direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×C2.D8, C8⋊1C60, C40⋊9C12, C24⋊3C20, C120⋊19C4, C30.59D8, C60.38Q8, C30.26Q16, C4⋊C4.3C30, (C2×C8).3C30, C4.7(C2×C60), C2.2(C15×D8), C6.14(C5×D8), C4.2(Q8×C15), C6.7(C5×Q16), (C2×C24).9C10, (C2×C40).13C6, C10.14(C3×D8), C30.65(C4⋊C4), C12.10(C5×Q8), C20.10(C3×Q8), C10.7(C3×Q16), C2.2(C15×Q16), (C2×C120).29C2, C12.44(C2×C20), C60.258(C2×C4), C20.65(C2×C12), (C2×C30).192D4, C22.11(D4×C15), (C2×C60).573C22, C2.4(C15×C4⋊C4), C6.13(C5×C4⋊C4), (C5×C4⋊C4).10C6, C10.20(C3×C4⋊C4), (C2×C6).49(C5×D4), (C3×C4⋊C4).10C10, (C15×C4⋊C4).24C2, (C2×C4).20(C2×C30), (C2×C10).49(C3×D4), (C2×C20).121(C2×C6), (C2×C12).124(C2×C10), SmallGroup(480,210)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C2.D8
G = < a,b,c,d | a15=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 104 in 72 conjugacy classes, 56 normal (40 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C20, C20, C2×C10, C24, C2×C12, C2×C12, C30, C2.D8, C40, C2×C20, C2×C20, C3×C4⋊C4, C2×C24, C60, C60, C2×C30, C5×C4⋊C4, C2×C40, C3×C2.D8, C120, C2×C60, C2×C60, C5×C2.D8, C15×C4⋊C4, C2×C120, C15×C2.D8
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, Q8, C10, C12, C2×C6, C15, C4⋊C4, D8, Q16, C20, C2×C10, C2×C12, C3×D4, C3×Q8, C30, C2.D8, C2×C20, C5×D4, C5×Q8, C3×C4⋊C4, C3×D8, C3×Q16, C60, C2×C30, C5×C4⋊C4, C5×D8, C5×Q16, C3×C2.D8, C2×C60, D4×C15, Q8×C15, C5×C2.D8, C15×C4⋊C4, C15×D8, C15×Q16, C15×C2.D8
►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)
(1 476)(2 477)(3 478)(4 479)(5 480)(6 466)(7 467)(8 468)(9 469)(10 470)(11 471)(12 472)(13 473)(14 474)(15 475)(16 185)(17 186)(18 187)(19 188)(20 189)(21 190)(22 191)(23 192)(24 193)(25 194)(26 195)(27 181)(28 182)(29 183)(30 184)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 106)(40 107)(41 108)(42 109)(43 110)(44 111)(45 112)(46 412)(47 413)(48 414)(49 415)(50 416)(51 417)(52 418)(53 419)(54 420)(55 406)(56 407)(57 408)(58 409)(59 410)(60 411)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(73 97)(74 98)(75 99)(76 226)(77 227)(78 228)(79 229)(80 230)(81 231)(82 232)(83 233)(84 234)(85 235)(86 236)(87 237)(88 238)(89 239)(90 240)(121 313)(122 314)(123 315)(124 301)(125 302)(126 303)(127 304)(128 305)(129 306)(130 307)(131 308)(132 309)(133 310)(134 311)(135 312)(136 343)(137 344)(138 345)(139 331)(140 332)(141 333)(142 334)(143 335)(144 336)(145 337)(146 338)(147 339)(148 340)(149 341)(150 342)(151 196)(152 197)(153 198)(154 199)(155 200)(156 201)(157 202)(158 203)(159 204)(160 205)(161 206)(162 207)(163 208)(164 209)(165 210)(166 444)(167 445)(168 446)(169 447)(170 448)(171 449)(172 450)(173 436)(174 437)(175 438)(176 439)(177 440)(178 441)(179 442)(180 443)(211 360)(212 346)(213 347)(214 348)(215 349)(216 350)(217 351)(218 352)(219 353)(220 354)(221 355)(222 356)(223 357)(224 358)(225 359)(241 367)(242 368)(243 369)(244 370)(245 371)(246 372)(247 373)(248 374)(249 375)(250 361)(251 362)(252 363)(253 364)(254 365)(255 366)(256 400)(257 401)(258 402)(259 403)(260 404)(261 405)(262 391)(263 392)(264 393)(265 394)(266 395)(267 396)(268 397)(269 398)(270 399)(271 459)(272 460)(273 461)(274 462)(275 463)(276 464)(277 465)(278 451)(279 452)(280 453)(281 454)(282 455)(283 456)(284 457)(285 458)(286 328)(287 329)(288 330)(289 316)(290 317)(291 318)(292 319)(293 320)(294 321)(295 322)(296 323)(297 324)(298 325)(299 326)(300 327)(376 427)(377 428)(378 429)(379 430)(380 431)(381 432)(382 433)(383 434)(384 435)(385 421)(386 422)(387 423)(388 424)(389 425)(390 426)
