direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×Dic15, C120⋊12C4, C24⋊4Dic5, C40⋊7Dic3, C30.27C42, C15⋊14(C4×C8), C6.8(C8×D5), C5⋊6(C8×Dic3), C3⋊3(C8×Dic5), C2.2(C8×D15), C15⋊3C8⋊17C4, C10.17(S3×C8), C20.90(C4×S3), C30.45(C2×C8), (C2×C40).11S3, (C2×C8).10D15, C12.58(C4×D5), (C2×C4).91D30, (C2×C24).14D5, C4.20(C4×D15), C60.236(C2×C4), (C2×C120).18C2, (C2×C20).405D6, C2.2(C4×Dic15), C6.10(C4×Dic5), C22.8(C4×D15), (C2×C12).409D10, C10.22(C4×Dic3), C4.11(C2×Dic15), C12.41(C2×Dic5), C20.62(C2×Dic3), (C2×C60).491C22, (C2×Dic15).24C4, (C4×Dic15).21C2, (C2×C6).26(C4×D5), (C2×C10).51(C4×S3), (C2×C15⋊3C8).24C2, (C2×C30).133(C2×C4), SmallGroup(480,173)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C8×Dic15 |
Generators and relations for C8×Dic15
G = < a,b,c | a8=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 308 in 88 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, C2×C4, C10, C10, Dic3, C12, C2×C6, C15, C42, C2×C8, C2×C8, Dic5, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C30, C4×C8, C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, Dic15, C60, C2×C30, C2×C5⋊2C8, C4×Dic5, C2×C40, C8×Dic3, C15⋊3C8, C120, C2×Dic15, C2×C60, C8×Dic5, C2×C15⋊3C8, C4×Dic15, C2×C120, C8×Dic15
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D5, Dic3, D6, C42, C2×C8, Dic5, D10, C4×S3, C2×Dic3, D15, C4×C8, C4×D5, C2×Dic5, S3×C8, C4×Dic3, Dic15, D30, C8×D5, C4×Dic5, C8×Dic3, C4×D15, C2×Dic15, C8×Dic5, C8×D15, C4×Dic15, C8×Dic15
►Regular action on 480 points
Generators in S480
(1 159 119 207 42 254 284 386)(2 160 120 208 43 255 285 387)(3 161 91 209 44 256 286 388)(4 162 92 210 45 257 287 389)(5 163 93 181 46 258 288 390)(6 164 94 182 47 259 289 361)(7 165 95 183 48 260 290 362)(8 166 96 184 49 261 291 363)(9 167 97 185 50 262 292 364)(10 168 98 186 51 263 293 365)(11 169 99 187 52 264 294 366)(12 170 100 188 53 265 295 367)(13 171 101 189 54 266 296 368)(14 172 102 190 55 267 297 369)(15 173 103 191 56 268 298 370)(16 174 104 192 57 269 299 371)(17 175 105 193 58 270 300 372)(18 176 106 194 59 241 271 373)(19 177 107 195 60 242 272 374)(20 178 108 196 31 243 273 375)(21 179 109 197 32 244 274 376)(22 180 110 198 33 245 275 377)(23 151 111 199 34 246 276 378)(24 152 112 200 35 247 277 379)(25 153 113 201 36 248 278 380)(26 154 114 202 37 249 279 381)(27 155 115 203 38 250 280 382)(28 156 116 204 39 251 281 383)(29 157 117 205 40 252 282 384)(30 158 118 206 41 253 283 385)(61 320 239 420 476 131 431 345)(62 321 240 391 477 132 432 346)(63 322 211 392 478 133 433 347)(64 323 212 393 479 134 434 348)(65 324 213 394 480 135 435 349)(66 325 214 395 451 136 436 350)(67 326 215 396 452 137 437 351)(68 327 216 397 453 138 438 352)(69 328 217 398 454 139 439 353)(70 329 218 399 455 140 440 354)(71 330 219 400 456 141 441 355)(72 301 220 401 457 142 442 356)(73 302 221 402 458 143 443 357)(74 303 222 403 459 144 444 358)(75 304 223 404 460 145 445 359)(76 305 224 405 461 146 446 360)(77 306 225 406 462 147 447 331)(78 307 226 407 463 148 448 332)(79 308 227 408 464 149 449 333)(80 309 228 409 465 150 450 334)(81 310 229 410 466 121 421 335)(82 311 230 411 467 122 422 336)(83 312 231 412 468 123 423 337)(84 313 232 413 469 124 424 338)(85 314 233 414 470 125 425 339)(86 315 234 415 471 126 426 340)(87 316 235 416 472 127 427 341)(88 317 236 417 473 128 428 342)(89 318 237 418 474 129 429 343)(90 319 238 419 475 130 430 344)
