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## G = C4×C15⋊3C8order 480 = 25·3·5

### Direct product of C4 and C15⋊3C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C4×C15⋊3C8
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C2×C15⋊3C8 — C4×C15⋊3C8
 Lower central C15 — C4×C15⋊3C8
 Upper central C1 — C42

Generators and relations for C4×C153C8
G = < a,b,c | a4=b15=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 228 in 88 conjugacy classes, 67 normal (21 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, C12, C2×C6, C15, C42, C2×C8, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C30, C30, C4×C8, C52C8, C2×C20, C2×C20, C2×C3⋊C8, C4×C12, C60, C2×C30, C2×C52C8, C4×C20, C4×C3⋊C8, C153C8, C2×C60, C2×C60, C4×C52C8, C2×C153C8, C4×C60, C4×C153C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D5, Dic3, D6, C42, C2×C8, Dic5, D10, C3⋊C8, C4×S3, C2×Dic3, D15, C4×C8, C52C8, C4×D5, C2×Dic5, C2×C3⋊C8, C4×Dic3, Dic15, D30, C2×C52C8, C4×Dic5, C4×C3⋊C8, C153C8, C4×D15, C2×Dic15, C4×C52C8, C2×C153C8, C4×Dic15, C4×C153C8

Smallest permutation representation of C4×C153C8
Regular action on 480 points
Generators in S480
(1 167 32 150)(2 168 33 136)(3 169 34 137)(4 170 35 138)(5 171 36 139)(6 172 37 140)(7 173 38 141)(8 174 39 142)(9 175 40 143)(10 176 41 144)(11 177 42 145)(12 178 43 146)(13 179 44 147)(14 180 45 148)(15 166 31 149)(16 157 60 135)(17 158 46 121)(18 159 47 122)(19 160 48 123)(20 161 49 124)(21 162 50 125)(22 163 51 126)(23 164 52 127)(24 165 53 128)(25 151 54 129)(26 152 55 130)(27 153 56 131)(28 154 57 132)(29 155 58 133)(30 156 59 134)(61 239 100 208)(62 240 101 209)(63 226 102 210)(64 227 103 196)(65 228 104 197)(66 229 105 198)(67 230 91 199)(68 231 92 200)(69 232 93 201)(70 233 94 202)(71 234 95 203)(72 235 96 204)(73 236 97 205)(74 237 98 206)(75 238 99 207)(76 218 115 186)(77 219 116 187)(78 220 117 188)(79 221 118 189)(80 222 119 190)(81 223 120 191)(82 224 106 192)(83 225 107 193)(84 211 108 194)(85 212 109 195)(86 213 110 181)(87 214 111 182)(88 215 112 183)(89 216 113 184)(90 217 114 185)(241 409 271 387)(242 410 272 388)(243 411 273 389)(244 412 274 390)(245 413 275 376)(246 414 276 377)(247 415 277 378)(248 416 278 379)(249 417 279 380)(250 418 280 381)(251 419 281 382)(252 420 282 383)(253 406 283 384)(254 407 284 385)(255 408 285 386)(256 399 290 372)(257 400 291 373)(258 401 292 374)(259 402 293 375)(260 403 294 361)(261 404 295 362)(262 405 296 363)(263 391 297 364)(264 392 298 365)(265 393 299 366)(266 394 300 367)(267 395 286 368)(268 396 287 369)(269 397 288 370)(270 398 289 371)(301 470 342 443)(302 471 343 444)(303 472 344 445)(304 473 345 446)(305 474 331 447)(306 475 332 448)(307 476 333 449)(308 477 334 450)(309 478 335 436)(310 479 336 437)(311 480 337 438)(312 466 338 439)(313 467 339 440)(314 468 340 441)(315 469 341 442)(316 455 350 421)(317 456 351 422)(318 457 352 423)(319 458 353 424)(320 459 354 425)(321 460 355 426)(322 461 356 427)(323 462 357 428)(324 463 358 429)(325 464 359 430)(326 465 360 431)(327 451 346 432)(328 452 347 433)(329 453 348 434)(330 454 349 435)
(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)
(1 440 120 410 19 433 96 403)(2 439 106 409 20 432 97 402)(3 438 107 408 21 431 98 401)(4 437 108 407 22 430 99 400)(5 436 109 406 23 429 100 399)(6 450 110 420 24 428 101 398)(7 449 111 419 25 427 102 397)(8 448 112 418 26 426 103 396)(9 447 113 417 27 425 104 395)(10 446 114 416 28 424 105 394)(11 445 115 415 29 423 91 393)(12 444 116 414 30 422 92 392)(13 443 117 413 16 421 93 391)(14 442 118 412 17 435 94 405)(15 441 119 411 18 434 95 404)(31 468 80 389 47 453 71 362)(32 467 81 388 48 452 72 361)(33 466 82 387 49 451 73 375)(34 480 83 386 50 465 74 374)(35 479 84 385 51 464 75 373)(36 478 85 384 52 463 61 372)(37 477 86 383 53 462 62 371)(38 476 87 382 54 461 63 370)(39 475 88 381 55 460 64 369)(40 474 89 380 56 459 65 368)(41 473 90 379 57 458 66 367)(42 472 76 378 58 457 67 366)(43 471 77 377 59 456 68 365)(44 470 78 376 60 455 69 364)(45 469 79 390 46 454 70 363)(121 349 233 262 148 341 221 244)(122 348 234 261 149 340 222 243)(123 347 235 260 150 339 223 242)(124 346 236 259 136 338 224 241)(125 360 237 258 137 337 225 255)(126 359 238 257 138 336 211 254)(127 358 239 256 139 335 212 253)(128 357 240 270 140 334 213 252)(129 356 226 269 141 333 214 251)(130 355 227 268 142 332 215 250)(131 354 228 267 143 331 216 249)(132 353 229 266 144 345 217 248)(133 352 230 265 145 344 218 247)(134 351 231 264 146 343 219 246)(135 350 232 263 147 342 220 245)(151 322 210 288 173 307 182 281)(152 321 196 287 174 306 183 280)(153 320 197 286 175 305 184 279)(154 319 198 300 176 304 185 278)(155 318 199 299 177 303 186 277)(156 317 200 298 178 302 187 276)(157 316 201 297 179 301 188 275)(158 330 202 296 180 315 189 274)(159 329 203 295 166 314 190 273)(160 328 204 294 167 313 191 272)(161 327 205 293 168 312 192 271)(162 326 206 292 169 311 193 285)(163 325 207 291 170 310 194 284)(164 324 208 290 171 309 195 283)(165 323 209 289 172 308 181 282)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4L 5A 5B 6A 6B 6C 8A ··· 8P 10A ··· 10F 12A ··· 12L 15A 15B 15C 15D 20A ··· 20X 30A ··· 30L 60A ··· 60AV order 1 2 2 2 3 4 ··· 4 5 5 6 6 6 8 ··· 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 1 ··· 1 2 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 Dic5 D10 C3⋊C8 C4×S3 D15 C5⋊2C8 C4×D5 Dic15 D30 C15⋊3C8 C4×D15 kernel C4×C15⋊3C8 C2×C15⋊3C8 C4×C60 C15⋊3C8 C2×C60 C60 C4×C20 C4×C12 C2×C20 C2×C20 C2×C12 C2×C12 C20 C20 C42 C12 C12 C2×C4 C2×C4 C4 C4 # reps 1 2 1 8 4 16 1 2 2 1 4 2 8 4 4 16 8 8 4 32 16

Matrix representation of C4×C153C8 in GL5(𝔽241)

 64 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 240 0 0 0 0 0 240
,
 1 0 0 0 0 0 190 190 0 0 0 51 240 0 0 0 0 0 0 240 0 0 0 1 240
,
 64 0 0 0 0 0 84 212 0 0 0 158 157 0 0 0 0 0 81 84 0 0 0 165 160

G:=sub<GL(5,GF(241))| [64,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,240,0,0,0,0,0,240],[1,0,0,0,0,0,190,51,0,0,0,190,240,0,0,0,0,0,0,1,0,0,0,240,240],[64,0,0,0,0,0,84,158,0,0,0,212,157,0,0,0,0,0,81,165,0,0,0,84,160] >;

C4×C153C8 in GAP, Magma, Sage, TeX

C_4\times C_{15}\rtimes_3C_8
% in TeX

G:=Group("C4xC15:3C8");
// GroupNames label

G:=SmallGroup(480,162);
// by ID

G=gap.SmallGroup(480,162);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,100,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^4=b^15=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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