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## G = C5×C2.Dic12order 480 = 25·3·5

### Direct product of C5 and C2.Dic12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C2.Dic12
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C60 — C5×C4⋊Dic3 — C5×C2.Dic12
 Lower central C3 — C6 — C12 — C5×C2.Dic12
 Upper central C1 — C2×C10 — C2×C20 — C2×C40

Generators and relations for C5×C2.Dic12
G = < a,b,c,d | a5=b6=c8=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b3c3 >

Subgroups: 196 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, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, C20, C20, C2×C10, C24, Dic6, Dic6, C2×Dic3, C2×C12, C30, Q8⋊C4, C40, C2×C20, C2×C20, C5×Q8, C4⋊Dic3, C2×C24, C2×Dic6, C5×Dic3, C60, C2×C30, C5×C4⋊C4, C2×C40, Q8×C10, C2.Dic12, C120, C5×Dic6, C5×Dic6, C10×Dic3, C2×C60, C5×Q8⋊C4, C5×C4⋊Dic3, C2×C120, C10×Dic6, C5×C2.Dic12
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, C24⋊C2, Dic12, D6⋊C4, S3×C10, C5×C22⋊C4, C5×SD16, C5×Q16, C2.Dic12, S3×C20, C5×D12, C5×C3⋊D4, C5×Q8⋊C4, C5×C24⋊C2, C5×Dic12, C5×D6⋊C4, C5×C2.Dic12

Smallest permutation representation of C5×C2.Dic12
Regular action on 480 points
Generators in S480
(1 14 71 183 47)(2 15 72 184 48)(3 16 65 177 41)(4 9 66 178 42)(5 10 67 179 43)(6 11 68 180 44)(7 12 69 181 45)(8 13 70 182 46)(17 346 470 322 446)(18 347 471 323 447)(19 348 472 324 448)(20 349 465 325 441)(21 350 466 326 442)(22 351 467 327 443)(23 352 468 328 444)(24 345 469 321 445)(25 134 190 49 166)(26 135 191 50 167)(27 136 192 51 168)(28 129 185 52 161)(29 130 186 53 162)(30 131 187 54 163)(31 132 188 55 164)(32 133 189 56 165)(33 157 172 57 148)(34 158 173 58 149)(35 159 174 59 150)(36 160 175 60 151)(37 153 176 61 152)(38 154 169 62 145)(39 155 170 63 146)(40 156 171 64 147)(73 264 238 97 214)(74 257 239 98 215)(75 258 240 99 216)(76 259 233 100 209)(77 260 234 101 210)(78 261 235 102 211)(79 262 236 103 212)(80 263 237 104 213)(81 245 220 105 196)(82 246 221 106 197)(83 247 222 107 198)(84 248 223 108 199)(85 241 224 109 200)(86 242 217 110 193)(87 243 218 111 194)(88 244 219 112 195)(89 201 369 113 225)(90 202 370 114 226)(91 203 371 115 227)(92 204 372 116 228)(93 205 373 117 229)(94 206 374 118 230)(95 207 375 119 231)(96 208 376 120 232)(121 462 340 438 316)(122 463 341 439 317)(123 464 342 440 318)(124 457 343 433 319)(125 458 344 434 320)(126 459 337 435 313)(127 460 338 436 314)(128 461 339 437 315)(137 475 357 451 333)(138 476 358 452 334)(139 477 359 453 335)(140 478 360 454 336)(141 479 353 455 329)(142 480 354 456 330)(143 473 355 449 331)(144 474 356 450 332)(249 297 421 273 397)(250 298 422 274 398)(251 299 423 275 399)(252 300 424 276 400)(253 301 417 277 393)(254 302 418 278 394)(255 303 419 279 395)(256 304 420 280 396)(265 363 411 289 387)(266 364 412 290 388)(267 365 413 291 389)(268 366 414 292 390)(269 367 415 293 391)(270 368 416 294 392)(271 361 409 295 385)(272 362 410 296 386)(281 383 431 305 407)(282 384 432 306 408)(283 377 425 307 401)(284 378 426 308 402)(285 379 427 309 403)(286 380 428 310 404)(287 381 429 311 405)(288 382 430 312 406)
(1 131 39 207 74 81)(2 132 40 208 75 82)(3 133 33 201 76 83)(4 134 34 202 77 84)(5 135 35 203 78 85)(6 136 36 204 79 86)(7 129 37 205 80 87)(8 130 38 206 73 88)(9 190 158 370 260 248)(10 191 159 371 261 241)(11 192 160 372 262 242)(12 185 153 373 263 243)(13 186 154 374 264 244)(14 187 155 375 257 245)(15 188 156 376 258 246)(16 189 157 369 259 247)(17 415 427 303 125 140)(18 416 428 304 126 141)(19 409 429 297 127 142)(20 410 430 298 128 143)(21 411 431 299 121 144)(22 412 432 300 122 137)(23 413 425 301 123 138)(24 414 426 302 124 139)(25 149 90 210 199 42)(26 150 91 211 200 43)(27 151 92 212 193 44)(28 152 93 213 194 45)(29 145 94 214 195 46)(30 146 95 215 196 47)(31 147 96 216 197 48)(32 148 89 209 198 41)(49 173 114 234 223 66)(50 174 115 235 224 67)(51 175 116 236 217 68)(52 176 117 237 218 69)(53 169 118 238 219 70)(54 170 119 239 220 71)(55 171 120 240 221 72)(56 172 113 233 222 65)(57 225 100 107 177 165)(58 226 101 108 178 166)(59 227 102 109 179 167)(60 228 103 110 180 168)(61 229 104 111 181 161)(62 230 97 112 182 162)(63 231 98 105 183 163)(64 232 99 106 184 164)(249 314 330 448 361 381)(250 315 331 441 362 382)(251 316 332 442 363 383)(252 317 333 443 364 384)(253 318 334 444 365 377)(254 319 335 445 366 378)(255 320 336 446 367 379)(256 313 329 447 368 380)(265 281 399 438 450 326)(266 282 400 439 451 327)(267 283 393 440 452 328)(268 284 394 433 453 321)(269 285 395 434 454 322)(270 286 396 435 455 323)(271 287 397 436 456 324)(272 288 398 437 449 325)(273 338 354 472 385 405)(274 339 355 465 386 406)(275 340 356 466 387 407)(276 341 357 467 388 408)(277 342 358 468 389 401)(278 343 359 469 390 402)(279 344 360 470 391 403)(280 337 353 471 392 404)(289 305 423 462 474 350)(290 306 424 463 475 351)(291 307 417 464 476 352)(292 308 418 457 477 345)(293 309 419 458 478 346)(294 310 420 459 479 347)(295 311 421 460 480 348)(296 312 422 461 473 349)
(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)
(1 254 207 445)(2 448 208 249)(3 252 201 443)(4 446 202 255)(5 250 203 441)(6 444 204 253)(7 256 205 447)(8 442 206 251)(9 17 370 303)(10 298 371 20)(11 23 372 301)(12 304 373 18)(13 21 374 299)(14 302 375 24)(15 19 376 297)(16 300 369 22)(25 454 210 285)(26 288 211 449)(27 452 212 283)(28 286 213 455)(29 450 214 281)(30 284 215 453)(31 456 216 287)(32 282 209 451)(33 364 83 317)(34 320 84 367)(35 362 85 315)(36 318 86 365)(37 368 87 313)(38 316 88 363)(39 366 81 319)(40 314 82 361)(41 400 89 327)(42 322 90 395)(43 398 91 325)(44 328 92 393)(45 396 93 323)(46 326 94 399)(47 394 95 321)(48 324 96 397)(49 478 234 309)(50 312 235 473)(51 476 236 307)(52 310 237 479)(53 474 238 305)(54 308 239 477)(55 480 240 311)(56 306 233 475)(57 388 107 341)(58 344 108 391)(59 386 109 339)(60 342 110 389)(61 392 111 337)(62 340 112 387)(63 390 105 343)(64 338 106 385)(65 424 113 351)(66 346 114 419)(67 422 115 349)(68 352 116 417)(69 420 117 347)(70 350 118 423)(71 418 119 345)(72 348 120 421)(73 383 130 332)(74 335 131 378)(75 381 132 330)(76 333 133 384)(77 379 134 336)(78 331 135 382)(79 377 136 334)(80 329 129 380)(97 407 162 356)(98 359 163 402)(99 405 164 354)(100 357 165 408)(101 403 166 360)(102 355 167 406)(103 401 168 358)(104 353 161 404)(121 244 411 154)(122 157 412 247)(123 242 413 160)(124 155 414 245)(125 248 415 158)(126 153 416 243)(127 246 409 156)(128 159 410 241)(137 189 432 259)(138 262 425 192)(139 187 426 257)(140 260 427 190)(141 185 428 263)(142 258 429 188)(143 191 430 261)(144 264 431 186)(145 438 195 265)(146 268 196 433)(147 436 197 271)(148 266 198 439)(149 434 199 269)(150 272 200 437)(151 440 193 267)(152 270 194 435)(169 462 219 289)(170 292 220 457)(171 460 221 295)(172 290 222 463)(173 458 223 293)(174 296 224 461)(175 464 217 291)(176 294 218 459)(177 276 225 467)(178 470 226 279)(179 274 227 465)(180 468 228 277)(181 280 229 471)(182 466 230 275)(183 278 231 469)(184 472 232 273)

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

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

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

150 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 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 20I ··· 20X 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF 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 12 12 15 15 15 15 20 ··· 20 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 2 2 2 12 12 12 12 1 1 1 1 2 2 2 2 2 2 2 1 ··· 1 2 2 2 2 2 2 2 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

150 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 2 2 2 2 type + + + + + + + + - + - image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 S3 D4 D4 D6 SD16 Q16 C4×S3 C3⋊D4 D12 C5×S3 C5×D4 C5×D4 C24⋊C2 Dic12 S3×C10 C5×SD16 C5×Q16 S3×C20 C5×C3⋊D4 C5×D12 C5×C24⋊C2 C5×Dic12 kernel C5×C2.Dic12 C5×C4⋊Dic3 C2×C120 C10×Dic6 C5×Dic6 C2.Dic12 C4⋊Dic3 C2×C24 C2×Dic6 Dic6 C2×C40 C60 C2×C30 C2×C20 C30 C30 C20 C20 C2×C10 C2×C8 C12 C2×C6 C10 C10 C2×C4 C6 C6 C4 C4 C22 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 4 4 8 8 8 8 8 16 16

Matrix representation of C5×C2.Dic12 in GL4(𝔽241) generated by

 91 0 0 0 0 91 0 0 0 0 205 0 0 0 0 205
,
 0 240 0 0 1 1 0 0 0 0 0 1 0 0 240 240
,
 171 101 0 0 140 70 0 0 0 0 213 147 0 0 94 66
,
 160 220 0 0 60 81 0 0 0 0 234 194 0 0 201 7
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,205,0,0,0,0,205],[0,1,0,0,240,1,0,0,0,0,0,240,0,0,1,240],[171,140,0,0,101,70,0,0,0,0,213,94,0,0,147,66],[160,60,0,0,220,81,0,0,0,0,234,201,0,0,194,7] >;

C5×C2.Dic12 in GAP, Magma, Sage, TeX

C_5\times C_2.{\rm Dic}_{12}
% in TeX

G:=Group("C5xC2.Dic12");
// GroupNames label

G:=SmallGroup(480,135);
// by ID

G=gap.SmallGroup(480,135);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,280,309,428,2803,136,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^6=c^8=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^3>;
// generators/relations

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