direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×C15⋊C8, C60⋊2C8, C30.10C42, C15⋊6(C4×C8), C12⋊2(C5⋊C8), C20⋊2(C3⋊C8), C6.3(C4×F5), C30.36(C2×C8), (C2×C60).14C4, C6.8(D5⋊C8), Dic5⋊2(C3⋊C8), (C3×Dic5)⋊4C8, (C2×C12).14F5, Dic5.19(C4×S3), (C4×Dic5).14S3, (C6×Dic5).17C4, C10.10(C4×Dic3), (C2×C20).10Dic3, C2.2(C60.C4), (C2×Dic5).8Dic3, (C12×Dic5).26C2, (C2×Dic5).200D6, (C6×Dic5).259C22, C3⋊1(C4×C5⋊C8), C5⋊2(C4×C3⋊C8), C6.8(C2×C5⋊C8), C2.2(C4×C3⋊F5), C10.3(C2×C3⋊C8), C2.2(C2×C15⋊C8), C22.9(C2×C3⋊F5), (C2×C4).10(C3⋊F5), (C2×C6).33(C2×F5), (C2×C15⋊C8).8C2, (C2×C30).27(C2×C4), (C2×C10).3(C2×Dic3), (C3×Dic5).49(C2×C4), SmallGroup(480,305)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C4×C15⋊C8 |
Generators and relations for C4×C15⋊C8
G = < a,b,c | a4=b15=c8=1, ab=ba, ac=ca, cbc-1=b2 >
Subgroups: 284 in 88 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C30, C4×C8, C5⋊C8, C2×Dic5, C2×C20, C2×C3⋊C8, C4×C12, C3×Dic5, C60, C2×C30, C4×Dic5, C2×C5⋊C8, C4×C3⋊C8, C15⋊C8, C6×Dic5, C2×C60, C4×C5⋊C8, C12×Dic5, C2×C15⋊C8, C4×C15⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, F5, C3⋊C8, C4×S3, C2×Dic3, C4×C8, C5⋊C8, C2×F5, C2×C3⋊C8, C4×Dic3, C3⋊F5, D5⋊C8, C4×F5, C2×C5⋊C8, C4×C3⋊C8, C15⋊C8, C2×C3⋊F5, C4×C5⋊C8, C60.C4, C4×C3⋊F5, C2×C15⋊C8, C4×C15⋊C8
►Regular action on 480 points
Generators in S480
(1 157 41 129)(2 158 42 130)(3 159 43 131)(4 160 44 132)(5 161 45 133)(6 162 31 134)(7 163 32 135)(8 164 33 121)(9 165 34 122)(10 151 35 123)(11 152 36 124)(12 153 37 125)(13 154 38 126)(14 155 39 127)(15 156 40 128)(16 176 51 142)(17 177 52 143)(18 178 53 144)(19 179 54 145)(20 180 55 146)(21 166 56 147)(22 167 57 148)(23 168 58 149)(24 169 59 150)(25 170 60 136)(26 171 46 137)(27 172 47 138)(28 173 48 139)(29 174 49 140)(30 175 50 141)(61 222 96 182)(62 223 97 183)(63 224 98 184)(64 225 99 185)(65 211 100 186)(66 212 101 187)(67 213 102 188)(68 214 103 189)(69 215 104 190)(70 216 105 191)(71 217 91 192)(72 218 92 193)(73 219 93 194)(74 220 94 195)(75 221 95 181)(76 239 111 206)(77 240 112 207)(78 226 113 208)(79 227 114 209)(80 228 115 210)(81 229 116 196)(82 230 117 197)(83 231 118 198)(84 232 119 199)(85 233 120 200)(86 234 106 201)(87 235 107 202)(88 236 108 203)(89 237 109 204)(90 238 110 205)(241 396 276 361)(242 397 277 362)(243 398 278 363)(244 399 279 364)(245 400 280 365)(246 401 281 366)(247 402 282 367)(248 403 283 368)(249 404 284 369)(250 405 285 370)(251 391 271 371)(252 392 272 372)(253 393 273 373)(254 394 274 374)(255 395 275 375)(256 406 296 381)(257 407 297 382)(258 408 298 383)(259 409 299 384)(260 410 300 385)(261 411 286 386)(262 412 287 387)(263 413 288 388)(264 414 289 389)(265 415 290 390)(266 416 291 376)(267 417 292 377)(268 418 293 378)(269 419 294 379)(270 420 295 380)(301 459 331 422)(302 460 332 423)(303 461 333 424)(304 462 334 425)(305 463 335 426)(306 464 336 427)(307 465 337 428)(308 451 338 429)(309 452 339 430)(310 453 340 431)(311 454 341 432)(312 455 342 433)(313 456 343 434)(314 457 344 435)(315 458 345 421)(316 474 346 436)(317 475 347 437)(318 476 348 438)(319 477 349 439)(320 478 350 440)(321 479 351 441)(322 480 352 442)(323 466 353 443)(324 467 354 444)(325 468 355 445)(326 469 356 446)(327 470 357 447)(328 471 358 448)(329 472 359 449)(330 473 360 450)