(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)
(1 147 16 132)(2 146 17 131)(3 145 18 130)(4 144 19 129)(5 143 20 128)(6 142 21 127)(7 141 22 126)(8 140 23 125)(9 139 24 124)(10 138 25 123)(11 137 26 122)(12 136 27 121)(13 135 28 150)(14 134 29 149)(15 133 30 148)(31 317 46 302)(32 316 47 301)(33 315 48 330)(34 314 49 329)(35 313 50 328)(36 312 51 327)(37 311 52 326)(38 310 53 325)(39 309 54 324)(40 308 55 323)(41 307 56 322)(42 306 57 321)(43 305 58 320)(44 304 59 319)(45 303 60 318)(61 208 76 193)(62 207 77 192)(63 206 78 191)(64 205 79 190)(65 204 80 189)(66 203 81 188)(67 202 82 187)(68 201 83 186)(69 200 84 185)(70 199 85 184)(71 198 86 183)(72 197 87 182)(73 196 88 181)(74 195 89 210)(75 194 90 209)(91 359 106 344)(92 358 107 343)(93 357 108 342)(94 356 109 341)(95 355 110 340)(96 354 111 339)(97 353 112 338)(98 352 113 337)(99 351 114 336)(100 350 115 335)(101 349 116 334)(102 348 117 333)(103 347 118 332)(104 346 119 331)(105 345 120 360)(151 425 166 440)(152 424 167 439)(153 423 168 438)(154 422 169 437)(155 421 170 436)(156 450 171 435)(157 449 172 434)(158 448 173 433)(159 447 174 432)(160 446 175 431)(161 445 176 430)(162 444 177 429)(163 443 178 428)(164 442 179 427)(165 441 180 426)(211 253 226 268)(212 252 227 267)(213 251 228 266)(214 250 229 265)(215 249 230 264)(216 248 231 263)(217 247 232 262)(218 246 233 261)(219 245 234 260)(220 244 235 259)(221 243 236 258)(222 242 237 257)(223 241 238 256)(224 270 239 255)(225 269 240 254)(271 419 286 404)(272 418 287 403)(273 417 288 402)(274 416 289 401)(275 415 290 400)(276 414 291 399)(277 413 292 398)(278 412 293 397)(279 411 294 396)(280 410 295 395)(281 409 296 394)(282 408 297 393)(283 407 298 392)(284 406 299 391)(285 405 300 420)(361 457 376 472)(362 456 377 471)(363 455 378 470)(364 454 379 469)(365 453 380 468)(366 452 381 467)(367 451 382 466)(368 480 383 465)(369 479 384 464)(370 478 385 463)(371 477 386 462)(372 476 387 461)(373 475 388 460)(374 474 389 459)(375 473 390 458)
G:=sub<Sym(480)| (1,159,119,207,42,254,284,386)(2,160,120,208,43,255,285,387)(3,161,91,209,44,256,286,388)(4,162,92,210,45,257,287,389)(5,163,93,181,46,258,288,390)(6,164,94,182,47,259,289,361)(7,165,95,183,48,260,290,362)(8,166,96,184,49,261,291,363)(9,167,97,185,50,262,292,364)(10,168,98,186,51,263,293,365)(11,169,99,187,52,264,294,366)(12,170,100,188,53,265,295,367)(13,171,101,189,54,266,296,368)(14,172,102,190,55,267,297,369)(15,173,103,191,56,268,298,370)(16,174,104,192,57,269,299,371)(17,175,105,193,58,270,300,372)(18,176,106,194,59,241,271,373)(19,177,107,195,60,242,272,374)(20,178,108,196,31,243,273,375)(21,179,109,197,32,244,274,376)(22,180,110,198,33,245,275,377)(23,151,111,199,34,246,276,378)(24,152,112,200,35,247,277,379)(25,153,113,201,36,248,278,380)(26,154,114,202,37,249,279,381)(27,155,115,203,38,250,280,382)(28,156,116,204,39,251,281,383)(29,157,117,205,40,252,282,384)(30,158,118,206,41,253,283,385)(61,320,239,420,476,131,431,345)(62,321,240,391,477,132,432,346)(63,322,211,392,478,133,433,347)(64,323,212,393,479,134,434,348)(65,324,213,394,480,135,435,349)(66,325,214,395,451,136,436,350)(67,326,215,396,452,137,437,351)(68,327,216,397,453,138,438,352)(69,328,217,398,454,139,439,353)(70,329,218,399,455,140,440,354)(71,330,219,400,456,141,441,355)(72,301,220,401,457,142,442,356)(73,302,221,402,458,143,443,357)(74,303,222,403,459,144,444,358)(75,304,223,404,460,145,445,359)(76,305,224,405,461,146,446,360)(77,306,225,406,462,147,447,331)(78,307,226,407,463,148,448,332)(79,308,227,408,464,149,449,333)(80,309,228,409,465,150,450,334)(81,310,229,410,466,121,421,335)(82,311,230,411,467,122,422,336)(83,312,231,412,468,123,423,337)(84,313,232,413,469,124,424,338)(85,314,233,414,470,125,425,339)(86,315,234,415,471,126,426,340)(87,316,235,416,472,127,427,341)(88,317,236,417,473,128,428,342)(89,318,237,418,474,129,429,343)(90,319,238,419,475,130,430,344), (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), (1,147,16,132)(2,146,17,131)(3,145,18,130)(4,144,19,129)(5,143,20,128)(6,142,21,127)(7,141,22,126)(8,140,23,125)(9,139,24,124)(10,138,25,123)(11,137,26,122)(12,136,27,121)(13,135,28,150)(14,134,29,149)(15,133,30,148)(31,317,46,302)(32,316,47,301)(33,315,48,330)(34,314,49,329)(35,313,50,328)(36,312,51,327)(37,311,52,326)(38,310,53,325)(39,309,54,324)(40,308,55,323)(41,307,56,322)(42,306,57,321)(43,305,58,320)(44,304,59,319)(45,303,60,318)(61,208,76,193)(62,207,77,192)(63,206,78,191)(64,205,79,190)(65,204,80,189)(66,203,81,188)(67,202,82,187)(68,201,83,186)(69,200,84,185)(70,199,85,184)(71,198,86,183)(72,197,87,182)(73,196,88,181)(74,195,89,210)(75,194,90,209)(91,359,106,344)(92,358,107,343)(93,357,108,342)(94,356,109,341)(95,355,110,340)(96,354,111,339)(97,353,112,338)(98,352,113,337)(99,351,114,336)(100,350,115,335)(101,349,116,334)(102,348,117,333)(103,347,118,332)(104,346,119,331)(105,345,120,360)(151,425,166,440)(152,424,167,439)(153,423,168,438)(154,422,169,437)(155,421,170,436)(156,450,171,435)(157,449,172,434)(158,448,173,433)(159,447,174,432)(160,446,175,431)(161,445,176,430)(162,444,177,429)(163,443,178,428)(164,442,179,427)(165,441,180,426)(211,253,226,268)(212,252,227,267)(213,251,228,266)(214,250,229,265)(215,249,230,264)(216,248,231,263)(217,247,232,262)(218,246,233,261)(219,245,234,260)(220,244,235,259)(221,243,236,258)(222,242,237,257)(223,241,238,256)(224,270,239,255)(225,269,240,254)(271,419,286,404)(272,418,287,403)(273,417,288,402)(274,416,289,401)(275,415,290,400)(276,414,291,399)(277,413,292,398)(278,412,293,397)(279,411,294,396)(280,410,295,395)(281,409,296,394)(282,408,297,393)(283,407,298,392)(284,406,299,391)(285,405,300,420)(361,457,376,472)(362,456,377,471)(363,455,378,470)(364,454,379,469)(365,453,380,468)(366,452,381,467)(367,451,382,466)(368,480,383,465)(369,479,384,464)(370,478,385,463)(371,477,386,462)(372,476,387,461)(373,475,388,460)(374,474,389,459)(375,473,390,458)>;
G:=Group( (1,159,119,207,42,254,284,386)(2,160,120,208,43,255,285,387)(3,161,91,209,44,256,286,388)(4,162,92,210,45,257,287,389)(5,163,93,181,46,258,288,390)(6,164,94,182,47,259,289,361)(7,165,95,183,48,260,290,362)(8,166,96,184,49,261,291,363)(9,167,97,185,50,262,292,364)(10,168,98,186,51,263,293,365)(11,169,99,187,52,264,294,366)(12,170,100,188,53,265,295,367)(13,171,101,189,54,266,296,368)(14,172,102,190,55,267,297,369)(15,173,103,191,56,268,298,370)(16,174,104,192,57,269,299,371)(17,175,105,193,58,270,300,372)(18,176,106,194,59,241,271,373)(19,177,107,195,60,242,272,374)(20,178,108,196,31,243,273,375)(21,179,109,197,32,244,274,376)(22,180,110,198,33,245,275,377)(23,151,111,199,34,246,276,378)(24,152,112,200,35,247,277,379)(25,153,113,201,36,248,278,380)(26,154,114,202,37,249,279,381)(27,155,115,203,38,250,280,382)(28,156,116,204,39,251,281,383)(29,157,117,205,40,252,282,384)(30,158,118,206,41,253,283,385)(61,320,239,420,476,131,431,345)(62,321,240,391,477,132,432,346)(63,322,211,392,478,133,433,347)(64,323,212,393,479,134,434,348)(65,324,213,394,480,135,435,349)(66,325,214,395,451,136,436,350)(67,326,215,396,452,137,437,351)(68,327,216,397,453,138,438,352)(69,328,217,398,454,139,439,353)(70,329,218,399,455,140,440,354)(71,330,219,400,456,141,441,355)(72,301,220,401,457,142,442,356)(73,302,221,402,458,143,443,357)(74,303,222,403,459,144,444,358)(75,304,223,404,460,145,445,359)(76,305,224,405,461,146,446,360)(77,306,225,406,462,147,447,331)(78,307,226,407,463,148,448,332)(79,308,227,408,464,149,449,333)(80,309,228,409,465,150,450,334)(81,310,229,410,466,121,421,335)(82,311,230,411,467,122,422,336)(83,312,231,412,468,123,423,337)(84,313,232,413,469,124,424,338)(85,314,233,414,470,125,425,339)(86,315,234,415,471,126,426,340)(87,316,235,416,472,127,427,341)(88,317,236,417,473,128,428,342)(89,318,237,418,474,129,429,343)(90,319,238,419,475,130,430,344), (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), (1,147,16,132)(2,146,17,131)(3,145,18,130)(4,144,19,129)(5,143,20,128)(6,142,21,127)(7,141,22,126)(8,140,23,125)(9,139,24,124)(10,138,25,123)(11,137,26,122)(12,136,27,121)(13,135,28,150)(14,134,29,149)(15,133,30,148)(31,317,46,302)(32,316,47,301)(33,315,48,330)(34,314,49,329)(35,313,50,328)(36,312,51,327)(37,311,52,326)(38,310,53,325)(39,309,54,324)(40,308,55,323)(41,307,56,322)(42,306,57,321)(43,305,58,320)(44,304,59,319)(45,303,60,318)(61,208,76,193)(62,207,77,192)(63,206,78,191)(64,205,79,190)(65,204,80,189)(66,203,81,188)(67,202,82,187)(68,201,83,186)(69,200,84,185)(70,199,85,184)(71,198,86,183)(72,197,87,182)(73,196,88,181)(74,195,89,210)(75,194,90,209)(91,359,106,344)(92,358,107,343)(93,357,108,342)(94,356,109,341)(95,355,110,340)(96,354,111,339)(97,353,112,338)(98,352,113,337)(99,351,114,336)(100,350,115,335)(101,349,116,334)(102,348,117,333)(103,347,118,332)(104,346,119,331)(105,345,120,360)(151,425,166,440)(152,424,167,439)(153,423,168,438)(154,422,169,437)(155,421,170,436)(156,450,171,435)(157,449,172,434)(158,448,173,433)(159,447,174,432)(160,446,175,431)(161,445,176,430)(162,444,177,429)(163,443,178,428)(164,442,179,427)(165,441,180,426)(211,253,226,268)(212,252,227,267)(213,251,228,266)(214,250,229,265)(215,249,230,264)(216,248,231,263)(217,247,232,262)(218,246,233,261)(219,245,234,260)(220,244,235,259)(221,243,236,258)(222,242,237,257)(223,241,238,256)(224,270,239,255)(225,269,240,254)(271,419,286,404)(272,418,287,403)(273,417,288,402)(274,416,289,401)(275,415,290,400)(276,414,291,399)(277,413,292,398)(278,412,293,397)(279,411,294,396)(280,410,295,395)(281,409,296,394)(282,408,297,393)(283,407,298,392)(284,406,299,391)(285,405,300,420)(361,457,376,472)(362,456,377,471)(363,455,378,470)(364,454,379,469)(365,453,380,468)(366,452,381,467)(367,451,382,466)(368,480,383,465)(369,479,384,464)(370,478,385,463)(371,477,386,462)(372,476,387,461)(373,475,388,460)(374,474,389,459)(375,473,390,458) );
G=PermutationGroup([[(1,159,119,207,42,254,284,386),(2,160,120,208,43,255,285,387),(3,161,91,209,44,256,286,388),(4,162,92,210,45,257,287,389),(5,163,93,181,46,258,288,390),(6,164,94,182,47,259,289,361),(7,165,95,183,48,260,290,362),(8,166,96,184,49,261,291,363),(9,167,97,185,50,262,292,364),(10,168,98,186,51,263,293,365),(11,169,99,187,52,264,294,366),(12,170,100,188,53,265,295,367),(13,171,101,189,54,266,296,368),(14,172,102,190,55,267,297,369),(15,173,103,191,56,268,298,370),(16,174,104,192,57,269,299,371),(17,175,105,193,58,270,300,372),(18,176,106,194,59,241,271,373),(19,177,107,195,60,242,272,374),(20,178,108,196,31,243,273,375),(21,179,109,197,32,244,274,376),(22,180,110,198,33,245,275,377),(23,151,111,199,34,246,276,378),(24,152,112,200,35,247,277,379),(25,153,113,201,36,248,278,380),(26,154,114,202,37,249,279,381),(27,155,115,203,38,250,280,382),(28,156,116,204,39,251,281,383),(29,157,117,205,40,252,282,384),(30,158,118,206,41,253,283,385),(61,320,239,420,476,131,431,345),(62,321,240,391,477,132,432,346),(63,322,211,392,478,133,433,347),(64,323,212,393,479,134,434,348),(65,324,213,394,480,135,435,349),(66,325,214,395,451,136,436,350),(67,326,215,396,452,137,437,351),(68,327,216,397,453,138,438,352),(69,328,217,398,454,139,439,353),(70,329,218,399,455,140,440,354),(71,330,219,400,456,141,441,355),(72,301,220,401,457,142,442,356),(73,302,221,402,458,143,443,357),(74,303,222,403,459,144,444,358),(75,304,223,404,460,145,445,359),(76,305,224,405,461,146,446,360),(77,306,225,406,462,147,447,331),(78,307,226,407,463,148,448,332),(79,308,227,408,464,149,449,333),(80,309,228,409,465,150,450,334),(81,310,229,410,466,121,421,335),(82,311,230,411,467,122,422,336),(83,312,231,412,468,123,423,337),(84,313,232,413,469,124,424,338),(85,314,233,414,470,125,425,339),(86,315,234,415,471,126,426,340),(87,316,235,416,472,127,427,341),(88,317,236,417,473,128,428,342),(89,318,237,418,474,129,429,343),(90,319,238,419,475,130,430,344)], [(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)], [(1,147,16,132),(2,146,17,131),(3,145,18,130),(4,144,19,129),(5,143,20,128),(6,142,21,127),(7,141,22,126),(8,140,23,125),(9,139,24,124),(10,138,25,123),(11,137,26,122),(12,136,27,121),(13,135,28,150),(14,134,29,149),(15,133,30,148),(31,317,46,302),(32,316,47,301),(33,315,48,330),(34,314,49,329),(35,313,50,328),(36,312,51,327),(37,311,52,326),(38,310,53,325),(39,309,54,324),(40,308,55,323),(41,307,56,322),(42,306,57,321),(43,305,58,320),(44,304,59,319),(45,303,60,318),(61,208,76,193),(62,207,77,192),(63,206,78,191),(64,205,79,190),(65,204,80,189),(66,203,81,188),(67,202,82,187),(68,201,83,186),(69,200,84,185),(70,199,85,184),(71,198,86,183),(72,197,87,182),(73,196,88,181),(74,195,89,210),(75,194,90,209),(91,359,106,344),(92,358,107,343),(93,357,108,342),(94,356,109,341),(95,355,110,340),(96,354,111,339),(97,353,112,338),(98,352,113,337),(99,351,114,336),(100,350,115,335),(101,349,116,334),(102,348,117,333),(103,347,118,332),(104,346,119,331),(105,345,120,360),(151,425,166,440),(152,424,167,439),(153,423,168,438),(154,422,169,437),(155,421,170,436),(156,450,171,435),(157,449,172,434),(158,448,173,433),(159,447,174,432),(160,446,175,431),(161,445,176,430),(162,444,177,429),(163,443,178,428),(164,442,179,427),(165,441,180,426),(211,253,226,268),(212,252,227,267),(213,251,228,266),(214,250,229,265),(215,249,230,264),(216,248,231,263),(217,247,232,262),(218,246,233,261),(219,245,234,260),(220,244,235,259),(221,243,236,258),(222,242,237,257),(223,241,238,256),(224,270,239,255),(225,269,240,254),(271,419,286,404),(272,418,287,403),(273,417,288,402),(274,416,289,401),(275,415,290,400),(276,414,291,399),(277,413,292,398),(278,412,293,397),(279,411,294,396),(280,410,295,395),(281,409,296,394),(282,408,297,393),(283,407,298,392),(284,406,299,391),(285,405,300,420),(361,457,376,472),(362,456,377,471),(363,455,378,470),(364,454,379,469),(365,453,380,468),(366,452,381,467),(367,451,382,466),(368,480,383,465),(369,479,384,464),(370,478,385,463),(371,477,386,462),(372,476,387,461),(373,475,388,460),(374,474,389,459),(375,473,390,458)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 15 | ··· | 15 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 15 | ··· | 15 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | S3 | D5 | Dic3 | D6 | Dic5 | D10 | C4×S3 | C4×S3 | D15 | C4×D5 | C4×D5 | S3×C8 | Dic15 | D30 | C8×D5 | C4×D15 | C4×D15 | C8×D15 |
| kernel | C8×Dic15 | C2×C15⋊3C8 | C4×Dic15 | C2×C120 | C15⋊3C8 | C120 | C2×Dic15 | Dic15 | C2×C40 | C2×C24 | C40 | C2×C20 | C24 | C2×C12 | C20 | C2×C10 | C2×C8 | C12 | C2×C6 | C10 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of C8×Dic15 ►in GL4(𝔽241) generated by
| 233 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 30 | 177 |
| 0 | 0 | 64 | 225 |
| 240 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 109 | 89 |
| 0 | 0 | 205 | 132 |
G:=sub<GL(4,GF(241))| [233,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,240,0,0,0,0,30,64,0,0,177,225],[240,0,0,0,0,177,0,0,0,0,109,205,0,0,89,132] >;
C8×Dic15 in GAP, Magma, Sage, TeX
C_8\times {\rm Dic}_{15} % in TeX
G:=Group("C8xDic15"); // GroupNames label
G:=SmallGroup(480,173);
// by ID
G=gap.SmallGroup(480,173);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^30=1,c^2=b^15,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations