direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C6.SD16, Dic6⋊3C20, C60.135D4, C20.62D12, C30.23Q16, C30.31SD16, C4.2(S3×C20), C12.2(C5×D4), C6.4(C5×Q16), C20.77(C4×S3), C12.4(C2×C20), C4.10(C5×D12), C6.4(C5×SD16), C60.174(C2×C4), (C5×Dic6)⋊15C4, (C2×C20).347D6, (C2×C30).175D4, C10.51(D6⋊C4), C15⋊13(Q8⋊C4), (C2×Dic6).5C10, C30.93(C22⋊C4), C10.11(D4.S3), (C2×C60).340C22, (C10×Dic6).15C2, C10.11(C3⋊Q16), (C2×C3⋊C8).3C10, C4⋊C4.3(C5×S3), C2.6(C5×D6⋊C4), C3⋊1(C5×Q8⋊C4), (C3×C4⋊C4).3C10, (C10×C3⋊C8).15C2, (C5×C4⋊C4).10S3, (C2×C6).31(C5×D4), C6.4(C5×C22⋊C4), (C15×C4⋊C4).17C2, C2.2(C5×D4.S3), C2.2(C5×C3⋊Q16), (C2×C4).35(S3×C10), (C2×C12).10(C2×C10), C22.15(C5×C3⋊D4), (C2×C10).87(C3⋊D4), SmallGroup(480,129)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C6.SD16
G = < a,b,c,d | a5=b6=c8=1, d2=b3c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c-1 >
Subgroups: 180 in 84 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, Q8, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, C20, C20, C2×C10, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C30, Q8⋊C4, C40, C2×C20, C2×C20, C5×Q8, C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C5×Dic3, C60, C60, C2×C30, C5×C4⋊C4, C2×C40, Q8×C10, C6.SD16, C5×C3⋊C8, C5×Dic6, C5×Dic6, C10×Dic3, C2×C60, C2×C60, C5×Q8⋊C4, C10×C3⋊C8, C15×C4⋊C4, C10×Dic6, C5×C6.SD16
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, D6, C22⋊C4, SD16, Q16, C20, C2×C10, C4×S3, D12, C3⋊D4, C5×S3, Q8⋊C4, C2×C20, C5×D4, D6⋊C4, D4.S3, C3⋊Q16, S3×C10, C5×C22⋊C4, C5×SD16, C5×Q16, C6.SD16, S3×C20, C5×D12, C5×C3⋊D4, C5×Q8⋊C4, C5×D6⋊C4, C5×D4.S3, C5×C3⋊Q16, C5×C6.SD16
►Regular action on 480 points
Generators in S480
(1 122 166 140 147)(2 123 167 141 148)(3 124 168 142 149)(4 125 161 143 150)(5 126 162 144 151)(6 127 163 137 152)(7 128 164 138 145)(8 121 165 139 146)(9 65 417 41 393)(10 66 418 42 394)(11 67 419 43 395)(12 68 420 44 396)(13 69 421 45 397)(14 70 422 46 398)(15 71 423 47 399)(16 72 424 48 400)(17 97 107 73 83)(18 98 108 74 84)(19 99 109 75 85)(20 100 110 76 86)(21 101 111 77 87)(22 102 112 78 88)(23 103 105 79 81)(24 104 106 80 82)(25 35 355 49 59)(26 36 356 50 60)(27 37 357 51 61)(28 38 358 52 62)(29 39 359 53 63)(30 40 360 54 64)(31 33 353 55 57)(32 34 354 56 58)(89 441 365 113 465)(90 442 366 114 466)(91 443 367 115 467)(92 444 368 116 468)(93 445 361 117 469)(94 446 362 118 470)(95 447 363 119 471)(96 448 364 120 472)(129 253 301 153 277)(130 254 302 154 278)(131 255 303 155 279)(132 256 304 156 280)(133 249 297 157 273)(134 250 298 158 274)(135 251 299 159 275)(136 252 300 160 276)(169 287 270 263 246)(170 288 271 264 247)(171 281 272 257 248)(172 282 265 258 241)(173 283 266 259 242)(174 284 267 260 243)(175 285 268 261 244)(176 286 269 262 245)(177 225 349 201 325)(178 226 350 202 326)(179 227 351 203 327)(180 228 352 204 328)(181 229 345 205 321)(182 230 346 206 322)(183 231 347 207 323)(184 232 348 208 324)(185 233 335 209 311)(186 234 336 210 312)(187 235 329 211 305)(188 236 330 212 306)(189 237 331 213 307)(190 238 332 214 308)(191 239 333 215 309)(192 240 334 216 310)(193 296 344 217 320)(194 289 337 218 313)(195 290 338 219 314)(196 291 339 220 315)(197 292 340 221 316)(198 293 341 222 317)(199 294 342 223 318)(200 295 343 224 319)(369 425 415 401 391)(370 426 416 402 392)(371 427 409 403 385)(372 428 410 404 386)(373 429 411 405 387)(374 430 412 406 388)(375 431 413 407 389)(376 432 414 408 390)(377 475 457 451 433)(378 476 458 452 434)(379 477 459 453 435)(380 478 460 454 436)(381 479 461 455 437)(382 480 462 456 438)(383 473 463 449 439)(384 474 464 450 440)