(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)
(1 466 107 381 24 458 105 371)(2 474 111 383 25 451 94 373)(3 467 115 385 26 459 98 375)(4 475 119 387 27 452 102 362)(5 468 108 389 28 460 91 364)(6 476 112 376 29 453 95 366)(7 469 116 378 30 461 99 368)(8 477 120 380 16 454 103 370)(9 470 109 382 17 462 92 372)(10 478 113 384 18 455 96 374)(11 471 117 386 19 463 100 361)(12 479 106 388 20 456 104 363)(13 472 110 390 21 464 93 365)(14 480 114 377 22 457 97 367)(15 473 118 379 23 465 101 369)(31 438 77 416 49 431 75 401)(32 446 81 418 50 424 64 403)(33 439 85 420 51 432 68 405)(34 447 89 407 52 425 72 392)(35 440 78 409 53 433 61 394)(36 448 82 411 54 426 65 396)(37 441 86 413 55 434 69 398)(38 449 90 415 56 427 73 400)(39 442 79 417 57 435 62 402)(40 450 83 419 58 428 66 404)(41 443 87 406 59 421 70 391)(42 436 76 408 60 429 74 393)(43 444 80 410 46 422 63 395)(44 437 84 412 47 430 67 397)(45 445 88 414 48 423 71 399)(121 319 233 295 142 311 214 285)(122 327 237 297 143 304 218 272)(123 320 226 299 144 312 222 274)(124 328 230 286 145 305 211 276)(125 321 234 288 146 313 215 278)(126 329 238 290 147 306 219 280)(127 322 227 292 148 314 223 282)(128 330 231 294 149 307 212 284)(129 323 235 296 150 315 216 271)(130 316 239 298 136 308 220 273)(131 324 228 300 137 301 224 275)(132 317 232 287 138 309 213 277)(133 325 236 289 139 302 217 279)(134 318 240 291 140 310 221 281)(135 326 229 293 141 303 225 283)(151 350 208 259 178 342 182 254)(152 358 197 261 179 335 186 241)(153 351 201 263 180 343 190 243)(154 359 205 265 166 336 194 245)(155 352 209 267 167 344 183 247)(156 360 198 269 168 337 187 249)(157 353 202 256 169 345 191 251)(158 346 206 258 170 338 195 253)(159 354 210 260 171 331 184 255)(160 347 199 262 172 339 188 242)(161 355 203 264 173 332 192 244)(162 348 207 266 174 340 181 246)(163 356 196 268 175 333 185 248)(164 349 200 270 176 341 189 250)(165 357 204 257 177 334 193 252)
G:=sub<Sym(480)| (1,157,41,129)(2,158,42,130)(3,159,43,131)(4,160,44,132)(5,161,45,133)(6,162,31,134)(7,163,32,135)(8,164,33,121)(9,165,34,122)(10,151,35,123)(11,152,36,124)(12,153,37,125)(13,154,38,126)(14,155,39,127)(15,156,40,128)(16,176,51,142)(17,177,52,143)(18,178,53,144)(19,179,54,145)(20,180,55,146)(21,166,56,147)(22,167,57,148)(23,168,58,149)(24,169,59,150)(25,170,60,136)(26,171,46,137)(27,172,47,138)(28,173,48,139)(29,174,49,140)(30,175,50,141)(61,222,96,182)(62,223,97,183)(63,224,98,184)(64,225,99,185)(65,211,100,186)(66,212,101,187)(67,213,102,188)(68,214,103,189)(69,215,104,190)(70,216,105,191)(71,217,91,192)(72,218,92,193)(73,219,93,194)(74,220,94,195)(75,221,95,181)(76,239,111,206)(77,240,112,207)(78,226,113,208)(79,227,114,209)(80,228,115,210)(81,229,116,196)(82,230,117,197)(83,231,118,198)(84,232,119,199)(85,233,120,200)(86,234,106,201)(87,235,107,202)(88,236,108,203)(89,237,109,204)(90,238,110,205)(241,396,276,361)(242,397,277,362)(243,398,278,363)(244,399,279,364)(245,400,280,365)(246,401,281,366)(