(1 38 250 442 121 327 262 209)(2 39 251 443 122 328 263 210)(3 40 252 444 123 329 264 196)(4 41 253 445 124 330 265 197)(5 42 254 446 125 316 266 198)(6 43 255 447 126 317 267 199)(7 44 241 448 127 318 268 200)(8 45 242 449 128 319 269 201)(9 31 243 450 129 320 270 202)(10 32 244 436 130 321 256 203)(11 33 245 437 131 322 257 204)(12 34 246 438 132 323 258 205)(13 35 247 439 133 324 259 206)(14 36 248 440 134 325 260 207)(15 37 249 441 135 326 261 208)(16 360 52 427 100 142 77 461)(17 346 53 428 101 143 78 462)(18 347 54 429 102 144 79 463)(19 348 55 430 103 145 80 464)(20 349 56 431 104 146 81 465)(21 350 57 432 105 147 82 451)(22 351 58 433 91 148 83 452)(23 352 59 434 92 149 84 453)(24 353 60 435 93 150 85 454)(25 354 46 421 94 136 86 455)(26 355 47 422 95 137 87 456)(27 356 48 423 96 138 88 457)(28 357 49 424 97 139 89 458)(29 358 50 425 98 140 90 459)(30 359 51 426 99 141 76 460)(61 334 227 273 185 211 418 376)(62 335 228 274 186 212 419 377)(63 336 229 275 187 213 420 378)(64 337 230 276 188 214 406 379)(65 338 231 277 189 215 407 380)(66 339 232 278 190 216 408 381)(67 340 233 279 191 217 409 382)(68 341 234 280 192 218 410 383)(69 342 235 281 193 219 411 384)(70 343 236 282 194 220 412 385)(71 344 237 283 195 221 413 386)(72 345 238 284 181 222 414 387)(73 331 239 285 182 223 415 388)(74 332 240 271 183 224 416 389)(75 333 226 272 184 225 417 390)(106 362 180 314 286 392 165 477)(107 363 166 315 287 393 151 478)(108 364 167 301 288 394 152 479)(109 365 168 302 289 395 153 480)(110 366 169 303 290 396 154 466)(111 367 170 304 291 397 155 467)(112 368 171 305 292 398 156 468)(113 369 172 306 293 399 157 469)(114 370 173 307 294 400 158 470)(115 371 174 308 295 401 159 471)(116 372 175 309 296 402 160 472)(117 373 176 310 297 403 161 473)(118 374 177 311 298 404 162 474)(119 375 178 312 299 405 163 475)(120 361 179 313 300 391 164 476)
(1 49 476 415)(2 50 477 416)(3 51 478 417)(4 52 479 418)(5 53 480 419)(6 54 466 420)(7 55 467 406)(8 56 468 407)(9 57 469 408)(10 58 470 409)(11 59 471 410)(12 60 472 411)(13 46 473 412)(14 47 474 413)(15 48 475 414)(16 364 185 253)(17 365 186 254)(18 366 187 255)(19 367 188 241)(20 368 189 242)(21 369 190 243)(22 370 191 244)(23 371 192 245)(24 372 193 246)(25 373 194 247)(26 374 195 248)(27 375 181 249)(28 361 182 250)(29 362 183 251)(30 363 184 252)(31 350 113 216)(32 351 114 217)(33 352 115 218)(34 353 116 219)(35 354 117 220)(36 355 118 221)(37 356 119 222)(38 357 120 223)(39 358 106 224)(40 359 107 225)(41 360 108 211)(42 346 109 212)(43 347 110 213)(44 348 111 214)(45 349 112 215)(61 265 100 394)(62 266 101 395)(63 267 102 396)(64 268 103 397)(65 269 104 398)(66 270 105 399)(67 256 91 400)(68 257 92 401)(69 258 93 402)(70 259 94 403)(71 260 95 404)(72 261 96 405)(73 262 97 391)(74 263 98 392)(75 264 99 393)(76 315 226 123)(77 301 227 124)(78 302 228 125)(79 303 229 126)(80 304 230 127)(81 305 231 128)(82 306 232 129)(83 307 233 130)(84 308 234 131)(85 309 235 132)(86 310 236 133)(87 311 237 134)(88 312 238 135)(89 313 239 121)(90 314 240 122)(136 297 343 324)(137 298 344 325)(138 299 345 326)(139 300 331 327)(140 286 332 328)(141 287 333 329)(142 288 