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## G = C5×C6.SD16order 480 = 25·3·5

### Direct product of C5 and C6.SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C6.SD16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C60 — C10×Dic6 — C5×C6.SD16
 Lower central C3 — C6 — C12 — C5×C6.SD16
 Upper central C1 — C2×C10 — C2×C20 — C5×C4⋊C4

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 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C10 [×3], Dic3 [×2], C12 [×2], C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, C20 [×2], C20 [×3], C2×C10, C3⋊C8, Dic6 [×2], Dic6, C2×Dic3, C2×C12, C2×C12, C30 [×3], Q8⋊C4, C40, C2×C20, C2×C20 [×2], C5×Q8 [×3], C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C5×Dic3 [×2], C60 [×2], C60, C2×C30, C5×C4⋊C4, C2×C40, Q8×C10, C6.SD16, C5×C3⋊C8, C5×Dic6 [×2], 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 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], D6, C22⋊C4, SD16, Q16, C20 [×2], C2×C10, C4×S3, D12, C3⋊D4, C5×S3, Q8⋊C4, C2×C20, C5×D4 [×2], 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

Smallest permutation representation of C5×C6.SD16
Regular action on 480 points
Generators in S480
(1 124 273 151 249)(2 125 274 152 250)(3 126 275 145 251)(4 127 276 146 252)(5 128 277 147 253)(6 121 278 148 254)(7 122 279 149 255)(8 123 280 150 256)(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)(17 118 470 94 446)(18 119 471 95 447)(19 120 472 96 448)(20 113 465 89 441)(21 114 466 90 442)(22 115 467 91 443)(23 116 468 92 444)(24 117 469 93 445)(25 374 430 49 406)(26 375 431 50 407)(27 376 432 51 408)(28 369 425 52 401)(29 370 426 53 402)(30 371 427 54 403)(31 372 428 55 404)(32 373 429 56 405)(33 353 412 57 388)(34 354 413 58 389)(35 355 414 59 390)(36 356 415 60 391)(37 357 416 61 392)(38 358 409 62 385)(39 359 410 63 386)(40 360 411 64 387)(73 380 478 97 454)(74 381 479 98 455)(75 382 480 99 456)(76 383 473 100 449)(77 384 474 101 450)(78 377 475 102 451)(79 378 476 103 452)(80 379 477 104 453)(81 362 460 105 436)(82 363 461 106 437)(83 364 462 107 438)(84 365 463 108 439)(85 366 464 109 440)(86 367 457 110 433)(87 368 458 111 434)(88 361 459 112 435)(129 247 301 153 271)(130 248 302 154 272)(131 241 303 155 265)(132 242 304 156 266)(133 243 297 157 267)(134 244 298 158 268)(135 245 299 159 269)(136 246 300 160 270)(137 169 287 161 263)(138 170 288 162 264)(139 171 281 163 257)(140 172 282 164 258)(141 173 283 165 259)(142 174 284 166 260)(143 175 285 167 261)(144 176 286 168 262)(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)(193 291 339 217 315)(194 292 340 218 316)(195 293 341 219 317)(196 294 342 220 318)(197 295 343 221 319)(198 296 344 222 320)(199 289 337 223 313)(200 290 338 224 314)
(1 247 137 441 74 87)(2 88 75 442 138 248)(3 241 139 443 76 81)(4 82 77 444 140 242)(5 243 141 445 78 83)(6 84 79 446 142 244)(7 245 143 447 80 85)(8 86 73 448 144 246)(9 371 355 290 178 192)(10 185 179 291 356 372)(11 373 357 292 180 186)(12 187 181 293 358 374)(13 375 359 294 182 188)(14 189 183 295 360 376)(15 369 353 296 184 190)(16 191 177 289 354 370)(17 174 298 121 365 378)(18 379 366 122 299 175)(19 176 300 123 367 380)(20 381 368 124 301 169)(21 170 302 125 361 382)(22 383 362 126 303 171)(23 172 304 127 363 384)(24 377 364 128 297 173)(25 396 305 321 195 38)(26 39 196 322 306 397)(27 398 307 323 197 40)(28 33 198 324 308 399)(29 400 309 325 199 34)(30 35 200 326 310 393)(31 394 311 327 193 36)(32 37 194 328 312 395)(41 403 390 314 202 216)(42 209 203 315 391 404)(43 405 392 316 204 210)(44 211 205 317 385 406)(45 407 386 318 206 212)(46 213 207 319 387 408)(47 401 388 320 208 214)(48 215 201 313 389 402)(49 420 329 345 219 62)(50 63 220 346 330 421)(51 422 331 347 221 64)(52 57 222 348 332 423)(53 424 333 349 223 58)(54 59 224 350 334 417)(55 418 335 351 217 60)(56 61 218 352 336 419)(65 427 414 338 226 240)(66 233 227 339 415 428)(67 429 416 340 228 234)(68 235 229 341 409 430)(69 431 410 342 230 236)(70 237 231 343 411 432)(71 425 412 344 232 238)(72 239 225 337 413 426)(89 455 434 249 129 263)(90 264 130 250 435 456)(91 449 436 251 131 257)(92 258 132 252 437 450)(93 451 438 253 133 259)(94 260 134 254 439 452)(95 453 440 255 135 261)(96 262 136 256 433 454)(97 472 168 270 150 110)(98 111 151 271 161 465)(99 466 162 272 152 112)(100 105 145 265 163 467)(101 468 164 266 146 106)(102 107 147 267 165 469)(103 470 166 268 148 108)(104 109 149 269 167 471)(113 479 458 273 153 287)(114 288 154 274 459 480)(115 473 460 275 155 281)(116 282 156 276 461 474)(117 475 462 277 157 283)(118 284 158 278 463 476)(119 477 464 279 159 285)(120 286 160 280 457 478)
(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)
(1 290 445 13)(2 16 446 293)(3 296 447 11)(4 14 448 291)(5 294 441 9)(6 12 442 289)(7 292 443 15)(8 10 444 295)(17 341 125 72)(18 67 126 344)(19 339 127 70)(20 65 128 342)(21 337 121 68)(22 71 122 340)(23 343 123 66)(24 69 124 338)(25 456 325 134)(26 129 326 451)(27 454 327 132)(28 135 328 449)(29 452 321 130)(30 133 322 455)(31 450 323 136)(32 131 324 453)(33 261 312 436)(34 439 305 264)(35 259 306 434)(36 437 307 262)(37 257 308 440)(38 435 309 260)(39 263 310 438)(40 433 311 258)(41 147 318 465)(42 468 319 150)(43 145 320 471)(44 466 313 148)(45 151 314 469)(46 472 315 146)(47 149 316 467)(48 470 317 152)(49 480 349 158)(50 153 350 475)(51 478 351 156)(52 159 352 473)(53 476 345 154)(54 157 346 479)(55 474 347 160)(56 155 348 477)(57 285 336 460)(58 463 329 288)(59 283 330 458)(60 461 331 286)(61 281 332 464)(62 459 333 284)(63 287 334 462)(64 457 335 282)(73 179 242 376)(74 371 243 182)(75 177 244 374)(76 369 245 180)(77 183 246 372)(78 375 247 178)(79 181 248 370)(80 373 241 184)(81 353 143 186)(82 189 144 356)(83 359 137 192)(84 187 138 354)(85 357 139 190)(86 185 140 360)(87 355 141 188)(88 191 142 358)(89 393 253 196)(90 199 254 396)(91 399 255 194)(92 197 256 394)(93 397 249 200)(94 195 250 400)(95 395 251 198)(96 193 252 398)(97 203 266 408)(98 403 267 206)(99 201 268 406)(100 401 269 204)(101 207 270 404)(102 407 271 202)(103 205 272 402)(104 405 265 208)(105 388 167 210)(106 213 168 391)(107 386 161 216)(108 211 162 389)(109 392 163 214)(110 209 164 387)(111 390 165 212)(112 215 166 385)(113 417 277 220)(114 223 278 420)(115 423 279 218)(116 221 280 418)(117 421 273 224)(118 219 274 424)(119 419 275 222)(120 217 276 422)(169 240 364 410)(170 413 365 235)(171 238 366 416)(172 411 367 233)(173 236 368 414)(174 409 361 239)(175 234 362 412)(176 415 363 237)(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)| (1,124,273,151,249)(2,125,274,152,250)(3,126,275,145,251)(4,127,276,146,252)(5,128,277,147,253)(6,121,278,148,254)(7,122,279,149,255)(8,123,280,150,256)(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)(17,118,470,94,446)(18,119,471,95,447)(19,120,472,96,448)(20,113,465,89,441)(21,114,466,90,442)(22,115,467,91,443)(23,116,468,92,444)(24,117,469,93,445)(25,374,430,49,406)(26,375,431,50,407)(27,376,432,51,408)(28,369,425,52,401)(29,370,426,53,402)(30,371,427,54,403)(31,372,428,55,404)(32,373,429,56,405)(33,353,412,57,388)(34,354,413,58,389)(35,355,414,59,390)(36,356,415,60,391)(37,357,416,61,392)(38,358,409,62,385)(39,359,410,63,386)(40,360,411,64,387)(73,380,478,97,454)(74,381,479,98,455)(75,382,480,99,456)(76,383,473,100,449)(77,384,474,101,450)(78,377,475,102,451)(79,378,476,103,452)(80,379,477,104,453)(81,362,460,105,436)(82,363,461,106,437)(83,364,462,107,438)(84,365,463,108,439)(85,366,464,109,440)(86,367,457,110,433)(87,368,458,111,434)(88,361,459,112,435)(129,247,301,153,271)(130,248,302,154,272)(131,241,303,155,265)(132,242,304,156,266)(133,243,297,157,267)(134,244,298,158,268)(135,245,299,159,269)(136,246,300,160,270)(137,169,287,161,263)(138,170,288,162,264)(139,171,281,163,257)(140,172,282,164,258)(141,173,283,165,259)(142,174,284,166,260)(143,175,285,167,261)(144,176,286,168,262)(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)(193,291,339,217,315)(194,292,340,218,316)(195,293,341,219,317)(196,294,342,220,318)(197,295,343,221,319)(198,296,344,222,320)(199,289,337,223,313)(200,290,338,224,314), (1,247,137,441,74,87)(2,88,75,442,138,248)(3,241,139,443,76,81)(4,82,77,444,140,242)(5,243,141,445,78,83)(6,84,79,446,142,244)(7,245,143,447,80,85)(8,86,73,448,144,246)(9,371,355,290,178,192)(10,185,179,291,356,372)(11,373,357,292,180,186)(12,187,181,293,358,374)(13,375,359,294,182,188)(14,189,183,295,360,376)(15,369,353,296,184,190)(16,191,177,289,354,370)(17,174,298,121,365,378)(18,379,366,122,299,175)(19,176,300,123,367,380)(20,381,368,124,301,169)(21,170,302,125,361,382)(22,383,362,126,303,171)(23,172,304,127,363,384)(24,377,364,128,297,173)(25,396,305,321,195,38)(26,39,196,322,306,397)(27,398,307,323,197,40)(28,33,198,324,308,399)(29,400,309,325,199,34)(30,35,200,326,310,393)(31,394,311,327,193,36)(32,37,194,328,312,395)(41,403,390,314,202,216)(42,209,203,315,391,404)(43,405,392,316,204,210)(44,211,205,317,385,406)(45,407,386,318,206,212)(46,213,207,319,387,408)(47,401,388,320,208,214)(48,215,201,313,389,402)(49,420,329,345,219,62)(50,63,220,346,330,421)(51,422,331,347,221,64)(52,57,222,348,332,423)(53,424,333,349,223,58)(54,59,224,350,334,417)(55,418,335,351,217,60)(56,61,218,352,336,419)(65,427,414,338,226,240)(66,233,227,339,415,428)(67,429,416,340,228,234)(68,235,229,341,409,430)(69,431,410,342,230,236)(70,237,231,343,411,432)(71,425,412,344,232,238)(72,239,225,337,413,426)(89,455,434,249,129,263)(90,264,130,250,435,456)(91,449,436,251,131,257)(92,258,132,252,437,450)(93,451,438,253,133,259)(94,260,134,254,439,452)(95,453,440,255,135,261)(96,262,136,256,433,454)(97,472,168,270,150,110)(98,111,151,271,161,465)(99,466,162,272,152,112)(100,105,145,265,163,467)(101,468,164,266,146,106)(102,107,147,267,165,469)(103,470,166,268,148,108)(104,109,149,269,167,471)(113,479,458,273,153,287)(114,288,154,274,459,480)(115,473,460,275,155,281)(116,282,156,276,461,474)(117,475,462,277,157,283)(118,284,158,278,463,476)(119,477,464,279,159,285)(120,286,160,280,457,478), (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), (1,290,445,13)(2,16,446,293)(3,296,447,11)(4,14,448,291)(5,294,441,9)(6,12,442,289)(7,292,443,15)(8,10,444,295)(17,341,125,72)(18,67,126,344)(19,339,127,70)(20,65,128,342)(21,337,121,68)(22,71,122,340)(23,343,123,66)(24,69,124,338)(25,456,325,134)(26,129,326,451)(27,454,327,132)(28,135,328,449)(29,452,321,130)(30,133,322,455)(31,450,323,136)(32,131,324,453)(33,261,312,436)(34,439,305,264)(35,259,306,434)(36,437,307,262)(37,257,308,440)(38,435,309,260)(39,263,310,438)(40,433,311,258)(41,147,318,465)(42,468,319,150)(43,145,320,471)(44,466,313,148)(45,151,314,469)(46,472,315,146)(47,149,316,467)(48,470,317,152)(49,480,349,158)(50,153,350,475)(51,478,351,156)(52,159,352,473)(53,476,345,154)(54,157,346,479)(55,474,347,160)(56,155,348,477)(57,285,336,460)(58,463,329,288)(59,283,330,458)(60,461,331,286)(61,281,332,464)(62,459,333,284)(63,287,334,462)(64,457,335,282)(73,179,242,376)(74,371,243,182)(75,177,244,374)(76,369,245,180)(77,183,246,372)(78,375,247,178)(79,181,248,370)(80,373,241,184)(81,353,143,186)(82,189,144,356)(83,359,137,192)(84,187,138,354)(85,357,139,190)(86,185,140,360)(87,355,141,188)(88,191,142,358)(89,393,253,196)(90,199,254,396)(91,399,255,194)(92,197,256,394)(93,397,249,200)(94,195,250,400)(95,395,251,198)(96,193,252,398)(97,203,266,408)(98,403,267,206)(99,201,268,406)(100,401,269,204)(101,207,270,404)(102,407,271,202)(103,205,272,402)(104,405,265,208)(105,388,167,210)(106,213,168,391)(107,386,161,216)(108,211,162,389)(109,392,163,214)(110,209,164,387)(111,390,165,212)(112,215,166,385)(113,417,277,220)(114,223,278,420)(115,423,279,218)(116,221,280,418)(117,421,273,224)(118,219,274,424)(119,419,275,222)(120,217,276,422)(169,240,364,410)(170,413,365,235)(171,238,366,416)(172,411,367,233)(173,236,368,414)(174,409,361,239)(175,234,362,412)(176,415,363,237)(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( (1,124,273,151,249)(2,125,274,152,250)(3,126,275,145,251)(4,127,276,146,252)(5,128,277,147,253)(6,121,278,148,254)(7,122,279,149,255)(8,123,280,150,256)(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)(17,118,470,94,446)(18,119,471,95,447)(19,120,472,96,448)(20,113,465,89,441)(21,114,466,90,442)(22,115,467,91,443)(23,116,468,92,444)(24,117,469,93,445)(25,374,430,49,406)(26,375,431,50,407)(27,376,432,51,408)(28,369,425,52,401)(29,370,426,53,402)(30,371,427,54,403)(31,372,428,55,404)(32,373,429,56,405)(33,353,412,57,388)(34,354,413,58,389)(35,355,414,59,390)(36,356,415,60,391)(37,357,416,61,392)(38,358,409,62,385)(39,359,410,63,386)(40,360,411,64,387)(73,380,478,97,454)(74,381,479,98,455)(75,382,480,99,456)(76,383,473,100,449)(77,384,474,101,450)(78,377,475,102,451)(79,378,476,103,452)(80,379,477,104,453)(81,362,460,105,436)(82,363,461,106,437)(83,364,462,107,438)(84,365,463,108,439)(85,366,464,109,440)(86,367,457,110,433)(87,368,458,111,434)(88,361,459,112,435)(129,247,301,153,271)(130,248,302,154,272)(131,241,303,155,265)(132,242,304,156,266)(133,243,297,157,267)(134,244,298,158,268)(135,245,299,159,269)(136,246,300,160,270)(137,169,287,161,263)(138,170,288,162,264)(139,171,281,163,257)(140,172,282,164,258)(141,173,283,165,259)(142,174,284,166,260)(143,175,285,167,261)(144,176,286,168,262)(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)(193,291,339,217,315)(194,292,340,218,316)(195,293,341,219,317)(196,294,342,220,318)(197,295,343,221,319)(198,296,344,222,320)(199,289,337,223,313)(200,290,338,224,314), (1,247,137,441,74,87)(2,88,75,442,138,248)(3,241,139,443,76,81)(4,82,77,444,140,242)(5,243,141,445,78,83)(6,84,79,446,142,244)(7,245,143,447,80,85)(8,86,73,448,144,246)(9,371,355,290,178,192)(10,185,179,291,356,372)(11,373,357,292,180,186)(12,187,181,293,358,374)(13,375,359,294,182,188)(14,189,183,295,360,376)(15,369,353,296,184,190)(16,191,177,289,354,370)(17,174,298,121,365,378)(18,379,366,122,299,175)(19,176,300,123,367,380)(20,381,368,124,301,169)(21,170,302,125,361,382)(22,383,362,126,303,171)(23,172,304,127,363,384)(24,377,364,128,297,173)(25,396,305,321,195,38)(26,39,196,322,306,397)(27,398,307,323,197,40)(28,33,198,324,308,399)(29,400,309,325,199,34)(30,35,200,326,310,393)(31,394,311,327,193,36)(32,37,194,328,312,395)(41,403,390,314,202,216)(42,209,203,315,391,404)(43,405,392,316,204,210)(44,211,205,317,385,406)(45,407,386,318,206,212)(46,213,207,319,387,408)(47,401,388,320,208,214)(48,215,201,313,389,402)(49,420,329,345,219,62)(50,63,220,346,330,421)(51,422,331,347,221,64)(52,57,222,348,332,423)(53,424,333,349,223,58)(54,59,224,350,334,417)(55,418,335,351,217,60)(56,61,218,352,336,419)(65,427,414,338,226,240)(66,233,227,339,415,428)(67,429,416,340,228,234)(68,235,229,341,409,430)(69,431,410,342,230,236)(70,237,231,343,411,432)(71,425,412,344,232,238)(72,239,225,337,413,426)(89,455,434,249,129,263)(90,264,130,250,435,456)(91,449,436,251,131,257)(92,258,132,252,437,450)(93,451,438,253,133,259)(94,260,134,254,439,452)(95,453,440,255,135,261)(96,262,136,256,433,454)(97,472,168,270,150,110)(98,111,151,271,161,465)(99,466,162,272,152,112)(100,105,145,265,163,467)(101,468,164,266,146,106)(102,107,147,267,165,469)(103,470,166,268,148,108)(104,109,149,269,167,471)(113,479,458,273,153,287)(114,288,154,274,459,480)(115,473,460,275,155,281)(116,282,156,276,461,474)(117,475,462,277,157,283)(118,284,158,278,463,476)(119,477,464,279,159,285)(120,286,160,280,457,478), (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), (1,290,445,13)(2,16,446,293)(3,296,447,11)(4,14,448,291)(5,294,441,9)(6,12,442,289)(7,292,443,15)(8,10,444,295)(17,341,125,72)(18,67,126,344)(19,339,127,70)(20,65,128,342)(21,337,121,68)(22,71,122,340)(23,343,123,66)(24,69,124,338)(25,456,325,134)(26,129,326,451)(27,454,327,132)(28,135,328,449)(29,452,321,130)(30,133,322,455)(31,450,323,136)(32,131,324,453)(33,261,312,436)(34,439,305,264)(35,259,306,434)(36,437,307,262)(37,257,308,440)(38,435,309,260)(39,263,310,438)(40,433,311,258)(41,147,318,465)(42,468,319,150)(43,145,320,471)(44,466,313,148)(45,151,314,469)(46,472,315,146)(47,149,316,467)(48,470,317,152)(49,480,349,158)(50,153,350,475)(51,478,351,156)(52,159,352,473)(53,476,345,154)(54,157,346,479)(55,474,347,160)(56,155,348,477)(57,285,336,460)(58,463,329,288)(59,283,330,458)(60,461,331,286)(61,281,332,464)(62,459,333,284)(63,287,334,462)(64,457,335,282)(73,179,242,376)(74,371,243,182)(75,177,244,374)(76,369,245,180)(77,183,246,372)(78,375,247,178)(79,181,248,370)(80,373,241,184)(81,353,143,186)(82,189,144,356)(83,359,137,192)(84,187,138,354)(85,357,139,190)(86,185,140,360)(87,355,141,188)(88,191,142,358)(89,393,253,196)(90,199,254,396)(91,399,255,194)(92,197,256,394)(93,397,249,200)(94,195,250,400)(95,395,251,198)(96,193,252,398)(97,203,266,408)(98,403,267,206)(99,201,268,406)(100,401,269,204)(101,207,270,404)(102,407,271,202)(103,205,272,402)(104,405,265,208)(105,388,167,210)(106,213,168,391)(107,386,161,216)(108,211,162,389)(109,392,163,214)(110,209,164,387)(111,390,165,212)(112,215,166,385)(113,417,277,220)(114,223,278,420)(115,423,279,218)(116,221,280,418)(117,421,273,224)(118,219,274,424)(119,419,275,222)(120,217,276,422)(169,240,364,410)(170,413,365,235)(171,238,366,416)(172,411,367,233)(173,236,368,414)(174,409,361,239)(175,234,362,412)(176,415,363,237)(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([(1,124,273,151,249),(2,125,274,152,250),(3,126,275,145,251),(4,127,276,146,252),(5,128,277,147,253),(6,121,278,148,254),(7,122,279,149,255),(8,123,280,150,256),(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),(17,118,470,94,446),(18,119,471,95,447),(19,120,472,96,448),(20,113,465,89,441),(21,114,466,90,442),(22,115,467,91,443),(23,116,468,92,444),(24,117,469,93,445),(25,374,430,49,406),(26,375,431,50,407),(27,376,432,51,408),(28,369,425,52,401),(29,370,426,53,402),(30,371,427,54,403),(31,372,428,55,404),(32,373,429,56,405),(33,353,412,57,388),(34,354,413,58,389),(35,355,414,59,390),(36,356,415,60,391),(37,357,416,61,392),(38,358,409,62,385),(39,359,410,63,386),(40,360,411,64,387),(73,380,478,97,454),(74,381,479,98,455),(75,382,480,99,456),(76,383,473,100,449),(77,384,474,101,450),(78,377,475,102,451),(79,378,476,103,452),(80,379,477,104,453),(81,362,460,105,436),(82,363,461,106,437),(83,364,462,107,438),(84,365,463,108,439),(85,366,464,109,440),(86,367,457,110,433),(87,368,458,111,434),(88,361,459,112,435),(129,247,301,153,271),(130,248,302,154,272),(131,241,303,155,265),(132,242,304,156,266),(133,243,297,157,267),(134,244,298,158,268),(135,245,299,159,269),(136,246,300,160,270),(137,169,287,161,263),(138,170,288,162,264),(139,171,281,163,257),(140,172,282,164,258),(141,173,283,165,259),(142,174,284,166,260),(143,175,285,167,261),(144,176,286,168,262),(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),(193,291,339,217,315),(194,292,340,218,316),(195,293,341,219,317),(196,294,342,220,318),(197,295,343,221,319),(198,296,344,222,320),(199,289,337,223,313),(200,290,338,224,314)], [(1,247,137,441,74,87),(2,88,75,442,138,248),(3,241,139,443,76,81),(4,82,77,444,140,242),(5,243,141,445,78,83),(6,84,79,446,142,244),(7,245,143,447,80,85),(8,86,73,448,144,246),(9,371,355,290,178,192),(10,185,179,291,356,372),(11,373,357,292,180,186),(12,187,181,293,358,374),(13,375,359,294,182,188),(14,189,183,295,360,376),(15,369,353,296,184,190),(16,191,177,289,354,370),(17,174,298,121,365,378),(18,379,366,122,299,175),(19,176,300,123,367,380),(20,381,368,124,301,169),(21,170,302,125,361,382),(22,383,362,126,303,171),(23,172,304,127,363,384),(24,377,364,128,297,173),(25,396,305,321,195,38),(26,39,196,322,306,397),(27,398,307,323,197,40),(28,33,198,324,308,399),(29,400,309,325,199,34),(30,35,200,326,310,393),(31,394,311,327,193,36),(32,37,194,328,312,395),(41,403,390,314,202,216),(42,209,203,315,391,404),(43,405,392,316,204,210),(44,211,205,317,385,406),(45,407,386,318,206,212),(46,213,207,319,387,408),(47,401,388,320,208,214),(48,215,201,313,389,402),(49,420,329,345,219,62),(50,63,220,346,330,421),(51,422,331,347,221,64),(52,57,222,348,332,423),(53,424,333,349,223,58),(54,59,224,350,334,417),(55,418,335,351,217,60),(56,61,218,352,336,419),(65,427,414,338,226,240),(66,233,227,339,415,428),(67,429,416,340,228,234),(68,235,229,341,409,430),(69,431,410,342,230,236),(70,237,231,343,411,432),(71,425,412,344,232,238),(72,239,225,337,413,426),(89,455,434,249,129,263),(90,264,130,250,435,456),(91,449,436,251,131,257),(92,258,132,252,437,450),(93,451,438,253,133,259),(94,260,134,254,439,452),(95,453,440,255,135,261),(96,262,136,256,433,454),(97,472,168,270,150,110),(98,111,151,271,161,465),(99,466,162,272,152,112),(100,105,145,265,163,467),(101,468,164,266,146,106),(102,107,147,267,165,469),(103,470,166,268,148,108),(104,109,149,269,167,471),(113,479,458,273,153,287),(114,288,154,274,459,480),(115,473,460,275,155,281),(116,282,156,276,461,474),(117,475,462,277,157,283),(118,284,158,278,463,476),(119,477,464,279,159,285),(120,286,160,280,457,478)], [(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)], [(1,290,445,13),(2,16,446,293),(3,296,447,11),(4,14,448,291),(5,294,441,9),(6,12,442,289),(7,292,443,15),(8,10,444,295),(17,341,125,72),(18,67,126,344),(19,339,127,70),(20,65,128,342),(21,337,121,68),(22,71,122,340),(23,343,123,66),(24,69,124,338),(25,456,325,134),(26,129,326,451),(27,454,327,132),(28,135,328,449),(29,452,321,130),(30,133,322,455),(31,450,323,136),(32,131,324,453),(33,261,312,436),(34,439,305,264),(35,259,306,434),(36,437,307,262),(37,257,308,440),(38,435,309,260),(39,263,310,438),(40,433,311,258),(41,147,318,465),(42,468,319,150),(43,145,320,471),(44,466,313,148),(45,151,314,469),(46,472,315,146),(47,149,316,467),(48,470,317,152),(49,480,349,158),(50,153,350,475),(51,478,351,156),(52,159,352,473),(53,476,345,154),(54,157,346,479),(55,474,347,160),(56,155,348,477),(57,285,336,460),(58,463,329,288),(59,283,330,458),(60,461,331,286),(61,281,332,464),(62,459,333,284),(63,287,334,462),(64,457,335,282),(73,179,242,376),(74,371,243,182),(75,177,244,374),(76,369,245,180),(77,183,246,372),(78,375,247,178),(79,181,248,370),(80,373,241,184),(81,353,143,186),(82,189,144,356),(83,359,137,192),(84,187,138,354),(85,357,139,190),(86,185,140,360),(87,355,141,188),(88,191,142,358),(89,393,253,196),(90,199,254,396),(91,399,255,194),(92,197,256,394),(93,397,249,200),(94,195,250,400),(95,395,251,198),(96,193,252,398),(97,203,266,408),(98,403,267,206),(99,201,268,406),(100,401,269,204),(101,207,270,404),(102,407,271,202),(103,205,272,402),(104,405,265,208),(105,388,167,210),(106,213,168,391),(107,386,161,216),(108,211,162,389),(109,392,163,214),(110,209,164,387),(111,390,165,212),(112,215,166,385),(113,417,277,220),(114,223,278,420),(115,423,279,218),(116,221,280,418),(117,421,273,224),(118,219,274,424),(119,419,275,222),(120,217,276,422),(169,240,364,410),(170,413,365,235),(171,238,366,416),(172,411,367,233),(173,236,368,414),(174,409,361,239),(175,234,362,412),(176,415,363,237),(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

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