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G = C22×C15⋊C8  order 480 = 25·3·5

Direct product of C22 and C15⋊C8

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C22×C15⋊C8, (C2×C30)⋊2C8, C30⋊8(C2×C8), C15⋊10(C22×C8), C23.5(C3⋊F5), (C22×C6).6F5, (C22×C30).6C4, C6.42(C22×F5), C30.80(C22×C4), (C6×Dic5).22C4, (C2×Dic5).209D6, (C22×C10).8Dic3, Dic5.18(C2×Dic3), (C2×Dic5).13Dic3, (C22×Dic5).11S3, Dic5.53(C22×S3), (C3×Dic5).67C23, C10.11(C22×Dic3), (C6×Dic5).268C22, C6⋊2(C2×C5⋊C8), C10⋊2(C2×C3⋊C8), C3⋊2(C22×C5⋊C8), C5⋊2(C22×C3⋊C8), (C2×C6)⋊3(C5⋊C8), (C2×C10)⋊5(C3⋊C8), C2.3(C22×C3⋊F5), (C2×C6).49(C2×F5), (C2×C30).43(C2×C4), C22.20(C2×C3⋊F5), (C2×C6×Dic5).16C2, (C3×Dic5).68(C2×C4), (C2×C10).19(C2×Dic3), SmallGroup(480,1070)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C15 — C22×C15⋊C8
Chief series
C1 — C5 — C15 — C30 — C3×Dic5 — C15⋊C8 — C2×C15⋊C8 — C22×C15⋊C8
Lower central
C15 — C22×C15⋊C8
Upper central
C1 — C23

Generators and relations for C22×C15⋊C8
 G = < a,b,c,d | a2=b2=c15=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c2 >

Subgroups: 460 in 152 conjugacy classes, 97 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C23, C10, C10, C12, C2×C6, C15, C2×C8, C22×C4, Dic5, Dic5, C2×C10, C3⋊C8, C2×C12, C22×C6, C30, C30, C22×C8, C5⋊C8, C2×Dic5, C22×C10, C2×C3⋊C8, C22×C12, C3×Dic5, C3×Dic5, C2×C30, C2×C5⋊C8, C22×Dic5, C22×C3⋊C8, C15⋊C8, C6×Dic5, C22×C30, C22×C5⋊C8, C2×C15⋊C8, C2×C6×Dic5, C22×C15⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, Dic3, D6, C2×C8, C22×C4, F5, C3⋊C8, C2×Dic3, C22×S3, C22×C8, C5⋊C8, C2×F5, C2×C3⋊C8, C22×Dic3, C3⋊F5, C2×C5⋊C8, C22×F5, C22×C3⋊C8, C15⋊C8, C2×C3⋊F5, C22×C5⋊C8, C2×C15⋊C8, C22×C3⋊F5, C22×C15⋊C8

Smallest permutation representation of C22×C15⋊C8
►Regular action on 480 points
Generators in S480
(1 134)(2 135)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 133)(16 137)(17 138)(18 139)(19 140)(20 141)(21 142)(22 143)(23 144)(24 145)(25 146)(26 147)(27 148)(28 149)(29 150)(30 136)(31 160)(32 161)(33 162)(34 163)(35 164)(36 165)(37 151)(38 152)(39 153)(40 154)(41 155)(42 156)(43 157)(44 158)(45 159)(46 171)(47 172)(48 173)(49 174)(50 175)(51 176)(52 177)(53 178)(54 179)(55 180)(56 166)(57 167)(58 168)(59 169)(60 170)(61 192)(62 193)(63 194)(64 195)(65 181)(66 182)(67 183)(68 184)(69 185)(70 186)(71 187)(72 188)(73 189)(74 190)(75 191)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 212)(92 213)(93 214)(94 215)(95 216)(96 217)(97 218)(98 219)(99 220)(100 221)(101 222)(102 223)(103 224)(104 225)(105 211)(106 232)(107 233)(108 234)(109 235)(110 236)(111 237)(112 238)(113 239)(114 240)(115 226)(116 227)(117 228)(118 229)(119 230)(120 231)(241 361)(242 362)(243 363)(244 364)(245 365)(246 366)(247 367)(248 368)(249 369)(250 370)(251 371)(252 372)(253 373)(254 374)(255 375)(256 381)(257 382)(258 383)(259 384)(260 385)(261 386)(262 387)(263 388)(264 389)(265 390)(266 376)(267 377)(268 378)(269 379)(270 380)(271 391)(272 392)(273 393)(274 394)(275 395)(276 396)(277 397)(278 398)(279 399)(280 400)(281 401)(282 402)(283 403)(284 404)(285 405)(286 412)(287 413)(288 414)(289 415)(290 416)(291 417)(292 418)(293 419)(294 420)(295 406)(296 407)(297 408)(298 409)(299 410)(300 411)(301 426)(302 427)(303 428)(304 429)(305 430)(306 431)(307 432)(308 433)(309 434)(310 435)(311 421)(312 422)(313 423)(314 424)(315 425)(316 436)(317 437)(318 438)(319 439)(320 440)(321 441)(322 442)(323 443)(324 444)(325 445)(326 446)(327 447)(328 448)(329 449)(330 450)(331 465)(332 451)(333 452)(334 453)(335 454)(336 455)(337 456)(338 457)(339 458)(340 459)(341 460)(342 461)(343 462)(344 463)(345 464)(346 475)(347 476)(348 477)(349 478)(350 479)(351 480)(352 466)(353 467)(354 468)(355 469)(356 470)(357 471)(358 472)(359 473)(360 474)

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

(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)

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

72 conjugacy classes

class 1 2A···2G 3 4A···4H 5 6A···6G8A···8P10A···10G12A···12H15A15B30A···30N
order12···234···456···68···810···1012···12151530···30
size11···125···542···215···154···410···10444···4

72 irreducible representations

dim11111122222444444
type++++-+-+-+
imageC1C2C2C4C4C8S3Dic3D6Dic3C3⋊C8F5C5⋊C8C2×F5C3⋊F5C15⋊C8C2×C3⋊F5
kernelC22×C15⋊C8C2×C15⋊C8C2×C6×Dic5C6×Dic5C22×C30C2×C30C22×Dic5C2×Dic5C2×Dic5C22×C10C2×C10C22×C6C2×C6C2×C6C23C22C22
# reps161621613318143286

Matrix representation of C22×C15⋊C8 ►in GL8(𝔽241)

10000000
01000000
0024000000
0002400000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
0000240000
0000024000
0000002400
0000000240
,
0240000000
1240000000
0002400000
0012400000
00000001
0000240240240240
00001000
00000100
,
236183000000
1785000000
0020930000
001132210000
0000744181217
0000366013464
0000171107143167
0000249828205

G:=sub<GL(8,GF(241))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[0,1,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,1,0,0,0,0,0,240,0,0,0,0,0,0,1,240,0,0],[236,178,0,0,0,0,0,0,183,5,0,0,0,0,0,0,0,0,20,113,0,0,0,0,0,0,93,221,0,0,0,0,0,0,0,0,74,36,171,24,0,0,0,0,4,60,107,98,0,0,0,0,181,134,143,28,0,0,0,0,217,64,167,205] >;
 

C22×C15⋊C8 in GAP, Magma, Sage, TeX 

C_2^2\times C_{15}\rtimes C_8
 
% in TeX
 
G:=Group("C2^2xC15:C8");
 
// GroupNames label
 
G:=SmallGroup(480,1070);
 
// by ID
 
G=gap.SmallGroup(480,1070);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,80,2693,14118,2379]);
 
// 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^2>;
 
// generators/relations
 

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