334 330)(143 289 335 316)(144 290 336 317)(145 291 337 318)(146 292 338 319)(147 293 339 320)(148 294 340 321)(149 295 341 322)(150 296 342 323)(151 390 196 426)(152 376 197 427)(153 377 198 428)(154 378 199 429)(155 379 200 430)(156 380 201 431)(157 381 202 432)(158 382 203 433)(159 383 204 434)(160 384 205 435)(161 385 206 421)(162 386 207 422)(163 387 208 423)(164 388 209 424)(165 389 210 425)(166 272 444 460)(167 273 445 461)(168 274 446 462)(169 275 447 463)(170 276 448 464)(171 277 449 465)(172 278 450 451)(173 279 436 452)(174 280 437 453)(175 281 438 454)(176 282 439 455)(177 283 440 456)(178 284 441 457)(179 285 442 458)(180 271 443 459)
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), (1,476)(2,477)(3,478)(4,479)(5,480)(6,466)(7,467)(8,468)(9,469)(10,470)(11,471)(12,472)(13,473)(14,474)(15,475)(16,185)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,193)(25,194)(26,195)(27,181)(28,182)(29,183)(30,184)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,412)(47,413)(48,414)(49,415)(50,416)(51,417)(52,418)(53,419)(54,420)(55,406)(56,407)(57,408)(58,409)(59,410)(60,411)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,226)(77,227)(78,228)(79,229)(80,230)(81,231)(82,232)(83,233)(84,234)(85,235)(86,236)(87,237)(88,238)(89,239)(90,240)(121,313)(122,314)(123,315)(124,301)(125,302)(126,303)(127,304)(128,305)(129,306)(130,307)(131,308)(132,309)(133,310)(134,311)(135,312)(136,343)(137,344)(138,345)(139,331)(140,332)(141,333)(142,334)(143,335)(144,336)(145,337)(146,338)(147,339)(148,340)(149,341)(150,342)(151,196)(152,197)(153,198)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,444)(167,445)(168,446)(169,447)(170,448)(171,449)(172,450)(173,436)(174,437)(175,438)(176,439)(177,440)(178,441)(179,442)(180,443)(211,360)(212,346)(213,347)(214,348)(215,349)(216,350)(217,351)(218,352)(219,353)(220,354)(221,355)(222,356)(223,357)(224,358)(225,359)(241,367)(242,368)(243,369)(244,370)(245,371)(246,372)(247,373)(248,374)(249,375)(250,361)(251,362)(252,363)(253,364)(254,365)(255,366)(256,400)(257,401)(258,402)(259,403)(260,404)(261,405)(262,391)(263,392)(264,393)(265,394)(266,395)(267,396)(268,397)(269,398)(270,399)(271,459)(272,460)(273,461)(274,462)(275,463)(276,464)(277,465)(278,451)(279,452)(280,453)(281,454)(282,455)(283,456)(284,457)(285,458)(286,328)(287,329)(288,330)(289,316)(290,317)(291,318)(292,319)(293,320)(294,321)(295,322)(296,323)(297,324)(298,325)(299,326)(300,327)(376,427)(377,428)(378,429)(379,430)(380,431)(381,432)(382,433)(383,434)(384,435)(385,421)(386,422)(387,423)(388,424)(389,425)(390,426), (1,38,250,442,121,327,262,209)(2,39,251,443,122,328,263,210)(3,40,252,444,123,329,264,196)(4,41,253,445,124,330,265,197)(5,42,254,446,125,316,266,198)(6,43,255,447,126,317,267,199)(7,44,241,448,127,318,268,200)(8,45,242,449,128,319,269,201)(9,31,243,450,129,320,270,202)(10,32,244,436,130,321,256,203)(11,33,245,437,131,322,257,204)(12,34,246,438,132,323,258,205)(13,35,247,439,133,324,259,206)(14,36,248,440,134,325,260,207)(15,37,249,441,135,326,261,208)(16,360,52,427,100,142,77,461)(17,346,53,428,101,143,78,462)(18,347,54,429,102,144,79,463)(19,348,55,430,103,145,80,464)(20,349,56,431,104,146,81,465)(21,350,57,432,105,147,82,451)(22,351,58,433,91,148,83,452)(23,352,59,434,92,149,84,453)(24,353,60,435,93,150,85,454)(25,354,46,421,94,136,86,455)(26,355,47,422,95,137,87,456)(27,356,48,423,96,138,88,457)(28,357,49,424,97,139,89,458)(29,358,50,425,98,140,90,459)(30,359,51,426,99,141,76,460)(61,334,227,273,185,211,418,376)(62,335,228,274,186,212,419,377)(63,336,229,275,187,213,420,378)(64,337,230,276,188,214,406,379)(65,338,231,277,189,215,407,380)(66,339,232,278,190,216,408,381)(67,340,233,279,191,217,409,382)(68,341,234,280,192,218,410,383)(69,342,235,281,193,219,411,384)(70,343,236,282,194,220,412,385)(71,344,237,283,195,221,413,386)(72,345,238,284,181,222,414,387)(73,331,239,285,182,223,415,388)(74,332,240,271,183,224,416,389)(75,333,226,272,184,225,417,390)(106,362,180,314,286,392,165,477)(107,363,166,315,287,393,151,478)(108,364,167,301,288,394,152,479)(109,365,168,302,289,395,153,480)(110,366,169,303,290,396,154,466)(111,367,170,304,291,397,155,467)(112,368,171,305,292,398,156,468)(113,369,172,306,293,399,157,469)(114,370,173,307,294,400,158,470)(115,371,174,308,295,401,159,471)(116,372,175,309,296,402,160,472)(117,373,176,310,297,403,161,473)(118,374,177,311,298,404,162,474)(119,375,178,312,299,405,163,475)(120,361,179,313,300,391,164,476), (1,49,476,415)(2,50,477,416)(3,51,478,417)(4,52,479,418)(5,53,480,419)(6,54,466,420)(7,55,467,406)(8,56,468,407)(9,57,469,408)(10,58,470,409)(11,59,471,410)(12,60,472,411)(13,46,473,412)(14,47,474,413)(15,48,475,414)(16,364,185,253)(17,365,186,254)(18,366,187,255)(19,367,188,241)(20,368,189,242)(21,369,190,243)(22,370,191,244)(23,371,192,245)(24,372,193,246)(25,373,194,247)(26,374,195,248)(27,375,181,249)(28,361,182,250)(29,362,183,251)(30,363,184,252)(31,350,113,216)(32,351,114,217)(33,352,115,218)(34,353,116,219)(35,354,117,220)(36,355,118,221)(37,356,119,222)(38,357,120,223)(39,358,106,224)(40,359,107,225)(41,360,108,211)(42,346,109,212)(43,347,110,213)(44,348,111,214)(45,349,112,215)(61,265,100,394)(62,266,101,395)(63,267,102,396)(64,268,103,397)(65,269,104,398)(66,270,105,399)(67,256,91,400)(68,257,92,401)(69,258,93,402)(70,259,94,403)(71,260,95,404)(72,261,96,405)(73,262,97,391)(74,263,98,392)(75,264,99,393)(76,315,226,123)(77,301,227,124)(78,302,228,125)(79,303,229,126)(80,304,230,127)(81,305,231,128)(82,306,232,129)(83,307,233,130)(84,308,234,131)(85,309,235,132)(86,310,236,133)(87,311,237,134)(88,312,238,135)(89,313,239,121)(90,314,240,122)(136,297,343,324)(137,298,344,325)(138,299,345,326)(139,300,331,327)(140,286,332,328)(141,287,333,329)(142,288,334,330)(143,289,335,316)(144,290,336,317)(145,291,337,318)(146,292,338,319)(147,293,339,320)(148,294,340,321)(149,295,341,322)(150,296,342,323)(151,390,196,426)(152,376,197,427)(153,377,198,428)(154,378,199,429)(155,379,200,430)(156,380,201,431)(157,381,202,432)(158,382,203,433)(159,383,204,434)(160,384,205,435)(161,385,206,421)(162,386,207,422)(163,387,208,423)(164,388,209,424)(165,389,210,425)(166,272,444,460)(167,273,445,461)(168,274,446,462)(169,275,447,463)(170,276,448,464)(171,277,449,465)(172,278,450,451)(173,279,436,452)(174,280,437,453)(175,281,438,454)(176,282,439,455)(177,283,440,456)(178,284,441,457)(179,285,442,458)(180,271,443,459)>;