247,402,282,367)(248,403,283,368)(249,404,284,369)(250,405,285,370)(251,391,271,371)(252,392,272,372)(253,393,273,373)(254,394,274,374)(255,395,275,375)(256,406,296,381)(257,407,297,382)(258,408,298,383)(259,409,299,384)(260,410,300,385)(261,411,286,386)(262,412,287,387)(263,413,288,388)(264,414,289,389)(265,415,290,390)(266,416,291,376)(267,417,292,377)(268,418,293,378)(269,419,294,379)(270,420,295,380)(301,459,331,422)(302,460,332,423)(303,461,333,424)(304,462,334,425)(305,463,335,426)(306,464,336,427)(307,465,337,428)(308,451,338,429)(309,452,339,430)(310,453,340,431)(311,454,341,432)(312,455,342,433)(313,456,343,434)(314,457,344,435)(315,458,345,421)(316,474,346,436)(317,475,347,437)(318,476,348,438)(319,477,349,439)(320,478,350,440)(321,479,351,441)(322,480,352,442)(323,466,353,443)(324,467,354,444)(325,468,355,445)(326,469,356,446)(327,470,357,447)(328,471,358,448)(329,472,359,449)(330,473,360,450), (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), (1,466,107,381,24,458,105,371)(2,474,111,383,25,451,94,373)(3,467,115,385,26,459,98,375)(4,475,119,387,27,452,102,362)(5,468,108,389,28,460,91,364)(6,476,112,376,29,453,95,366)(7,469,116,378,30,461,99,368)(8,477,120,380,16,454,103,370)(9,470,109,382,17,462,92,372)(10,478,113,384,18,455,96,374)(11,471,117,386,19,463,100,361)(12,479,106,388,20,456,104,363)(13,472,110,390,21,464,93,365)(14,480,114,377,22,457,97,367)(15,473,118,379,23,465,101,369)(31,438,77,416,49,431,75,401)(32,446,81,418,50,424,64,403)(33,439,85,420,51,432,68,405)(34,447,89,407,52,425,72,392)(35,440,78,409,53,433,61,394)(36,448,82,411,54,426,65,396)(37,441,86,413,55,434,69,398)(38,449,90,415,56,427,73,400)(39,442,79,417,57,435,62,402)(40,450,83,419,58,428,66,404)(41,443,87,406,59,421,70,391)(42,436,76,408,60,429,74,393)(43,444,80,410,46,422,63,395)(44,437,84,412,47,430,67,397)(45,445,88,414,48,423,71,399)(121,319,233,295,142,311,214,285)(122,327,237,297,143,304,218,272)(123,320,226,299,144,312,222,274)(124,328,230,286,145,305,211,276)(125,321,234,288,146,313,215,278)(126,329,238,290,147,306,219,280)(127,322,227,292,148,314,223,282)(128,330,231,294,149,307,212,284)(129,323,235,296,150,315,216,271)(130,316,239,298,136,308,220,273)(131,324,228,300,137,301,224,275)(132,317,232,287,138,309,213,277)(133,325,236,289,139,302,217,279)(134,318,240,291,140,310,221,281)(135,326,229,293,141,303,225,283)(151,350,208,259,178,342,182,254)(152,358,197,261,179,335,186,241)(153,351,201,263,180,343,190,243)(154,359,205,265,166,336,194,245)(155,352,209,267,167,344,183,247)(156,360,198,269,168,337,187,249)(157,353,202,256,169,345,191,251)(158,346,206,258,170,338,195,253)(159,354,210,260,171,331,184,255)(160,347,199,262,172,339,188,242)(161,355,203,264,173,332,192,244)(162,348,207,266,174,340,181,246)(163,356,196,268,175,333,185,248)(164,349,200,270,176,341,189,250)(165,357,204,257,177,334,193,252)>;
