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## G = C5×C12⋊2Q8order 480 = 25·3·5

### Direct product of C5 and C12⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C12⋊2Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — C10×Dic6 — C5×C12⋊2Q8
 Lower central C3 — C2×C6 — C5×C12⋊2Q8
 Upper central C1 — C2×C10 — C4×C20

Generators and relations for C5×C122Q8
G = < a,b,c,d | a5=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 276 in 136 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C4 [×4], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10, C10 [×2], Dic3 [×4], C12 [×6], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], C20 [×6], C20 [×4], C2×C10, Dic6 [×4], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C30, C30 [×2], C4⋊Q8, C2×C20, C2×C20 [×2], C2×C20 [×4], C5×Q8 [×4], C4⋊Dic3 [×4], C4×C12, C2×Dic6 [×2], C5×Dic3 [×4], C60 [×6], C2×C30, C4×C20, C5×C4⋊C4 [×4], Q8×C10 [×2], C122Q8, C5×Dic6 [×4], C10×Dic3 [×4], C2×C60, C2×C60 [×2], C5×C4⋊Q8, C5×C4⋊Dic3 [×4], C4×C60, C10×Dic6 [×2], C5×C122Q8
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], Q8 [×4], C23, C10 [×7], D6 [×3], C2×D4, C2×Q8 [×2], C2×C10 [×7], Dic6 [×4], D12 [×2], C22×S3, C5×S3, C4⋊Q8, C5×D4 [×2], C5×Q8 [×4], C22×C10, C2×Dic6 [×2], C2×D12, S3×C10 [×3], D4×C10, Q8×C10 [×2], C122Q8, C5×Dic6 [×4], C5×D12 [×2], S3×C2×C10, C5×C4⋊Q8, C10×Dic6 [×2], C10×D12, C5×C122Q8

Smallest permutation representation of C5×C122Q8
Regular action on 480 points
Generators in S480
(1 122 407 95 332)(2 123 408 96 333)(3 124 397 85 334)(4 125 398 86 335)(5 126 399 87 336)(6 127 400 88 325)(7 128 401 89 326)(8 129 402 90 327)(9 130 403 91 328)(10 131 404 92 329)(11 132 405 93 330)(12 121 406 94 331)(13 69 372 310 81)(14 70 361 311 82)(15 71 362 312 83)(16 72 363 301 84)(17 61 364 302 73)(18 62 365 303 74)(19 63 366 304 75)(20 64 367 305 76)(21 65 368 306 77)(22 66 369 307 78)(23 67 370 308 79)(24 68 371 309 80)(25 115 178 393 162)(26 116 179 394 163)(27 117 180 395 164)(28 118 169 396 165)(29 119 170 385 166)(30 120 171 386 167)(31 109 172 387 168)(32 110 173 388 157)(33 111 174 389 158)(34 112 175 390 159)(35 113 176 391 160)(36 114 177 392 161)(37 354 471 231 317)(38 355 472 232 318)(39 356 473 233 319)(40 357 474 234 320)(41 358 475 235 321)(42 359 476 236 322)(43 360 477 237 323)(44 349 478 238 324)(45 350 479 239 313)(46 351 480 240 314)(47 352 469 229 315)(48 353 470 230 316)(49 464 432 223 288)(50 465 421 224 277)(51 466 422 225 278)(52 467 423 226 279)(53 468 424 227 280)(54 457 425 228 281)(55 458 426 217 282)(56 459 427 218 283)(57 460 428 219 284)(58 461 429 220 285)(59 462 430 221 286)(60 463 431 222 287)(97 454 146 215 382)(98 455 147 216 383)(99 456 148 205 384)(100 445 149 206 373)(101 446 150 207 374)(102 447 151 208 375)(103 448 152 209 376)(104 449 153 210 377)(105 450 154 211 378)(106 451 155 212 379)(107 452 156 213 380)(108 453 145 214 381)(133 249 191 437 201)(134 250 192 438 202)(135 251 181 439 203)(136 252 182 440 204)(137 241 183 441 193)(138 242 184 442 194)(139 243 185 443 195)(140 244 186 444 196)(141 245 187 433 197)(142 246 188 434 198)(143 247 189 435 199)(144 248 190 436 200)(253 266 412 341 293)(254 267 413 342 294)(255 268 414 343 295)(256 269 415 344 296)(257 270 416 345 297)(258 271 417 346 298)(259 272 418 347 299)(260 273 419 348 300)(261 274 420 337 289)(262 275 409 338 290)(263 276 410 339 291)(264 265 411 340 292)
(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)
(1 466 188 266)(2 467 189 267)(3 468 190 268)(4 457 191 269)(5 458 192 270)(6 459 181 271)(7 460 182 272)(8 461 183 273)(9 462 184 274)(10 463 185 275)(11 464 186 276)(12 465 187 265)(13 163 353 207)(14 164 354 208)(15 165 355 209)(16 166 356 210)(17 167 357 211)(18 168 358 212)(19 157 359 213)(20 158 360 214)(21 159 349 215)(22 160 350 216)(23 161 351 205)(24 162 352 206)(25 469 373 68)(26 470 374 69)(27 471 375 70)(28 472 376 71)(29 473 377 72)(30 474 378 61)(31 475 379 62)(32 476 380 63)(33 477 381 64)(34 478 382 65)(35 479 383 66)(36 480 384 67)(37 151 82 395)(38 152 83 396)(39 153 84 385)(40 154 73 386)(41 155 74 387)(42 156 75 388)(43 145 76 389)(44 146 77 390)(45 147 78 391)(46 148 79 392)(47 149 80 393)(48 150 81 394)(49 244 263 330)(50 245 264 331)(51 246 253 332)(52 247 254 333)(53 248 255 334)(54 249 256 335)(55 250 257 336)(56 251 258 325)(57 252 259 326)(58 241 260 327)(59 242 261 328)(60 243 262 329)(85 280 144 295)(86 281 133 296)(87 282 134 297)(88 283 135 298)(89 284 136 299)(90 285 137 300)(91 286 138 289)(92 287 139 290)(93 288 140 291)(94 277 141 292)(95 278 142 293)(96 279 143 294)(97 368 112 238)(98 369 113 239)(99 370 114 240)(100 371 115 229)(101 372 116 230)(102 361 117 231)(103 362 118 232)(104 363 119 233)(105 364 120 234)(106 365 109 235)(107 366 110 236)(108 367 111 237)(121 421 433 411)(122 422 434 412)(123 423 435 413)(124 424 436 414)(125 425 437 415)(126 426 438 416)(127 427 439 417)(128 428 440 418)(129 429 441 419)(130 430 442 420)(131 431 443 409)(132 432 444 410)(169 318 448 312)(170 319 449 301)(171 320 450 302)(172 321 451 303)(173 322 452 304)(174 323 453 305)(175 324 454 306)(176 313 455 307)(177 314 456 308)(178 315 445 309)(179 316 446 310)(180 317 447 311)(193 348 402 220)(194 337 403 221)(195 338 404 222)(196 339 405 223)(197 340 406 224)(198 341 407 225)(199 342 408 226)(200 343 397 227)(201 344 398 228)(202 345 399 217)(203 346 400 218)(204 347 401 219)
(1 15 188 355)(2 14 189 354)(3 13 190 353)(4 24 191 352)(5 23 192 351)(6 22 181 350)(7 21 182 349)(8 20 183 360)(9 19 184 359)(10 18 185 358)(11 17 186 357)(12 16 187 356)(25 425 373 415)(26 424 374 414)(27 423 375 413)(28 422 376 412)(29 421 377 411)(30 432 378 410)(31 431 379 409)(32 430 380 420)(33 429 381 419)(34 428 382 418)(35 427 383 417)(36 426 384 416)(37 333 82 247)(38 332 83 246)(39 331 84 245)(40 330 73 244)(41 329 74 243)(42 328 75 242)(43 327 76 241)(44 326 77 252)(45 325 78 251)(46 336 79 250)(47 335 80 249)(48 334 81 248)(49 154 263 386)(50 153 264 385)(51 152 253 396)(52 151 254 395)(53 150 255 394)(54 149 256 393)(55 148 257 392)(56 147 258 391)(57 146 259 390)(58 145 260 389)(59 156 261 388)(60 155 262 387)(61 444 474 132)(62 443 475 131)(63 442 476 130)(64 441 477 129)(65 440 478 128)(66 439 479 127)(67 438 480 126)(68 437 469 125)(69 436 470 124)(70 435 471 123)(71 434 472 122)(72 433 473 121)(85 310 144 316)(86 309 133 315)(87 308 134 314)(88 307 135 313)(89 306 136 324)(90 305 137 323)(91 304 138 322)(92 303 139 321)(93 302 140 320)(94 301 141 319)(95 312 142 318)(96 311 143 317)(97 347 112 219)(98 346 113 218)(99 345 114 217)(100 344 115 228)(101 343 116 227)(102 342 117 226)(103 341 118 225)(104 340 119 224)(105 339 120 223)(106 338 109 222)(107 337 110 221)(108 348 111 220)(157 462 213 274)(158 461 214 273)(159 460 215 272)(160 459 216 271)(161 458 205 270)(162 457 206 269)(163 468 207 268)(164 467 208 267)(165 466 209 266)(166 465 210 265)(167 464 211 276)(168 463 212 275)(169 278 448 293)(170 277 449 292)(171 288 450 291)(172 287 451 290)(173 286 452 289)(174 285 453 300)(175 284 454 299)(176 283 455 298)(177 282 456 297)(178 281 445 296)(179 280 446 295)(180 279 447 294)(193 237 402 367)(194 236 403 366)(195 235 404 365)(196 234 405 364)(197 233 406 363)(198 232 407 362)(199 231 408 361)(200 230 397 372)(201 229 398 371)(202 240 399 370)(203 239 400 369)(204 238 401 368)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4F 4G 4H 4I 4J 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 12A ··· 12L 15A 15B 15C 15D 20A ··· 20X 20Y ··· 20AN 30A ··· 30L 60A ··· 60AV order 1 2 2 2 3 4 ··· 4 4 4 4 4 5 5 5 5 6 6 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 ··· 2 12 12 12 12 1 1 1 1 2 2 2 1 ··· 1 2 ··· 2 2 2 2 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + image C1 C2 C2 C2 C5 C10 C10 C10 S3 D4 Q8 D6 Dic6 D12 C5×S3 C5×D4 C5×Q8 S3×C10 C5×Dic6 C5×D12 kernel C5×C12⋊2Q8 C5×C4⋊Dic3 C4×C60 C10×Dic6 C12⋊2Q8 C4⋊Dic3 C4×C12 C2×Dic6 C4×C20 C60 C60 C2×C20 C20 C20 C42 C12 C12 C2×C4 C4 C4 # reps 1 4 1 2 4 16 4 8 1 2 4 3 8 4 4 8 16 12 32 16

Matrix representation of C5×C122Q8 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 58 0 0 0 0 58
,
 23 38 0 0 23 46 0 0 0 0 1 1 0 0 60 0
,
 23 46 0 0 15 38 0 0 0 0 23 46 0 0 15 38
,
 43 52 0 0 9 18 0 0 0 0 8 49 0 0 41 53
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,58,0,0,0,0,58],[23,23,0,0,38,46,0,0,0,0,1,60,0,0,1,0],[23,15,0,0,46,38,0,0,0,0,23,15,0,0,46,38],[43,9,0,0,52,18,0,0,0,0,8,41,0,0,49,53] >;

C5×C122Q8 in GAP, Magma, Sage, TeX

C_5\times C_{12}\rtimes_2Q_8
% in TeX

G:=Group("C5xC12:2Q8");
// GroupNames label

G:=SmallGroup(480,748);
// by ID

G=gap.SmallGroup(480,748);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,1149,568,926,226,15686]);
// Polycyclic

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

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