(1 253 246 441 437 87)(2 88 438 442 247 254)(3 255 248 443 439 81)(4 82 440 444 241 256)(5 249 242 445 433 83)(6 84 434 446 243 250)(7 251 244 447 435 85)(8 86 436 448 245 252)(9 371 355 295 178 192)(10 185 179 296 356 372)(11 373 357 289 180 186)(12 187 181 290 358 374)(13 375 359 291 182 188)(14 189 183 292 360 376)(15 369 353 293 184 190)(16 191 177 294 354 370)(17 126 297 173 361 377)(18 378 362 174 298 127)(19 128 299 175 363 379)(20 380 364 176 300 121)(21 122 301 169 365 381)(22 382 366 170 302 123)(23 124 303 171 367 383)(24 384 368 172 304 125)(25 319 202 216 41 403)(26 404 42 209 203 320)(27 313 204 210 43 405)(28 406 44 211 205 314)(29 315 206 212 45 407)(30 408 46 213 207 316)(31 317 208 214 47 401)(32 402 48 215 201 318)(33 198 324 308 399 391)(34 392 400 309 325 199)(35 200 326 310 393 385)(36 386 394 311 327 193)(37 194 328 312 395 387)(38 388 396 305 321 195)(39 196 322 306 397 389)(40 390 398 307 323 197)(49 343 226 240 65 427)(50 428 66 233 227 344)(51 337 228 234 67 429)(52 430 68 235 229 338)(53 339 230 236 69 431)(54 432 70 237 231 340)(55 341 232 238 71 425)(56 426 72 239 225 342)(57 222 348 332 423 415)(58 416 424 333 349 223)(59 224 350 334 417 409)(60 410 418 335 351 217)(61 218 352 336 419 411)(62 412 420 329 345 219)(63 220 346 330 421 413)(64 414 422 331 347 221)(73 151 133 259 93 451)(74 452 94 260 134 152)(75 145 135 261 95 453)(76 454 96 262 136 146)(77 147 129 263 89 455)(78 456 90 264 130 148)(79 149 131 257 91 449)(80 450 92 258 132 150)(97 162 157 283 117 475)(98 476 118 284 158 163)(99 164 159 285 119 477)(100 478 120 286 160 165)(101 166 153 287 113 479)(102 480 114 288 154 167)(103 168 155 281 115 473)(104 474 116 282 156 161)(105 142 279 272 467 463)(106 464 468 265 280 143)(107 144 273 266 469 457)(108 458 470 267 274 137)(109 138 275 268 471 459)(110 460 472 269 276 139)(111 140 277 270 465 461)(112 462 466 271 278 141)
(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)
(1 295 445 13)(2 16 446 290)(3 293 447 11)(4 14 448 296)(5 291 441 9)(6 12 442 294)(7 289 443 15)(8 10 444 292)(17 53 169 240)(18 235 170 56)(19 51 171 238)(20 233 172 54)(21 49 173 236)(22 239 174 52)(23 55 175 234)(24 237 176 50)(25 266 212 111)(26 106 213 269)(27 272 214 109)(28 112 215 267)(29 270 216 107)(30 110 209 265)(31 268 210 105)(32 108 211 271)(33 261 312 79)(34 74 305 264)(35 259 306 77)(36 80 307 262)(37 257 308 75)(38 78 309 260)(39 263 310 73)(40 76 311 258)(41 144 315 465)(42 468 316 139)(43 142 317 471)(44 466 318 137)(45 140 319 469)(46 472 320 143)(47 138 313 467)(48 470 314 141)(57 285 336 103)(58 98 329 288)(59 283 330 101)(60 104 331 286)(61 281 332 99)(62 102 333 284)(63 287 334 97)(64 100 335 282)(65 126 339 365)(66 368 340 121)(67 124 341 363)(68 366 342 127)(69 122 343 361)(70 364 344 125)(71 128 337 367)(72 362 338 123)(81 353 244 186)(82 189 245 356)(83 359 246 192)(84 187 247 354)(85 357 248 190)(86 185 241 360)(87 355 242 188)(88 191 243 358)(89 393 151 196)(90 199 152 396)(91 399 145 194)(92 197 146 394)(93 397 147 200)(94 195 148 400)(95 395 149 198)(96 193 150 398)(113 417 162 220)(114 223 163 420)(115 423 164 218)(116 221 165 418)(117 421 166 224)(118 219 167 