G:=Group( (1,157,41,129)(2,158,42,130)(3,159,43,131)(4,160,44,132)(5,161,45,133)(6,162,31,134)(7,163,32,135)(8,164,33,121)(9,165,34,122)(10,151,35,123)(11,152,36,124)(12,153,37,125)(13,154,38,126)(14,155,39,127)(15,156,40,128)(16,176,51,142)(17,177,52,143)(18,178,53,144)(19,179,54,145)(20,180,55,146)(21,166,56,147)(22,167,57,148)(23,168,58,149)(24,169,59,150)(25,170,60,136)(26,171,46,137)(27,172,47,138)(28,173,48,139)(29,174,49,140)(30,175,50,141)(61,222,96,182)(62,223,97,183)(63,224,98,184)(64,225,99,185)(65,211,100,186)(66,212,101,187)(67,213,102,188)(68,214,103,189)(69,215,104,190)(70,216,105,191)(71,217,91,192)(72,218,92,193)(73,219,93,194)(74,220,94,195)(75,221,95,181)(76,239,111,206)(77,240,112,207)(78,226,113,208)(79,227,114,209)(80,228,115,210)(81,229,116,196)(82,230,117,197)(83,231,118,198)(84,232,119,199)(85,233,120,200)(86,234,106,201)(87,235,107,202)(88,236,108,203)(89,237,109,204)(90,238,110,205)(241,396,276,361)(242,397,277,362)(243,398,278,363)(244,399,279,364)(245,400,280,365)(246,401,281,366)(247,402,282,367)(248,403,283,368)(249,404,284,369)(250,405,285,370)(251,391,271,371)(252,392,272,372)(253,393,273,373)(254,394,274,374)(255,395,275,375)(256,406,296,381)(257,407,297,382)(258,408,298,383)(259,409,299,384)(260,410,300,385)(261,411,286,386)(262,412,287,387)(263,413,288,388)(264,414,289,389)(265,415,290,390)(266,416,291,376)(267,417,292,377)(268,418,293,378)(269,419,294,379)(270,420,295,380)(301,459,331,422)(302,460,332,423)(303,461,333,424)(304,462,334,425)(305,463,335,426)(306,464,336,427)(307,465,337,428)(308,451,338,429)(309,452,339,430)(310,453,340,431)(311,454,341,432)(312,455,342,433)(313,456,343,434)(314,457,344,435)(315,458,345,421)(316,474,346,436)(317,475,347,437)(318,476,348,438)(319,477,349,439)(320,478,350,440)(321,479,351,441)(322,480,352,442)(323,466,353,443)(324,467,354,444)(325,468,355,445)(326,469,356,446)(327,470,357,447)(328,471,358,448)(329,472,359,449)(330,473,360,450), (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), (1,466,107,381,24,458,105,371)(2,474,111,383,25,451,94,373)(3,467,115,385,26,459,98,375)(4,475,119,387,27,452,102,362)(5,468,108,389,28,460,91,364)(6,476,112,376,29,453,95,366)(7,469,116,378,30,461,99,368)(8,477,120,380,16,454,103,370)(9,470,109,382,17,462,92,372)(10,478,113,384,18,455,96,374)(11,471,117,386,19,463,100,361)(12,479,106,388,20,456,104,363)(13,472,110,390,21,464,93,365)(14,480,114,377,22,457,97,367)(15,473,118,379,23,465,101,369)(31,438,77,416,49,431,75,401)(32,446,81,418,50,424,64,403)(33,439,85,420,51,432,68,405)(34,447,89,407,52,425,72,392)(35,440,78,409,53,433,61,394)(36,448,82,411,54,426,65,396)(37,441,86,413,55,434,69,398)(38,449,90,415,56,427,73,400)(39,442,79,417,57,435,62,402)(40,450,83,419,58,428,66,404)(41,443,87,406,59,421,70,391)(42,436,76,408,60,429,74,393)(43,444,80,410,46,422,63,395)(44,437,84,412,47,430,67,397)(45,445,88,414,48,423,71,399)(121,319,233,295,142,311,214,285)(122,327,237,297,143,304,218,272)(123,320,226,299,144,312,222,274)(124,328,230,286,145,305,211,276)(125,321,234,288,146,313,215,278)(126,329,238,290,147,306,219,280)(127,322,227,292,148,314,223,282)(128,330,231,294,149,307,212,284)(129,323,235,296,150,315,216,271)(130,316,239,298,136,308,220,273)(131,324,228,300,137,301,224,275)(132,317,232,287,138,309,213,277)(133,325,236,289,139,302,217,279)(134,318,240,291,140,310,221,281)(135,326,229,293,141,