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## G = C22×C15⋊3C8order 480 = 25·3·5

### Direct product of C22 and C15⋊3C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C22×C15⋊3C8
 Chief series C1 — C5 — C15 — C30 — C60 — C15⋊3C8 — C2×C15⋊3C8 — C22×C15⋊3C8
 Lower central C15 — C22×C15⋊3C8
 Upper central C1 — C22×C4

Generators and relations for C22×C153C8
G = < a,b,c,d | a2=b2=c15=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 372 in 152 conjugacy classes, 119 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C23, C10, C10, C12, C12, C2×C6, C15, C2×C8, C22×C4, C20, C20, C2×C10, C3⋊C8, C2×C12, C22×C6, C30, C30, C22×C8, C52C8, C2×C20, C22×C10, C2×C3⋊C8, C22×C12, C60, C60, C2×C30, C2×C52C8, C22×C20, C22×C3⋊C8, C153C8, C2×C60, C22×C30, C22×C52C8, C2×C153C8, C22×C60, C22×C153C8
Quotients:

Smallest permutation representation of C22×C153C8
Regular action on 480 points
Generators in S480
(1 203)(2 204)(3 205)(4 206)(5 207)(6 208)(7 209)(8 210)(9 196)(10 197)(11 198)(12 199)(13 200)(14 201)(15 202)(16 189)(17 190)(18 191)(19 192)(20 193)(21 194)(22 195)(23 181)(24 182)(25 183)(26 184)(27 185)(28 186)(29 187)(30 188)(31 236)(32 237)(33 238)(34 239)(35 240)(36 226)(37 227)(38 228)(39 229)(40 230)(41 231)(42 232)(43 233)(44 234)(45 235)(46 216)(47 217)(48 218)(49 219)(50 220)(51 221)(52 222)(53 223)(54 224)(55 225)(56 211)(57 212)(58 213)(59 214)(60 215)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 136)(71 137)(72 138)(73 139)(74 140)(75 141)(76 130)(77 131)(78 132)(79 133)(80 134)(81 135)(82 121)(83 122)(84 123)(85 124)(86 125)(87 126)(88 127)(89 128)(90 129)(91 180)(92 166)(93 167)(94 168)(95 169)(96 170)(97 171)(98 172)(99 173)(100 174)(101 175)(102 176)(103 177)(104 178)(105 179)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 151)(113 152)(114 153)(115 154)(116 155)(117 156)(118 157)(119 158)(120 159)(241 448)(242 449)(243 450)(244 436)(245 437)(246 438)(247 439)(248 440)(249 441)(250 442)(251 443)(252 444)(253 445)(254 446)(255 447)(256 429)(257 430)(258 431)(259 432)(260 433)(261 434)(262 435)(263 421)(264 422)(265 423)(266 424)(267 425)(268 426)(269 427)(270 428)(271 466)(272 467)(273 468)(274 469)(275 470)(276 471)(277 472)(278 473)(279 474)(280 475)(281 476)(282 477)(283 478)(284 479)(285 480)(286 451)(287 452)(288 453)(289 454)(290 455)(291 456)(292 457)(293 458)(294 459)(295 460)(296 461)(297 462)(298 463)(299 464)(300 465)(301 376)(302 377)(303 378)(304 379)(305 380)(306 381)(307 382)(308 383)(309 384)(310 385)(311 386)(312 387)(313 388)(314 389)(315 390)(316 373)(317 374)(318 375)(319 361)(320 362)(321 363)(322 364)(323 365)(324 366)(325 367)(326 368)(327 369)(328 370)(329 371)(330 372)(331 413)(332 414)(333 415)(334 416)(335 417)(336 418)(337 419)(338 420)(339 406)(340 407)(341 408)(342 409)(343 410)(344 411)(345 412)(346 392)(347 393)(348 394)(349 395)(350 396)(351 397)(352 398)(353 399)(354 400)(355 401)(356 402)(357 403)(358 404)(359 405)(360 391)
(1 88)(2 89)(3 90)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 61)(27 62)(28 63)(29 64)(30 65)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 106)(41 107)(42 108)(43 109)(44 110)(45 111)(46 105)(47 91)(48 92)(49 93)(50 94)(51 95)(52 96)(53 97)(54 98)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(121 197)(122 198)(123 199)(124 200)(125 201)(126 202)(127 203)(128 204)(129 205)(130 206)(131 207)(132 208)(133 209)(134 210)(135 196)(136 193)(137 194)(138 195)(139 181)(140 182)(141 183)(142 184)(143 185)(144 186)(145 187)(146 188)(147 189)(148 190)(149 191)(150 192)(151 236)(152 237)(153 238)(154 239)(155 240)(156 226)(157 227)(158 228)(159 229)(160 230)(161 231)(162 232)(163 233)(164 234)(165 235)(166 218)(167 219)(168 220)(169 221)(170 222)(171 223)(172 224)(173 225)(174 211)(175 212)(176 213)(177 214)(178 215)(179 216)(180 217)(241 316)(242 317)(243 318)(244 319)(245 320)(246 321)(247 322)(248 323)(249 324)(250 325)(251 326)(252 327)(253 328)(254 329)(255 330)(256 313)(257 314)(258 315)(259 301)(260 302)(261 303)(262 304)(263 305)(264 306)(265 307)(266 308)(267 309)(268 310)(269 311)(270 312)(271 353)(272 354)(273 355)(274 356)(275 357)(276 358)(277 359)(278 360)(279 346)(280 347)(281 348)(282 349)(283 350)(284 351)(285 352)(286 342)(287 343)(288 344)(289 345)(290 331)(291 332)(292 333)(293 334)(294 335)(295 336)(296 337)(297 338)(298 339)(299 340)(300 341)(361 436)(362 437)(363 438)(364 439)(365 440)(366 441)(367 442)(368 443)(369 444)(370 445)(371 446)(372 447)(373 448)(374 449)(375 450)(376 432)(377 433)(378 434)(379 435)(380 421)(381 422)(382 423)(383 424)(384 425)(385 426)(386 427)(387 428)(388 429)(389 430)(390 431)(391 473)(392 474)(393 475)(394 476)(395 477)(396 478)(397 479)(398 480)(399 466)(400 467)(401 468)(402 469)(403 470)(404 471)(405 472)(406 463)(407 464)(408 465)(409 451)(410 452)(411 453)(412 454)(413 455)(414 456)(415 457)(416 458)(417 459)(418 460)(419 461)(420 462)
(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)
(1 474 50 437 25 463 31 429)(2 473 51 436 26 462 32 428)(3 472 52 450 27 461 33 427)(4 471 53 449 28 460 34 426)(5 470 54 448 29 459 35 425)(6 469 55 447 30 458 36 424)(7 468 56 446 16 457 37 423)(8 467 57 445 17 456 38 422)(9 466 58 444 18 455 39 421)(10 480 59 443 19 454 40 435)(11 479 60 442 20 453 41 434)(12 478 46 441 21 452 42 433)(13 477 47 440 22 451 43 432)(14 476 48 439 23 465 44 431)(15 475 49 438 24 464 45 430)(61 420 113 387 89 391 95 361)(62 419 114 386 90 405 96 375)(63 418 115 385 76 404 97 374)(64 417 116 384 77 403 98 373)(65 416 117 383 78 402 99 372)(66 415 118 382 79 401 100 371)(67 414 119 381 80 400 101 370)(68 413 120 380 81 399 102 369)(69 412 106 379 82 398 103 368)(70 411 107 378 83 397 104 367)(71 410 108 377 84 396 105 366)(72 409 109 376 85 395 91 365)(73 408 110 390 86 394 92 364)(74 407 111 389 87 393 93 363)(75 406 112 388 88 392 94 362)(121 352 177 326 150 345 160 304)(122 351 178 325 136 344 161 303)(123 350 179 324 137 343 162 302)(124 349 180 323 138 342 163 301)(125 348 166 322 139 341 164 315)(126 347 167 321 140 340 165 314)(127 346 168 320 141 339 151 313)(128 360 169 319 142 338 152 312)(129 359 170 318 143 337 153 311)(130 358 171 317 144 336 154 310)(131 357 172 316 145 335 155 309)(132 356 173 330 146 334 156 308)(133 355 174 329 147 333 157 307)(134 354 175 328 148 332 158 306)(135 353 176 327 149 331 159 305)(181 300 234 258 201 281 218 247)(182 299 235 257 202 280 219 246)(183 298 236 256 203 279 220 245)(184 297 237 270 204 278 221 244)(185 296 238 269 205 277 222 243)(186 295 239 268 206 276 223 242)(187 294 240 267 207 275 224 241)(188 293 226 266 208 274 225 255)(189 292 227 265 209 273 211 254)(190 291 228 264 210 272 212 253)(191 290 229 263 196 271 213 252)(192 289 230 262 197 285 214 251)(193 288 231 261 198 284 215 250)(194 287 232 260 199 283 216 249)(195 286 233 259 200 282 217 248)

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

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

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

144 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 5A 5B 6A ··· 6G 8A ··· 8P 10A ··· 10N 12A ··· 12H 15A 15B 15C 15D 20A ··· 20P 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 3 4 ··· 4 5 5 6 ··· 6 8 ··· 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 2 1 ··· 1 2 2 2 ··· 2 15 ··· 15 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - - + - + - + - image C1 C2 C2 C4 C4 C8 S3 D5 Dic3 D6 Dic3 Dic5 D10 Dic5 C3⋊C8 D15 C5⋊2C8 Dic15 D30 Dic15 C15⋊3C8 kernel C22×C15⋊3C8 C2×C15⋊3C8 C22×C60 C2×C60 C22×C30 C2×C30 C22×C20 C22×C12 C2×C20 C2×C20 C22×C10 C2×C12 C2×C12 C22×C6 C2×C10 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 16 1 2 3 3 1 6 6 2 8 4 16 12 12 4 32

Matrix representation of C22×C153C8 in GL4(𝔽241) generated by

 240 0 0 0 0 1 0 0 0 0 240 0 0 0 0 240
,
 240 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 161 131 0 0 110 94
,
 1 0 0 0 0 240 0 0 0 0 5 118 0 0 132 236
G:=sub<GL(4,GF(241))| [240,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,161,110,0,0,131,94],[1,0,0,0,0,240,0,0,0,0,5,132,0,0,118,236] >;

C22×C153C8 in GAP, Magma, Sage, TeX

C_2^2\times C_{15}\rtimes_3C_8
% in TeX

G:=Group("C2^2xC15:3C8");
// GroupNames label

G:=SmallGroup(480,885);
// by ID

G=gap.SmallGroup(480,885);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^15=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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