424)(119 419 168 222)(120 217 161 422)(129 326 451 389)(130 392 452 321)(131 324 453 387)(132 390 454 327)(133 322 455 385)(134 388 456 325)(135 328 449 391)(136 386 450 323)(153 350 475 413)(154 416 476 345)(155 348 477 411)(156 414 478 351)(157 346 479 409)(158 412 480 349)(159 352 473 415)(160 410 474 347)(177 250 374 438)(178 433 375 253)(179 256 376 436)(180 439 369 251)(181 254 370 434)(182 437 371 249)(183 252 372 440)(184 435 373 255)(201 274 406 462)(202 457 407 277)(203 280 408 460)(204 463 401 275)(205 278 402 458)(206 461 403 273)(207 276 404 464)(208 459 405 279)(225 298 430 382)(226 377 431 301)(227 304 432 380)(228 383 425 299)(229 302 426 378)(230 381 427 297)(231 300 428 384)(232 379 429 303)
G:=sub<Sym(480)| (1,122,166,140,147)(2,123,167,141,148)(3,124,168,142,149)(4,125,161,143,150)(5,126,162,144,151)(6,127,163,137,152)(7,128,164,138,145)(8,121,165,139,146)(9,65,417,41,393)(10,66,418,42,394)(11,67,419,43,395)(12,68,420,44,396)(13,69,421,45,397)(14,70,422,46,398)(15,71,423,47,399)(16,72,424,48,400)(17,97,107,73,83)(18,98,108,74,84)(19,99,109,75,85)(20,100,110,76,86)(21,101,111,77,87)(22,102,112,78,88)(23,103,105,79,81)(24,104,106,80,82)(25,35,355,49,59)(26,36,356,50,60)(27,37,357,51,61)(28,38,358,52,62)(29,39,359,53,63)(30,40,360,54,64)(31,33,353,55,57)(32,34,354,56,58)(89,441,365,113,465)(90,442,366,114,466)(91,443,367,115,467)(92,444,368,116,468)(93,445,361,117,469)(94,446,362,118,470)(95,447,363,119,471)(96,448,364,120,472)(129,253,301,153,277)(130,254,302,154,278)(131,255,303,155,279)(132,256,304,156,280)(133,249,297,157,273)(134,250,298,158,274)(135,251,299,159,275)(136,252,300,160,276)(169,287,270,263,246)(170,288,271,264,247)(171,281,272,257,248)(172,282,265,258,241)(173,283,266,259,242)(174,284,267,260,243)(175,285,268,261,244)(176,286,269,262,245)(177,225,349,201,325)(178,226,350,202,326)(179,227,351,203,327)(180,228,352,204,328)(181,229,345,205,321)(182,230,346,206,322)(183,231,347,207,323)(184,232,348,208,324)(185,233,335,209,311)(186,234,336,210,312)(187,235,329,211,305)(188,236,330,212,306)(189,237,331,213,307)(190,238,332,214,308)(191,239,333,215,309)(192,240,334,216,310)(193,296,344,217,320)(194,289,337,218,313)(195,290,338,219,314)(196,291,339,220,315)(197,292,340,221,316)(198,293,341,222,317)(199,294,342,223,318)(200,295,343,224,319)(369,425,415,401,391)(370,426,416,402,392)(371,427,409,403,385)(372,428,410,404,386)(373,429,411,405,387)(374,430,412,406,388)(375,431,413,407,389)(376,432,414,408,390)(377,475,457,451,433)(378,476,458,452,434)(379,477,459,453,435)(380,478,460,454,436)(381,479,461,455,437)(382,480,462,456,438)(383,473,463,449,439)(384,474,464,450,440), (1,253,246,441,437,87)(2,88,438,442,247,254)(3,255,248,443,439,81)(4,82,440,444,241,256)(5,249,242,445,433,83)(6,84,434,446,243,250)(7,251,244,447,435,85)(8,86,436,448,245,252)(9,371,355,295,178,192)(10,185,179,296,356,372)(11,373,357,289,180,186)(12,187,181,290,358,374)(13,375,359,291,182,188)(14,189,183,292,360,376)(15,369,353,293,184,190)(16,191,177,294,354,370)(17,126,297,173,361,377)(18,378,362,174,298,127)(19,128,299,175,363,379)(20,380,364,176,300,121)(21,122,301,169,365,381)(22,382,366,170,302,123)(23,124,303,171,367,383)(24,384,368,172,304,125)(25,319,202,216,41,403)(26,404,42,209,203,320)(27,313,204,210,43,405)(28,406,44,211,205,314)(29,315,206,212,45,407)(30,408,46,213,207,316)(31,317,208,214,47,401)(32,