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), (1,476)(2,477)(3,478)(4,479)(5,480)(6,466)(7,467)(8,468)(9,469)(10,470)(11,471)(12,472)(13,473)(14,474)(15,475)(16,185)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,193)(25,194)(26,195)(27,181)(28,182)(29,183)(30,184)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,412)(47,413)(48,414)(49,415)(50,416)(51,417)(52,418)(53,419)(54,420)(55,406)(56,407)(57,408)(58,409)(59,410)(60,411)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,226)(77,227)(78,228)(79,229)(80,230)(81,231)(82,232)(83,233)(84,234)(85,235)(86,236)(87,237)(88,238)(89,239)(90,240)(121,313)(122,314)(123,315)(124,301)(125,302)(126,303)(127,304)(128,305)(129,306)(130,307)(131,308)(132,309)(133,310)(134,311)(135,312)(136,343)(137,344)(138,345)(139,331)(140,332)(141,333)(142,334)(143,335)(144,336)(145,337)(146,338)(147,339)(148,340)(149,341)(150,342)(151,196)(152,197)(153,198)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(166,444)(167,445)(168,446)(169,447)(170,448)(171,449)(172,450)(173,436)(174,437)(175,438)(176,439)(177,440)(178,441)(179,442)(180,443)(211,360)(212,346)(213,347)(214,348)(215,349)(216,350)(217,351)(218,352)(219,353)(220,354)(221,355)(222,356)(223,357)(224,358)(225,359)(241,367)(242,368)(243,369)(244,370)(245,371)(246,372)(247,373)(248,374)(249,375)(250,361)(251,362)(252,363)(253,364)(254,365)(255,366)(256,400)(257,401)(258,402)(259,403)(260,404)(261,405)(262,391)(263,392)(264,393)(265,394)(266,395)(267,396)(268,397)(269,398)(270,399)(271,459)(272,460)(273,461)(274,462)(275,463)(276,464)(277,465)(278,451)(279,452)(280,453)(281,454)(282,455)(283,456)(284,457)(285,458)(286,328)(287,329)(288,330)(289,316)(290,317)(291,318)(292,319)(293,320)(294,321)(295,322)(296,323)(297,324)(298,325)(299,326)(300,327)(376,427)(377,428)(378,429)(379,430)(380,431)(381,432)(382,433)(383,434)(384,435)(385,421)(386,422)(387,423)(388,424)(389,425)(390,426), (1,38,250,442,121,327,262,209)(2,39,251,443,122,328,263,210)(3,40,252,444,123,329,264,196)(4,41,253,445,124,330,265,197)(5,42,254,446,125,316,266,198)(6,43,255,447,126,317,267,199)(7,44,241,448,127,318,268,200)(8,45,242,449,128,319,269,201)(9,31,243,450,129,320,270,202)(10,32,244,436,130,321,256,203)(11,33,245,437,131,322,257,204)(12,34,246,438,132,323,258,205)(13,35,247,439,133,324,259,206)(14,36,248,440,134,325,260,207)(15,37,249,441,135,326,261,208)(16,360,52,427,100,142,77,461)(17,346,53,428,101,143,78,462)(18,347,54,429,102,144,79,463)(19,348,55,430,103,145,80,464)(20,349,56,431,104,146,81,465)(21,350,57,432,105,147,82,451)(22,351,58,433,91,148,83,452)(23,352,59,434,92,149,84,453)(24,353,60,435,93,150,85,454)(25,354,46,421,94,136,86,455)(26,355,47,422,95,137,87,456)(27,356,48,423,96,138,88,457)(28,357,49,424,97,139,89,458)(29,358,50,425,98,140,90,459)(30,359,51,426,99,141,76,460)(61,334,227,273,185,211,418,376)(62,335,228,274,186,212,419,377)(63,336,229,275,187,213,420,378)(64,337,230,276,188,214,406,379)(65,338,231,277,189,215,407,380)(66,339,232,278,190,216,408,381)(67,340,233,279,191,217,409,382)(68,341,234,280,192,218,410,383)(69,342,235,281,193,219,411,384)(70,343,236,282,194,220,412,385)(71,344,237,283,195,221,413,386)(72,345,238,284,181,222,414,387)(73,331,239,285,182,223,415,388)(74,332,240,271,183,224,416,389)(75,333,226,272,184,225,417,390)(106,362,180,314,286,392,165,477)(107,363,166,315,287,393,151,478)(108,364,167,301,288,394,152,479)(109,365,168,302,289,395,153,480)(110,366,169,303,290,396,154,466)(111,367,170,304,291,397,155,467)(112,368,171,305,292,398,156,468)(113,369,172,306,293,399,157,469)(114,370,173,307,294,400,158,470)(115,371,174,308,295,401,159,471)(116,372,175,309,296,402,160,472)(117,373,176,310,297,403,161,473)(118,374,177,311,298,404,162,474)(119,375,178,312,299,405,163,475)(120,361,179,313,300,391,164,476), (1,49,476,415)(2,50,477,416)(3,51,478,417)(4,52,479,418)(5,53,480,419)(6,54,466,420)(7,55,467,406)(8,56,468,407)(9,57,469,408)(10,58,470,409)(11,59,471,410)(12,60,472,411)(13,46,473,412)(14,47,474,413)(15,48,475,414)(16,364,185,253)(17,365,186,254)(18,366,187,255)(19,367,188,241)(20,368,189,242)(21,369,190,243)(22,370,191,244)(23,371,192,245)(24,372,193,246)(25,373,194,247)(26,374,195,248)(27,375,181,249)(28,361,182,250)(29,362,183,251)(30,363,184,252)(31,350,113,216)(32,351,114,217)(33,352,115,218)(34,353,116,219)(35,354,117,220)(36,355,118,221)(37,356,119,222)(38,357,120,223)(39,358,106,224)(40,359,107,225)(41,360,108,211)(42,346,109,212)(43,347,110,213)(44,348,111,214)(45,349,112,215)(61,265,100,394)(62,266,101,395)(63,267,102,396)(64,268,103,397)(65,269,104,398)(66,270,105,399)(67,256,91,400)(68,257,92,401)(69,258,93,402)(70,259,94,403)(71,260,95,404)(72,261,96,405)(73,262,97,391)(74,263,98,392)(75,264,99,393)(76,315,226,123)(77,301,227,124)(78,302,228,125)(79,303,229,126)(80,304,230,127)(81,305,231,128)(82,306,232,129)(83,307,233,130)(84,308,234,131)(85,309,235,132)(86,310,236,133)(87,311,237,134)(88,312,238,135)(89,313,239,121)(90,314,240,122)(136,297,343,324)(137,298,344,325)(138,299,345,326)(139,300,331,327)(140,286,332,328)(141,287,333,329)(142,288,334,330)(143,289,335,316)(144,290,336,317)(145,291,337,318)(146,292,338,319)(147,293,339,320)(148,294,340,321)(149,295,341,322)(150,296,342,323)(151,390,196,426)(152,376,197,427)(153,377,198,428)(154,378,199,429)(155,379,200,430)(156,380,201,431)(157,381,202,432)(158,382,203,433)(159,383,204,434)(160,384,205,435)(161,385,206,421)(162,386,207,422)(163,387,208,423)(164,388,209,424)(165,389,210,425)(166,272,444,460)(167,273,445,461)(168,274,446,462)(169,275,447,463)(170,276,448,464)(171,277,449,465)(172,278,450,451)(173,279,436,452)(174,280,437,453)(175,281,438,454)(176,282,439,455)(177,283,440,456)(178,284,441,457)(179,285,442,458)(180,271,443,459) );
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)], [(1,476),(2,477),(3,478),(4,479),(5,480),(6,466),(7,467),(8,468),(9,469),(10,470),(11,471),(12,472),(13,473),(14,474),(15,475),(16,185),(17,186),(18,187),(19,188),(20,189),(21,190),(22,191),(23,192),(24,193),(25,194),(26,195),(27,181),(28,182),(29,183),(30,184),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,119),(38,120),(39,106),(40,107),(41,108),(42,109),(43,110),(44,111),(45,112),(46,412),(47,413),(48,414),(49,415),(50,416),(51,417),(52,418),(53,419),(54,420),(55,406),(56,407),(57,408),(58,409),(59,410),(60,411),(61,100),(62,101),(63,102),(64,103),(65,104),(66,105),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96),(73,97),(74,98),(75,99),(76,226),(77,227),(78,228),(79,229),(80,230),(81,231),(82,232),(83,233),(84,234),(85,235),(86,236),(87,237),(88,238),(89,239),(90,240),(121,313),(122,314),(123,315),(124,301),(125,302),(126,303),(127,304),(128,305),(129,306),(130,307),(131,308),(132,309),(133,310),(134,311),(135,312),(136,343),(137,344),(138,345),(139,331),(140,332),(141,333),(142,334),(143,335),(144,336),(145,337),(146,338),(147,339),(148,340),(149,341),(150,342),(151,196),(152,197