic10, C30.35M4(2), C4⋊(C153C8), C203(C3⋊C8), C1510(C4⋊C8), (C4×C20).4S3, (C4×C60).4C2, (C4×C12).4D5, C32(C203C8), C121(C52C8), C54(C12⋊C8), (C2×C60).33C4, C30.59(C2×C8), (C2×C4).90D30, C30.35(C4⋊C4), (C2×C20).404D6, C6.6(C4⋊Dic5), (C2×C12).6Dic5, (C2×C4).3Dic15, C2.1(C605C4), (C2×C12).408D10, (C2×C20).17Dic3, C6.5(C4.Dic5), C2.2(C60.7C4), (C2×C60).490C22, C10.13(C4⋊Dic3), C22.8(C2×Dic15), C10.10(C4.Dic3), C10.16(C2×C3⋊C8), C6.7(C2×C52C8), C2.3(C2×C153C8), (C2×C153C8).8C2, (C2×C30).171(C2×C4), (C2×C6).27(C2×Dic5), (C2×C10).47(C2×Dic3), SmallGroup(480,164)

Series: Derived Chief Lower central Upper central

C1C30 — C605C8
C1C5C15C30C60C2×C60C2×C153C8 — C605C8
C15C30 — C605C8
C1C2×C4C42

Generators and relations for C605C8
 G = < a,b | a60=b8=1, bab-1=a-1 >

Subgroups: 228 in 76 conjugacy classes, 55 normal (45 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C4, C22, C5, C6 [×3], C8 [×2], C2×C4 [×3], C10 [×3], C12 [×2], C12 [×2], C12, C2×C6, C15, C42, C2×C8 [×2], C20 [×2], C20 [×2], C20, C2×C10, C3⋊C8 [×2], C2×C12 [×3], C30 [×3], C4⋊C8, C52C8 [×2], C2×C20 [×3], C2×C3⋊C8 [×2], C4×C12, C60 [×2], C60 [×2], C60, C2×C30, C2×C52C8 [×2], C4×C20, C12⋊C8, C153C8 [×2], C2×C60 [×3], C203C8, C2×C153C8 [×2], C4×C60, C605C8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4, Q8, D5, Dic3 [×2], D6, C4⋊C4, C2×C8, M4(2), Dic5 [×2], D10, C3⋊C8 [×2], Dic6, D12, C2×Dic3, D15, C4⋊C8, C52C8 [×2], Dic10, D20, C2×Dic5, C2×C3⋊C8, C4.Dic3, C4⋊Dic3, Dic15 [×2], D30, C2×C52C8, C4.Dic5, C4⋊Dic5, C12⋊C8, C153C8 [×2], Dic30, D60, C2×Dic15, C203C8, C2×C153C8, C60.7C4, C605C4, C605C8

Smallest permutation representation of C605C8
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)
(1 443 87 416 166 219 284 328)(2 442 88 415 167 218 285 327)(3 441 89 414 168 217 286 326)(4 440 90 413 169 216 287 325)(5 439 91 412 170 215 288 324)(6 438 92 411 171 214 289 323)(7 437 93 410 172 213 290 322)(8 436 94 409 173 212 291 321)(9 435 95 408 174 211 292 320)(10 434 96 407 175 210 293 319)(11 433 97 406 176 209 294 318)(12 432 98 405 177 208 295 317)(13 431 99 404 178 207 296 316)(14 430 100 403 179 206 297 315)(15 429 101 402 180 205 298 314)(16 428 102 401 121 204 299 313)(17 427 103 400 122 203 300 312)(18 426 104 399 123 202 241 311)(19 425 105 398 124 201 242 310)(20 424 106 397 125 200 243 309)(21 423 107 396 126 199 244 308)(22 422 108 395 127 198 245 307)(23 421 109 394 128 197 246 306)(24 480 110 393 129 196 247 305)(25 479 111 392 130 195 248 304)(26 478 112 391 131 194 249 303)(27 477 113 390 132 193 250 302)(28 476 114 389 133 192 251 301)(29 475 115 388 134 191 252 360)(30 474 116 387 135 190 253 359)(31 473 117 386 136 189 254 358)(32 472 118 385 137 188 255 357)(33 471 119 384 138 187 256 356)(34 470 120 383 139 186 257 355)(35 469 61 382 140 185 258 354)(36 468 62 381 141 184 259 353)(37 467 63 380 142 183 260 352)(38 466 64 379 143 182 261 351)(39 465 65 378 144 181 262 350)(40 464 66 377 145 240 263 349)(41 463 67 376 146 239 264 348)(42 462 68 375 147 238 265 347)(43 461 69 374 148 237 266 346)(44 460 70 373 149 236 267 345)(45 459 71 372 150 235 268 344)(46 458 72 371 151 234 269 343)(47 457 73 370 152 233 270 342)(48 456 74 369 153 232 271 341)(49 455 75 368 154 231 272 340)(50 454 76 367 155 230 273 339)(51 453 77 366 156 229 274 338)(52 452 78 365 157 228 275 337)(53 451 79 364 158 227 276 336)(54 450 80 363 159 226 277 335)(55 449 81 362 160 225 278 334)(56 448 82 361 161 224 279 333)(57 447 83 420 162 223 280 332)(58 446 84 419 163 222 281 331)(59 445 85 418 164 221 282 330)(60 444 86 417 165 220 283 329)

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

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

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

132 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B6C8A···8H10A···10F12A···12L15A15B15C15D20A···20X30A···30L60A···60AV
order1222344444444556668···810···1012···121515151520···2030···3060···60
size11112111122222222230···302···22···222222···22···22···2

132 irreducible representations

dim11111222222222222222222222222
type+++++-+-+-+-++-+-+-+
imageC1C2C2C4C8S3D4Q8D5Dic3D6M4(2)Dic5D10C3⋊C8Dic6D12D15C52C8Dic10D20C4.Dic3Dic15D30C4.Dic5C153C8Dic30D60C60.7C4
kernelC605C8C2×C153C8C4×C60C2×C60C60C4×C20C60C60C4×C12C2×C20C2×C20C30C2×C12C2×C12C20C20C20C42C12C12C12C10C2×C4C2×C4C6C4C4C4C2
# reps1214811122124242248444848168816

Matrix representation of C605C8 in GL4(𝔽241) generated by

1989900
1429900
0016357
0011251
,
3816400
12620300
0019057
007751
G:=sub<GL(4,GF(241))| [198,142,0,0,99,99,0,0,0,0,163,112,0,0,57,51],[38,126,0,0,164,203,0,0,0,0,190,77,0,0,57,51] >;

C605C8 in GAP, Magma, Sage, TeX

C_{60}\rtimes_5C_8
% in TeX

G:=Group("C60:5C8");
// GroupNames label

G:=SmallGroup(480,164);
// by ID

G=gap.SmallGroup(480,164);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,100,2693,18822]);
// Polycyclic

G:=Group<a,b|a^60=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

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