402,48,215,201,318)(33,198,324,308,399,391)(34,392,400,309,325,199)(35,200,326,310,393,385)(36,386,394,311,327,193)(37,194,328,312,395,387)(38,388,396,305,321,195)(39,196,322,306,397,389)(40,390,398,307,323,197)(49,343,226,240,65,427)(50,428,66,233,227,344)(51,337,228,234,67,429)(52,430,68,235,229,338)(53,339,230,236,69,431)(54,432,70,237,231,340)(55,341,232,238,71,425)(56,426,72,239,225,342)(57,222,348,332,423,415)(58,416,424,333,349,223)(59,224,350,334,417,409)(60,410,418,335,351,217)(61,218,352,336,419,411)(62,412,420,329,345,219)(63,220,346,330,421,413)(64,414,422,331,347,221)(73,151,133,259,93,451)(74,452,94,260,134,152)(75,145,135,261,95,453)(76,454,96,262,136,146)(77,147,129,263,89,455)(78,456,90,264,130,148)(79,149,131,257,91,449)(80,450,92,258,132,150)(97,162,157,283,117,475)(98,476,118,284,158,163)(99,164,159,285,119,477)(100,478,120,286,160,165)(101,166,153,287,113,479)(102,480,114,288,154,167)(103,168,155,281,115,473)(104,474,116,282,156,161)(105,142,279,272,467,463)(106,464,468,265,280,143)(107,144,273,266,469,457)(108,458,470,267,274,137)(109,138,275,268,471,459)(110,460,472,269,276,139)(111,140,277,270,465,461)(112,462,466,271,278,141), (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), (1,295,445,13)(2,16,446,290)(3,293,447,11)(4,14,448,296)(5,291,441,9)(6,12,442,294)(7,289,443,15)(8,10,444,292)(17,53,169,240)(18,235,170,56)(19,51,171,238)(20,233,172,54)(21,49,173,236)(22,239,174,52)(23,55,175,234)(24,237,176,50)(25,266,212,111)(26,106,213,269)(27,272,214,109)(28,112,215,267)(29,270,216,107)(30,110,209,265)(31,268,210,105)(32,108,211,271)(33,261,312,79)(34,74,305,264)(35,259,306,77)(36,80,307,262)(37,257,308,75)(38,78,309,260)(39,263,310,73)(40,76,311,258)(41,144,315,465)(42,468,316,139)(43,142,317,471)(44,466,318,137)(45,140,319,469)(46,472,320,143)(47,138,313,467)(48,470,314,141)(57,285,336,103)(58,98,329,288)(59,283,330,101)(60,104,331,286)(61,281,332,99)(62,102,333,284)(63,287,334,97)(64,100,335,282)(65,126,339,365)(66,368,340,121)(67,124,341,363)(68,366,342,127)(69,122,343,361)(70,364,344,125)(71,128,337,367)(72,362,338,123)(81,353,244,186)(82,189,245,356)(83,359,246,192)(84,187,247,354)(85,357,248,190)(86,185,241,360)(87,355,242,188)(88,191,243,358)(89,393,151,196)(90,199,152,396)(91,399,145,194)(92,197,146,394)(93,397,147,200)(94,195,148,400)(95,395,149,198)(96,193,150,398)(113,417,162,220)(114,223,163,420)(115,423,164,218)(116,221,165,418)(117,421,166,224)(118,219,167,424)(119,419,168,222)(120,217,161,422)(129,326,451,389)(130,392,452,321)(131,324,453,387)(132,390,454,327)(133,322,455,385)(134,388,456,325)(135,328,449,391)(136,386,450,323)(153,350,475,413)(154,416,476,345)(155,348,477,411)(156,414,478,351)(157,346,479,409)(158,412,480,349)(159,352,473,415)(160,410,474,347)(177,250,374,438)(178,433,375,253)(179,256,376,436)(180,439,369,251)(181,254,370,434)(182,437,371,249)(183,252,372,440)(184,435,373,255)(201,274,406,462)(202,457,407,277)(203,280,408,460)(204,463,401,275)(205,278,402,458)(206,461,403,273)(207,276,404,464)(208,459,405,279)(225,298,430,382)(226,377,431,301)(227,304,432,380)(228,383,425,299)(229,302,426,378)(230,381,427,297)(231,300,428,384)(232,379,429,303)>;
G:=Group( (1,122,166,140,147)(2,123,167,141,148)(3,124,168,142,149)(4,125,161,143,150)(5,126,162,144,151)(6,127,163,137,152)(7,128,164,138,145)(8,121,165,139,146)(9,65,417,41,393)(10,66,418,42,394)(11,67,419,43,395)(12,68,420,44,396)(13,69,421,45,397)(14,70,422,46,398)(15,71,423,47,399)(16,72,424,48,400)(17,97,107,73,83)(18,98,108,74,84)(19,99,109,75,85)(20,100,110,76,86)(21,101,111,77,87)(22,102,112,78,88)(23,103,105,79,81)(24,104,106,80,82)(25,35,355,49,59)(26,36,356,50,60)(27,37,357,51,61)(28,38,358,52,62)(29,39,359,53,63)(30,40,360,54,64)(31,33,353,55,57)(32,34,354,56,58)(89,441,365,113,465)(90,442,366,114,466)(91,443,367,115,467)(92,444,368,116,468)(93,445,361,117,469)(94,446,362,118,470)(95,447,363,119,471)(96,448,364,120,472)(129,253,301,153,277)(130,254,302,154,278)(131,255,303,155,279)(132,256,304,156,280)(133,249,297,157,273)(134,250,298,158,274)(135,251,299,159,275)(136,252,300,160,276)(169,287,270,263,246)(170,288,271,264,247)(171,281,272,257,248)(172,282,265,258,241)(173,283,266,259,242)(174,284,267,260,243)(175,285,268,261,244)(176,286,269,262,245)(177,225,349,201,325)(178,226,350,202,326)(179,227,351,203,327)(180,228,352,204,328)(181,229,345,205,321)(182,230,346,206,322)(183,231,347,207,323)(184,232,348,208,324)(185,233,335,209,311)(186,234,336,210,312)(187,235,329,211,305)(188,236,330,212,306)(189,237,331,213,307)(190,238,332,214,308)(191,239,333,215,309)(192,240,334,216,310)(193,296,344,217,320)(194,289,337,218,313)(195,290,338,219,314)(196,291,339,220,315)(197,292,340,221,316)(198,293,341,222,317)(199,294,342,223,318)(200,295,343,224,319)(369,425,415,401,391)(370,426,416,402,392)(371,427,409,403,385)(372,428,410,404,386)(373,429,411,405,387)(374,430,412,406,388)(375,431,413,407,389)(376,432,414,408,390)(377,475,457,451,433)(378,476,458,452,434)(379,477,459,453,435)(380,478,460,454,436)(381,479,461,455,437)(382,480,462,456,438)(383,473,463,449,439)(384,474,464,450,440), (1,253,246,441,437,87)(2,88,438,442,247,254)(3,255,248,443,439,81)(4,82,440,444,241,256)(5,249,242,445,433,83)(6,84,434,446,243,250)(7,251,244,447,435,85)(8,86,436,448,245,252)(9,371,355,295,178,192)(10,185,179,296,356,372)(11,373,357,289,180,186)(12,187,181,290,358,374)(13,375,359,291,182,188)(14,189,183,292,360,376)(15,369,353,293,184,190)(16,191,177,294,354,370)(17,126,297,173,361,377)(18,378,362,174,298,127)(19,128,299,175,363,379)(20,380,364,176,300,121)(21,122,301,169,365,381)(22,382,366,170,302,123)(23,124,303,171,367,383)(24,384,368,172,304,125)(25,319,202,216,41,403)(26,404,42,209,203,320)(27,313,204,210,43,405)(28,406,44,211,205,314)(29,315,206,212,45,407)(30,408,46,213,207,316)(31,317,208,214,47,401)(32,402,48,215,201,318)(33,198,324,308,399,391)(34,392,400,309,325,199)(35,200,326,310,393,385)(36,386,394,311,327,193)(37,194,328,312,395,387)(38,388,396,305,321,195)(39,196,322,306,397,389)(40,390,398,307,323,197)(49,343,226,240,65,427)(50,428,66,233,227,344)(51,337,228,234,67,429)(52,430,68,235,229,338)(53,339,230,236,69,431)(54,432,70,237,231,340)(55,341,232,238,71,425)(56,426,72,239,225,342)(57,222,348,332,423,415)(58,416,424,333,349,223)(59,224,350,334,417,409)(60,410,418,335,351,217)(61,218,352,336,419,411)(62,412,420,329,345,219)(63,220,346,330,421,413)(64,414,422,331,347,221)(73,151,133,259,93,451)(74,452,94,260,134,152)(75,145,135,261,95,453)(76,454,96,262,136,146)(77,147,129,263,89,455)(78,456,90,264,130,148)(79,149,131,257,91,449)(80,450,92,258,132,150)(97,162,157,283,117,475)(98,476,118,284,158,163)(99,164,159,285,119,477)(100,478,120,286,160,165)(101,166,153,287,113,479)(102,480,114,288,154,167)(103,168,155,281,115,473)(104,474,116,282,156,161)(105,142,279,272,467,463)(106,464,468,265,280,143)(107,144,273,266,469,457)(108,458,470,267,274,137)(109,138,275,268,471,459)(110,460,472,269,276,139)(111,140,277,270,465,461)(112,462,466,271,278,141), (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), (1,295,445,13)(2,16,446,290)(3,293,447,11)(4,14,448,296)(5,291,441,9)(6,12,442,294)(7,289,443,15)(8,10,444,292)(17,53,169,240)(18,235,170,56)(19,51,171,238)(20,233,172,54)(21,49,173,236)(22,239,174,52)(23,55,175,234)(24,237,176,50)(25,266,212,111)(26,106,213,269)(27,272,214,109)(28,112,215,267)(29,270,216,107)(30,110,209,265)(31,268,210,105)(32,108,211,271)(33,261,312,79)(34,74,305,264)(35,259,306,77)(36,80,307,262)(37,257,308,75)(38,78,309,260)(39,263,310,73)(40,76,311,258)(41,144,315,465)(42,468,316,139)(43,142,317,471)(44,466,318,137)(45,140,319,469)(46,472,320,143)(47,138,313,467)(48,470,314,141)(57,285,336,103)(58,98,329,288)(59,283,330,101)(60,104,331,286)(61,281,332,99)(62,102,333,284)(63,287,334,97)(64,100,335,282)(65,126,339,365)(66,368,340,121)(67,124,341,363)(68,366,342,127)(69,122,343,361)(70,364,344,125)(71,128,337,367)(72,362,338,123)(81,353,244,186)(82,189,245,356)(83,359,246,192)(84,187,247,354)(85,357,248,190)(86,185,241,360)(87,355,242,188)(88,191,243,358)(89,393,151,196)(90,199,152,396)(91,399,145,194)(92,197,146,394)(93,397,147,200)(94,195,148,400)(95,395,149,198)(96,193,150,398)(113,417,162,220)(114,223,163,420)(115,423,164,218)(116,221,165,418)(117,421,166,224)(118,219,167,424)(119,419,168,222)(120,217,161,422)(129,326,451,389)(130,392,452,321)(131,324,453,387)(132,390,454,327)(133,322,455,385)(134,388,456,325)(135,328,449,391)(136,386,450,323)(153,350,475,413)(154,416,476,345)(155,348,477,411)(156,414,478,351)(157,346,479,409)(158,412,480,349)(159,352,473,415)(160,410,474,347)(177,250,374,438)(178,433,375,253)(179,256,376,436)(180,439,369,251)(181,254,370,434)(182,437,371,249)(183,252,372,440)(184,435,373,255)(201,274,406,462)(202,457,407,277)(203,280,408,460)(204,463,401,275)(205,278,402,458)(206,461,403,273)(207,276,404,464)(208,459,405,279)(225,298,430,382)(226,377,431,301)(227,304,432,380)(228,383,425,299)(229,302,426,378)(230,381,427,297)(231,300,428,384)(232,379,429,303) );
G=PermutationGroup([[(1,122,166,140,147),(2,123,167,141,148),(3,124,168,142,149),(4,125,161,143,150),(5,126,162,144,151),(6,127,163,137,152),(7,128,164,138,145),(8,121,165,139,146),(9,65,417,41,393),(10,66,418,42,394),(11,67,419,43,395),(12,68,420,44,396),(13,69,421,45,397),(14,70,422,46,398),(15,71,423,47,399),(16,72,424,48,400),(17,97,107,73,83),(18,98,108,74,84),(19,99,109,75,85),(20,100,110,76,86),(21,101,111,77,87),(22,102,112,78,88),(23,103,105,79,81),(24,104,106,80,82),(25,35,355,49,59),(26,36,356,50,60),(27,37,357,51,61),(28,38,358,52,62),(29,39,359,53,63),(30,40,360,54,64),(31,33,353,55,57),(32,34,354,56,58),(89,441,365,113,465),(90,442,366,114,466),(91,443,367,115,467),(92,444,368,116,468),(93,445,361,117,469),(94,446,362,118,470),(95,447,363,119,471),(96,448,364,120,472),(129,253,301,153,277),(130,254,302,154,278),(131,255,303,155,279),(132,256,304,156,280),(133,249,297,157,273),(134,250,298,158,274),(135,251,299,159,275),(136,252,300,160,276),(169,287,270,263,246),(170,288,271,264,247),(171,281,272,257,248),(172,282,265,258,241),(173,283,266,259,242),(174,284,267,260,243),(175,285,268,261,244),(176,286,269,262,245),(177,225,349,201,325),(178,226,350,202,326),(179,227,351,203,327),(180,228,352,204,328),(181,229,345,205,321),(182,230,346,206,322),(183,231,347,207,323),(184,232,348,208,324),(185,233,335,209,311),(186,234,336,210,312),(187,235,329,211,305),(188,236,330,212,306),(189,237,331,213,307),(190,238,332,214,308),(191,239,333,215,309),(192,240,334,216,310),(193,296,344,217,320),(194,289,337,218,313),(195,290,338,219,314),(196,291,339,220,315),(197,292,340,221,316),(198,293,341,222,317),(199,294,342,223,318),(200,295,343,224,319),(369,425,415,401,391),(370,426,416,402,392),(371,427,409,403,385),(372,428,410,404,386),(373,429,411,405,387),(374,430,412,406,388),(375,431,413,407,389),(376,432,414,408,390),(377,475,457,451,433),(378,476,458,452,434),(379,477,459,453,435),(380,478,460,454,436),(381,479,461,455,437),(382,480,462,456,438),(383,473,463,449,439),(384,474,464,450,440)], [(1,253,246,441,437,87),(2,88,438,442,247,254),(3,255,248,443,439,81),(4,82,440,444,241,256),(5,249,242,445,433,83),(6,84,434,446,243,250),(7,251,244,447,435,85),(8,86,436,448,245,252),(9,371,355,295,178,192),(10,185,179,296,356,372),(11,373,357,289,180,186),(12,187,181,290,358,374),(13,375,359,291,182,188),(14,189,183,292,360,376),(15,369,353,293,184,190),(16,191,177,294,354,370),(17,126,297,173,361,377),(18,378,362,174,298,127),(19,128,299,175,363,379),(20,380,364,176,300,121),(21,122,301,169,365,381),(22,382,366,170,302,123),(23,124,303,171,367,383),(24,384,368,172,304,125),(25,319,202,216,41,403),(26,404,42,209,203,320),(27,313,204,210,43,405),(28,406,44,211,205,314),(29,315,206,212,45,407),(30,408,46,213,207,316),(31,317,208,214,47,401),(32,402,48,215,201,318),(33,198,324,308,399,391),(34,392,400,309,325,199),(35,200,326,310,393,385),(36,386,394,311,327,193),(37,194,328,312,395,387),(38,388,396,305,321,195),(39,196,322,306,397,389),(40,390,398,307,323,197),(49,343,226,240,65,427),(50,428,66,233,227,344),(51,337,228,234,67,429),(52,430,68,235,229,338),(53,339,230,236,69,431),(54,432,70,237,231,340),(55,341,232,238,71,425),(56,426,72,239,225,342),(57,222,348,332,423,415),(58,416,424,333,349,223),(59,224,350,334,417,409),(60,410,418,335,351,217),(61,218,352,336,419,411),(62,412,420,329,345,219),(63,220,346,330,421,413),(64,414,422,331,347,221),(73,151,133,259,93,451),(74,452,94,260,134,152),(75,145,135,261,95,453),(76,454,96,262,136,146),(77,147,129,263,89,455),(78,456,90,264,130,148),(79,149,131,257,91,449),(80,450,92,258,132,150),(97,162,157,283,117,475),(98,476,118,284,158,163),(99,164,159,285,119,477),(100,478,120,286,160,165),(101,166,153,287,113,479),(102,480,114,288,154,167),(103,168,155,281,115,473),(104,474,116,282,156,161),(105,142,279,272,467,463),(106,464,468,265,280,143),(107,144,273,266,469,457),(108,458,470,267,274,137),(109,138,275,268,471,459),(110,460,472,269,276,139),(111,140,277,270,465,461),(112,462,466,271,278,141)], [(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)], [(1,295,445,13),(2,16,446,290),(3,293,447,11),(4,14,448,296),(5,291,441,9),(6,12,442,294),(7,289,443,15),(8,10,444,292),(17,53,169,240),(18,235,170,56),(19,51,171,238),(20,233,172,54),(21,49,173,236),(22,239,174,52),(23,55,175,234),(24,237,176,50),(25,266,212,111),(26,106,213,269),(27,272,214,109),(28,112,215,267),(29,270,216,107),(30,110,209,265),(31,268,210,105),(32,108,211,271),(33,261,312,79),(34,74,305,264),(35,259,306,77),(36,80,307,262),(37,257,308,75),(38,78,309,260),(39,263,310,73),(40,76,311,258),(41,144,315,465),(42,468,316,139),(43,142,317,471),(44,466,318,137),(45,140,319,469),(46,472,320,143),(47,138,313,467),(48,470,314,141),(57,285,336,103),(58,98,329,288),(59,283,330,101),(60,104,331,286),(61,281,332,99),(62,102,333,284),(63,287,334,97),(64,100,335,282),(65,126,339,365),(66,368,340,121),(67,124,341,363),(68,366,342,127),(69,122,343,361),(70,364,344,125),(71,128,337,367),(72,362,338,123),(81,353,244,186),(82,189,245,356),(83,359,246,192),(84,187,247,354),(85,357,248,190),(86,185,241,360),(87,355,242,188),(88,191,243,358),(89,393,151,196),(90,199,152,396),(91,399,145,194),(92,197,146,394),(93,397,147,200),(94,195,148,400),(95,395,149,198),(96,193,150,398),(113,417,162,220),(114,223,163,420),(115,423,164,218),(116,221,165,418),(117,421,166,224),(118,219,167,424),(119,419,168,222),(120,217,161,422),(129,326,451,389),(130,392,452,321),(131,324,453,387),(132,390,454,327),(133,322,455,385),(134,388,456,325),(135,328,449,391),(136,386,450,323),(153,350,475,413),(154,416,476,345),(155,348,477,411),(156,414,478,351),(157,346,479,409),(158,412,480,349),(159,352,473,415),(160,410,474,347),(177,250,374,438),(178,433,375,253),(179,256,376,436),(180,439,369,251),(181,254,370,434),(182,437,371,249),(183,252,372,440),(184,435,373,255),(201,274,406,462),(202,457,407,277),(203,280,408,460),(204,463,401,275),(205,278,402,458),(206,461,403,273),(207,276,404,464),(208,459,405,279),(225,298,430,382),(226,377,431,301),(227,304,432,380),(228,383,425,299),(229,302,426,378),(230,381,427,297),(231,300,428,384),(232,379,429,303)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20X | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
120 irreducible representations
| dim | 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 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | - | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D4 | D4 | D6 | SD16 | Q16 | C4×S3 | D12 | C3⋊D4 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | C5×SD16 | C5×Q16 | S3×C20 | C5×D12 | C5×C3⋊D4 | D4.S3 | C3⋊Q16 | C5×D4.S3 | C5×C3⋊Q16 |
| kernel | C5×C6.SD16 | C10×C3⋊C8 | C15×C4⋊C4 | C10×Dic6 | C5×Dic6 | C6.SD16 | C2×C3⋊C8 | C3×C4⋊C4 | C2×Dic6 | Dic6 | C5×C4⋊C4 | C60 | C2×C30 | C2×C20 | C30 | C30 | C20 | C20 | C2×C10 | C4⋊C4 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C6.SD16 ►in GL4(𝔽241) generated by
| 205 | 0 | 0 | 0 |
| 0 | 205 | 0 | 0 |
| 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 87 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 240 |
| 0 | 0 | 1 | 0 |
| 230 | 11 | 0 | 0 |
| 230 | 230 | 0 | 0 |
| 0 | 0 | 118 | 225 |
| 0 | 0 | 102 | 123 |
| 45 | 216 | 0 | 0 |
| 216 | 196 | 0 | 0 |
| 0 | 0 | 99 | 43 |
| 0 | 0 | 198 | 142 |
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,87,0,0,0,0,87],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,240,0],[230,230,0,0,11,230,0,0,0,0,118,102,0,0,225,123],[45,216,0,0,216,196,0,0,0,0,99,198,0,0,43,142] >;
C5×C6.SD16 in GAP, Magma, Sage, TeX
C_5\times C_6.{\rm SD}_{16} % in TeX
G:=Group("C5xC6.SD16"); // GroupNames label
G:=SmallGroup(480,129);
// by ID
G=gap.SmallGroup(480,129);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,560,589,148,4204,2111,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^6=c^8=1,d^2=b^3*c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c^-1>;
// generators/relations