),(153,198),(154,199),(155,200),(156,201),(157,202),(158,203),(159,204),(160,205),(161,206),(162,207),(163,208),(164,209),(165,210),(166,444),(167,445),(168,446),(169,447),(170,448),(171,449),(172,450),(173,436),(174,437),(175,438),(176,439),(177,440),(178,441),(179,442),(180,443),(211,360),(212,346),(213,347),(214,348),(215,349),(216,350),(217,351),(218,352),(219,353),(220,354),(221,355),(222,356),(223,357),(224,358),(225,359),(241,367),(242,368),(243,369),(244,370),(245,371),(246,372),(247,373),(248,374),(249,375),(250,361),(251,362),(252,363),(253,364),(254,365),(255,366),(256,400),(257,401),(258,402),(259,403),(260,404),(261,405),(262,391),(263,392),(264,393),(265,394),(266,395),(267,396),(268,397),(269,398),(270,399),(271,459),(272,460),(273,461),(274,462),(275,463),(276,464),(277,465),(278,451),(279,452),(280,453),(281,454),(282,455),(283,456),(284,457),(285,458),(286,328),(287,329),(288,330),(289,316),(290,317),(291,318),(292,319),(293,320),(294,321),(295,322),(296,323),(297,324),(298,325),(299,326),(300,327),(376,427),(377,428),(378,429),(379,430),(380,431),(381,432),(382,433),(383,434),(384,435),(385,421),(386,422),(387,423),(388,424),(389,425),(390,426)], [(1,38,250,442,121,327,262,209),(2,39,251,443,122,328,263,210),(3,40,252,444,123,329,264,196),(4,41,253,445,124,330,265,197),(5,42,254,446,125,316,266,198),(6,43,255,447,126,317,267,199),(7,44,241,448,127,318,268,200),(8,45,242,449,128,319,269,201),(9,31,243,450,129,320,270,202),(10,32,244,436,130,321,256,203),(11,33,245,437,131,322,257,204),(12,34,246,438,132,323,258,205),(13,35,247,439,133,324,259,206),(14,36,248,440,134,325,260,207),(15,37,249,441,135,326,261,208),(16,360,52,427,100,142,77,461),(17,346,53,428,101,143,78,462),(18,347,54,429,102,144,79,463),(19,348,55,430,103,145,80,464),(20,349,56,431,104,146,81,465),(21,350,57,432,105,147,82,451),(22,351,58,433,91,148,83,452),(23,352,59,434,92,149,84,453),(24,353,60,435,93,150,85,454),(25,354,46,421,94,136,86,455),(26,355,47,422,95,137,87,456),(27,356,48,423,96,138,88,457),(28,357,49,424,97,139,89,458),(29,358,50,425,98,140,90,459),(30,359,51,426,99,141,76,460),(61,334,227,273,185,211,418,376),(62,335,228,274,186,212,419,377),(63,336,229,275,187,213,420,378),(64,337,230,276,188,214,406,379),(65,338,231,277,189,215,407,380),(66,339,232,278,190,216,408,381),(67,340,233,279,191,217,409,382),(68,341,234,280,192,218,410,383),(69,342,235,281,193,219,411,384),(70,343,236,282,194,220,412,385),(71,344,237,283,195,221,413,386),(72,345,238,284,181,222,414,387),(73,331,239,285,182,223,415,388),(74,332,240,271,183,224,416,389),(75,333,226,272,184,225,417,390),(106,362,180,314,286,392,165,477),(107,363,166,315,287,393,151,478),(108,364,167,301,288,394,152,479),(109,365,168,302,289,395,153,480),(110,366,169,303,290,396,154,466),(111,367,170,304,291,397,155,467),(112,368,171,305,292,398,156,468),(113,369,172,306,293,399,157,469),(114,370,173,307,294,400,158,470),(115,371,174,308,295,401,159,471),(116,372,175,309,296,402,160,472),(117,373,176,310,297,403,161,473),(118,374,177,311,298,404,162,474),(119,375,178,312,299,405,163,475),(120,361,179,313,300,391,164,476)], [(1,49,476,415),(2,50,477,416),(3,51,478,417),(4,52,479,418),(5,53,480,419),(6,54,466,420),(7,55,467,406),(8,56,468,407),(9,57,469,408),(10,58,470,409),(11,59,471,410),(12,60,472,411),(13,46,473,412),