303,225,283)(151,350,208,259,178,342,182,254)(152,358,197,261,179,335,186,241)(153,351,201,263,180,343,190,243)(154,359,205,265,166,336,194,245)(155,352,209,267,167,344,183,247)(156,360,198,269,168,337,187,249)(157,353,202,256,169,345,191,251)(158,346,206,258,170,338,195,253)(159,354,210,260,171,331,184,255)(160,347,199,262,172,339,188,242)(161,355,203,264,173,332,192,244)(162,348,207,266,174,340,181,246)(163,356,196,268,175,333,185,248)(164,349,200,270,176,341,189,250)(165,357,204,257,177,334,193,252) );
G=PermutationGroup([[(1,157,41,129),(2,158,42,130),(3,159,43,131),(4,160,44,132),(5,161,45,133),(6,162,31,134),(7,163,32,135),(8,164,33,121),(9,165,34,122),(10,151,35,123),(11,152,36,124),(12,153,37,125),(13,154,38,126),(14,155,39,127),(15,156,40,128),(16,176,51,142),(17,177,52,143),(18,178,53,144),(19,179,54,145),(20,180,55,146),(21,166,56,147),(22,167,57,148),(23,168,58,149),(24,169,59,150),(25,170,60,136),(26,171,46,137),(27,172,47,138),(28,173,48,139),(29,174,49,140),(30,175,50,141),(61,222,96,182),(62,223,97,183),(63,224,98,184),(64,225,99,185),(65,211,100,186),(66,212,101,187),(67,213,102,188),(68,214,103,189),(69,215,104,190),(70,216,105,191),(71,217,91,192),(72,218,92,193),(73,219,93,194),(74,220,94,195),(75,221,95,181),(76,239,111,206),(77,240,112,207),(78,226,113,208),(79,227,114,209),(80,228,115,210),(81,229,116,196),(82,230,117,197),(83,231,118,198),(84,232,119,199),(85,233,120,200),(86,234,106,201),(87,235,107,202),(88,236,108,203),(89,237,109,204),(90,238,110,205),(241,396,276,361),(242,397,277,362),(243,398,278,363),(244,399,279,364),(245,400,280,365),(246,401,281,366),(247,402,282,367),(248,403,283,368),(249,404,284,369),(250,405,285,370),(251,391,271,371),(252,392,272,372),(253,393,273,373),(254,394,274,374),(255,395,275,375),(256,406,296,381),(257,407,297,382),(258,408,298,383),(259,409,299,384),(260,410,300,385),(261,411,286,386),(262,412,287,387),(263,413,288,388),(264,414,289,389),(265,415,290,390),(266,416,291,376),(267,417,292,377),(268,418,293,378),(269,419,294,379),(270,420,295,380),(301,459,331,422),(302,460,332,423),(303,461,333,424),(304,462,334,425),(305,463,335,426),(306,464,336,427),(307,465,337,428),(308,451,338,429),(309,452,339,430),(310,453,340,431),(311,454,341,432),(312,455,342,433),(313,456,343,434),(314,457,344,435),(315,458,345,421),(316,474,346,436),(317,475,347,437),(318,476,348,438),(319,477,349,439),(320,478,350,440),(321,479,351,441),(322,480,352,442),(323,466,353,443),(324,467,354,444),(325,468,355,445),(326,469,356,446),(327,470,357,447),(328,471,358,448),(329,472,359,449),(330,473,360,450)], [(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)], [(1,466,107,381,24,458,105,371),(2,474,111,383,25,451,94,373),(3,467,115,385,26,459,98,375),(4,475,119,387,27,452,102,362),(5,468,108,389,28,460,91,364),(6,476,112,376,29,453,95,366),(7,469,116,378,30,461,99,368),(8,477,120,380,16,454,103,370),(9,470,109,382,17,462,92,372),(10,478,113,384,18,455,96,374),(11,471,117,386,19,463,100,361),(12,479,106,388,20,456,104,363),(13,472,110,390,21,464,93,365),(14,480,114,377,22,457,97,367),(15,473,118,379,23,465,101,369),(31,438,77,416,49,431,75,401),(32,446,81,418,50,424,64,403),(33,439,85,420,51,432,68,405),(34,447,89,407,52,425,72,392),(35,440,78,409,53,433,61,394),(36,448,82,411,54,426,65,396),(37,441,86,413,55,434,69,398),(38,449,90,415,56,427,73,400),(39,442,79,417,57,435,62,402),(40,450,83,419,58,428,66,404),(41,443,87,406,59,421,70,391),(42,436,76,408,60,429,74,393),(43,444,80,410,46,422,63,395),(44,437,84,412,47,430,67,397),(45,445,88,414,48,423,71,399),(121,319,233,295,142,311,214,285),(122,327,237,297,143,304,218,272),(123,320,226,299,144,312,222,274),(124,328,230,286,145,305,211,276),(125,321,234,288,146,313,215,278),(126,329,238,290,147,306,219,280),(127,322,227,292,148,314,223,282),(128,330,231,294,149,307,212,284),(129,323,235,296,150,315,216,271),(130,316,239,298,136,308,220,273),(131,324,228,300,137,301,224,275),(132,317,232,287,138,309,213,277),(133,325,236,289,139,302,217,279),(134,318,240,291,140,310,221,281),(135,326,229,293,141,303,225,283),(151,350,208,259,178,342,182,254),(152,358,197,261,179,335,186,241),(153,351,201,263,180,343,190,243),(154,359,205,265,166,336,194,245),(155,352,209,267,167,344,183,247),(156,360,198,269,168,337,187,249),(157,353,202,256,169,345,191,251),(158,346,206,258,170,338,195,253),(159,354,210,260,171,331,184,255),(160,347,199,262,172,339,188,242),(161,355,203,264,173,332,192,244),(162,348,207,266,174,340,181,246),(163,356,196,268,175,333,185,248),(164,349,200,270,176,341,189,250),(165,357,204,257,177,334,193,252)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5 | 6A | 6B | 6C | 8A | ··· | 8P | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 4 | 2 | 2 | 2 | 15 | ··· | 15 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | - | + | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C4×S3 | C3⋊C8 | F5 | C5⋊C8 | C2×F5 | C3⋊F5 | D5⋊C8 | C4×F5 | C15⋊C8 | C2×C3⋊F5 | C60.C4 | C4×C3⋊F5 |
| kernel | C4×C15⋊C8 | C12×Dic5 | C2×C15⋊C8 | C15⋊C8 | C6×Dic5 | C2×C60 | C3×Dic5 | C60 | C4×Dic5 | C2×Dic5 | C2×Dic5 | C2×C20 | Dic5 | Dic5 | C20 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C4×C15⋊C8 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 240 | 240 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 115 | 126 |
| 0 | 0 | 229 | 12 | 127 | 0 |
| 0 | 0 | 114 | 0 | 127 | 12 |
| 0 | 0 | 229 | 126 | 115 | 12 |
| 195 | 212 | 0 | 0 | 0 | 0 |
| 17 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 113 | 124 | 139 | 187 |
| 0 | 0 | 59 | 43 | 11 | 70 |
| 0 | 0 | 183 | 182 | 198 | 230 |
| 0 | 0 | 102 | 54 | 81 | 128 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[240,1,0,0,0,0,240,0,0,0,0,0,0,0,0,229,114,229,0,0,12,12,0,126,0,0,115,127,127,115,0,0,126,0,12,12],[195,17,0,0,0,0,212,46,0,0,0,0,0,0,113,59,183,102,0,0,124,43,182,54,0,0,139,11,198,81,0,0,187,70,230,128] >;
C4×C15⋊C8 in GAP, Magma, Sage, TeX
C_4\times C_{15}\rtimes C_8 % in TeX
G:=Group("C4xC15:C8"); // GroupNames label
G:=SmallGroup(480,305);
// by ID
G=gap.SmallGroup(480,305);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,100,2693,14118,4724]);
// Polycyclic
G:=Group<a,b,c|a^4=b^15=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations