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## G = C5×C6.Q16order 480 = 25·3·5

### Direct product of C5 and C6.Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C6.Q16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C60 — C10×C3⋊C8 — C5×C6.Q16
 Lower central C3 — C6 — C12 — C5×C6.Q16
 Upper central C1 — C2×C10 — C2×C20 — C5×C4⋊C4

Generators and relations for C5×C6.Q16
G = < a,b,c,d | a5=b12=c4=1, d2=b9c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b9c-1 >

Subgroups: 148 in 72 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C4⋊C4, C2×C8, C20, C20, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C30, C2.D8, C40, C2×C20, C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C60, C60, C2×C30, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C6.Q16, C5×C3⋊C8, C10×Dic3, C2×C60, C2×C60, C5×C2.D8, C10×C3⋊C8, C5×C4⋊Dic3, C15×C4⋊C4, C5×C6.Q16
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, Q8, C10, D6, C4⋊C4, D8, Q16, C20, C2×C10, Dic6, C4×S3, C3⋊D4, C5×S3, C2.D8, C2×C20, C5×D4, C5×Q8, Dic3⋊C4, D4⋊S3, C3⋊Q16, S3×C10, C5×C4⋊C4, C5×D8, C5×Q16, C6.Q16, C5×Dic6, S3×C20, C5×C3⋊D4, C5×C2.D8, C5×Dic3⋊C4, C5×D4⋊S3, C5×C3⋊Q16, C5×C6.Q16

Smallest permutation representation of C5×C6.Q16
Regular action on 480 points
Generators in S480
(1 410 355 199 388)(2 411 356 200 389)(3 412 357 201 390)(4 413 358 202 391)(5 414 359 203 392)(6 415 360 204 393)(7 416 349 193 394)(8 417 350 194 395)(9 418 351 195 396)(10 419 352 196 385)(11 420 353 197 386)(12 409 354 198 387)(13 424 67 369 245)(14 425 68 370 246)(15 426 69 371 247)(16 427 70 372 248)(17 428 71 361 249)(18 429 72 362 250)(19 430 61 363 251)(20 431 62 364 252)(21 432 63 365 241)(22 421 64 366 242)(23 422 65 367 243)(24 423 66 368 244)(25 190 143 104 403)(26 191 144 105 404)(27 192 133 106 405)(28 181 134 107 406)(29 182 135 108 407)(30 183 136 97 408)(31 184 137 98 397)(32 185 138 99 398)(33 186 139 100 399)(34 187 140 101 400)(35 188 141 102 401)(36 189 142 103 402)(37 296 236 75 320)(38 297 237 76 321)(39 298 238 77 322)(40 299 239 78 323)(41 300 240 79 324)(42 289 229 80 313)(43 290 230 81 314)(44 291 231 82 315)(45 292 232 83 316)(46 293 233 84 317)(47 294 234 73 318)(48 295 235 74 319)(49 124 110 275 379)(50 125 111 276 380)(51 126 112 265 381)(52 127 113 266 382)(53 128 114 267 383)(54 129 115 268 384)(55 130 116 269 373)(56 131 117 270 374)(57 132 118 271 375)(58 121 119 272 376)(59 122 120 273 377)(60 123 109 274 378)(85 147 221 472 454)(86 148 222 473 455)(87 149 223 474 456)(88 150 224 475 445)(89 151 225 476 446)(90 152 226 477 447)(91 153 227 478 448)(92 154 228 479 449)(93 155 217 480 450)(94 156 218 469 451)(95 145 219 470 452)(96 146 220 471 453)(157 467 333 215 277)(158 468 334 216 278)(159 457 335 205 279)(160 458 336 206 280)(161 459 325 207 281)(162 460 326 208 282)(163 461 327 209 283)(164 462 328 210 284)(165 463 329 211 285)(166 464 330 212 286)(167 465 331 213 287)(168 466 332 214 288)(169 440 259 308 341)(170 441 260 309 342)(171 442 261 310 343)(172 443 262 311 344)(173 444 263 312 345)(174 433 264 301 346)(175 434 253 302 347)(176 435 254 303 348)(177 436 255 304 337)(178 437 256 305 338)(179 438 257 306 339)(180 439 258 307 340)
(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)
(1 427 308 320)(2 422 309 315)(3 429 310 322)(4 424 311 317)(5 431 312 324)(6 426 301 319)(7 421 302 314)(8 428 303 321)(9 423 304 316)(10 430 305 323)(11 425 306 318)(12 432 307 313)(13 262 84 391)(14 257 73 386)(15 264 74 393)(16 259 75 388)(17 254 76 395)(18 261 77 390)(19 256 78 385)(20 263 79 392)(21 258 80 387)(22 253 81 394)(23 260 82 389)(24 255 83 396)(25 166 470 266)(26 161 471 273)(27 168 472 268)(28 163 473 275)(29 158 474 270)(30 165 475 265)(31 160 476 272)(32 167 477 267)(33 162 478 274)(34 157 479 269)(35 164 480 276)(36 159 469 271)(37 410 70 341)(38 417 71 348)(39 412 72 343)(40 419 61 338)(41 414 62 345)(42 409 63 340)(43 416 64 347)(44 411 65 342)(45 418 66 337)(46 413 67 344)(47 420 68 339)(48 415 69 346)(49 134 327 86)(50 141 328 93)(51 136 329 88)(52 143 330 95)(53 138 331 90)(54 133 332 85)(55 140 333 92)(56 135 334 87)(57 142 335 94)(58 137 336 89)(59 144 325 96)(60 139 326 91)(97 211 150 126)(98 206 151 121)(99 213 152 128)(100 208 153 123)(101 215 154 130)(102 210 155 125)(103 205 156 132)(104 212 145 127)(105 207 146 122)(106 214 147 129)(107 209 148 124)(108 216 149 131)(109 399 282 227)(110 406 283 222)(111 401 284 217)(112 408 285 224)(113 403 286 219)(114 398 287 226)(115 405 288 221)(116 400 277 228)(117 407 278 223)(118 402 279 218)(119 397 280 225)(120 404 281 220)(169 296 355 372)(170 291 356 367)(171 298 357 362)(172 293 358 369)(173 300 359 364)(174 295 360 371)(175 290 349 366)(176 297 350 361)(177 292 351 368)(178 299 352 363)(179 294 353 370)(180 289 354 365)(181 461 455 379)(182 468 456 374)(183 463 445 381)(184 458 446 376)(185 465 447 383)(186 460 448 378)(187 467 449 373)(188 462 450 380)(189 457 451 375)(190 464 452 382)(191 459 453 377)(192 466 454 384)(193 242 434 230)(194 249 435 237)(195 244 436 232)(196 251 437 239)(197 246 438 234)(198 241 439 229)(199 248 440 236)(200 243 441 231)(201 250 442 238)(202 245 443 233)(203 252 444 240)(204 247 433 235)
(1 125 305 207 7 131 311 213)(2 130 306 212 8 124 312 206)(3 123 307 205 9 129 301 211)(4 128 308 210 10 122 302 216)(5 121 309 215 11 127 303 209)(6 126 310 208 12 132 304 214)(13 87 81 144 19 93 75 138)(14 92 82 137 20 86 76 143)(15 85 83 142 21 91 77 136)(16 90 84 135 22 96 78 141)(17 95 73 140 23 89 79 134)(18 88 74 133 24 94 80 139)(25 370 479 291 31 364 473 297)(26 363 480 296 32 369 474 290)(27 368 469 289 33 362 475 295)(28 361 470 294 34 367 476 300)(29 366 471 299 35 372 477 293)(30 371 472 292 36 365 478 298)(37 398 67 223 43 404 61 217)(38 403 68 228 44 397 62 222)(39 408 69 221 45 402 63 227)(40 401 70 226 46 407 64 220)(41 406 71 219 47 400 65 225)(42 399 72 224 48 405 66 218)(49 263 336 389 55 257 330 395)(50 256 325 394 56 262 331 388)(51 261 326 387 57 255 332 393)(52 254 327 392 58 260 333 386)(53 259 328 385 59 253 334 391)(54 264 329 390 60 258 335 396)(97 426 147 316 103 432 153 322)(98 431 148 321 104 425 154 315)(99 424 149 314 105 430 155 320)(100 429 150 319 106 423 156 313)(101 422 151 324 107 428 145 318)(102 427 152 317 108 421 146 323)(109 340 279 418 115 346 285 412)(110 345 280 411 116 339 286 417)(111 338 281 416 117 344 287 410)(112 343 282 409 118 337 288 415)(113 348 283 414 119 342 277 420)(114 341 284 419 120 347 278 413)(157 353 266 176 163 359 272 170)(158 358 267 169 164 352 273 175)(159 351 268 174 165 357 274 180)(160 356 269 179 166 350 275 173)(161 349 270 172 167 355 276 178)(162 354 271 177 168 360 265 171)(181 249 452 234 187 243 446 240)(182 242 453 239 188 248 447 233)(183 247 454 232 189 241 448 238)(184 252 455 237 190 246 449 231)(185 245 456 230 191 251 450 236)(186 250 445 235 192 244 451 229)(193 374 443 465 199 380 437 459)(194 379 444 458 200 373 438 464)(195 384 433 463 201 378 439 457)(196 377 434 468 202 383 440 462)(197 382 435 461 203 376 441 467)(198 375 436 466 204 381 442 460)

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

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

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

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 Q8 D4 D6 D8 Q16 Dic6 C4×S3 C3⋊D4 C5×S3 C5×Q8 C5×D4 S3×C10 C5×D8 C5×Q16 C5×Dic6 S3×C20 C5×C3⋊D4 D4⋊S3 C3⋊Q16 C5×D4⋊S3 C5×C3⋊Q16 kernel C5×C6.Q16 C10×C3⋊C8 C5×C4⋊Dic3 C15×C4⋊C4 C5×C3⋊C8 C6.Q16 C2×C3⋊C8 C4⋊Dic3 C3×C4⋊C4 C3⋊C8 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.Q16 in GL5(𝔽241)

 1 0 0 0 0 0 98 0 0 0 0 0 98 0 0 0 0 0 205 0 0 0 0 0 205
,
 240 0 0 0 0 0 240 240 0 0 0 1 0 0 0 0 0 0 1 3 0 0 0 160 240
,
 177 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 15 44 0 0 0 214 226
,
 1 0 0 0 0 0 142 43 0 0 0 142 99 0 0 0 0 0 22 33 0 0 0 73 0

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,98,0,0,0,0,0,98,0,0,0,0,0,205,0,0,0,0,0,205],[240,0,0,0,0,0,240,1,0,0,0,240,0,0,0,0,0,0,1,160,0,0,0,3,240],[177,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,15,214,0,0,0,44,226],[1,0,0,0,0,0,142,142,0,0,0,43,99,0,0,0,0,0,22,73,0,0,0,33,0] >;

C5×C6.Q16 in GAP, Magma, Sage, TeX

C_5\times C_6.Q_{16}
% in TeX

G:=Group("C5xC6.Q16");
// GroupNames label

G:=SmallGroup(480,126);
// by ID

G=gap.SmallGroup(480,126);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,280,589,148,2111,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=c^4=1,d^2=b^9*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=b^9*c^-1>;
// generators/relations

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