(14,47,474,413),(15,48,475,414),(16,364,185,253),(17,365,186,254),(18,366,187,255),(19,367,188,241),(20,368,189,242),(21,369,190,243),(22,370,191,244),(23,371,192,245),(24,372,193,246),(25,373,194,247),(26,374,195,248),(27,375,181,249),(28,361,182,250),(29,362,183,251),(30,363,184,252),(31,350,113,216),(32,351,114,217),(33,352,115,218),(34,353,116,219),(35,354,117,220),(36,355,118,221),(37,356,119,222),(38,357,120,223),(39,358,106,224),(40,359,107,225),(41,360,108,211),(42,346,109,212),(43,347,110,213),(44,348,111,214),(45,349,112,215),(61,265,100,394),(62,266,101,395),(63,267,102,396),(64,268,103,397),(65,269,104,398),(66,270,105,399),(67,256,91,400),(68,257,92,401),(69,258,93,402),(70,259,94,403),(71,260,95,404),(72,261,96,405),(73,262,97,391),(74,263,98,392),(75,264,99,393),(76,315,226,123),(77,301,227,124),(78,302,228,125),(79,303,229,126),(80,304,230,127),(81,305,231,128),(82,306,232,129),(83,307,233,130),(84,308,234,131),(85,309,235,132),(86,310,236,133),(87,311,237,134),(88,312,238,135),(89,313,239,121),(90,314,240,122),(136,297,343,324),(137,298,344,325),(138,299,345,326),(139,300,331,327),(140,286,332,328),(141,287,333,329),(142,288,334,330),(143,289,335,316),(144,290,336,317),(145,291,337,318),(146,292,338,319),(147,293,339,320),(148,294,340,321),(149,295,341,322),(150,296,342,323),(151,390,196,426),(152,376,197,427),(153,377,198,428),(154,378,199,429),(155,379,200,430),(156,380,201,431),(157,381,202,432),(158,382,203,433),(159,383,204,434),(160,384,205,435),(161,385,206,421),(162,386,207,422),(163,387,208,423),(164,388,209,424),(165,389,210,425),(166,272,444,460),(167,273,445,461),(168,274,446,462),(169,275,447,463),(170,276,448,464),(171,277,449,465),(172,278,450,451),(173,279,436,452),(174,280,437,453),(175,281,438,454),(176,282,439,455),(177,283,440,456),(178,284,441,457),(179,285,442,458),(180,271,443,459)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | ··· | 15H | 20A | ··· | 20H | 20I | ··· | 20X | 24A | ··· | 24H | 30A | ··· | 30X | 40A | ··· | 40P | 60A | ··· | 60P | 60Q | ··· | 60AV | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C60 | Q8 | D4 | D8 | Q16 | C3×Q8 | C3×D4 | C5×Q8 | C5×D4 | C3×D8 | C3×Q16 | C5×D8 | C5×Q16 | Q8×C15 | D4×C15 | C15×D8 | C15×Q16 |
| kernel | C15×C2.D8 | C15×C4⋊C4 | C2×C120 | C5×C2.D8 | C120 | C3×C2.D8 | C5×C4⋊C4 | C2×C40 | C3×C4⋊C4 | C2×C24 | C40 | C2.D8 | C24 | C4⋊C4 | C2×C8 | C8 | C60 | C2×C30 | C30 | C30 | C20 | C2×C10 | C12 | C2×C6 | C10 | C10 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 4 | 8 | 8 | 16 | 16 | 8 | 32 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of C15×C2.D8 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 100 | 0 |
| 0 | 0 | 100 |
| 240 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 240 |
| 1 | 0 | 0 |
| 0 | 0 | 219 |
| 0 | 11 | 219 |
| 64 | 0 | 0 |
| 0 | 104 | 64 |
| 0 | 136 | 137 |
G:=sub<GL(3,GF(241))| [1,0,0,0,100,0,0,0,100],[240,0,0,0,240,0,0,0,240],[1,0,0,0,0,11,0,219,219],[64,0,0,0,104,136,0,64,137] >;
C15×C2.D8 in GAP, Magma, Sage, TeX
C_{15}\times C_2.D_8 % in TeX
G:=Group("C15xC2.D8"); // GroupNames label
G:=SmallGroup(480,210);
// by ID
G=gap.SmallGroup(480,210);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,2108,10504,172]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^2=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations