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## G = C4×C7⋊C16order 448 = 26·7

### Direct product of C4 and C7⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C4×C7⋊C16
 Chief series C1 — C7 — C14 — C28 — C56 — C2×C56 — C2×C7⋊C16 — C4×C7⋊C16
 Lower central C7 — C4×C7⋊C16
 Upper central C1 — C4×C8

Generators and relations for C4×C7⋊C16
G = < a,b,c | a4=b7=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C4×C7⋊C16
Regular action on 448 points
Generators in S448
(1 190 382 301)(2 191 383 302)(3 192 384 303)(4 177 369 304)(5 178 370 289)(6 179 371 290)(7 180 372 291)(8 181 373 292)(9 182 374 293)(10 183 375 294)(11 184 376 295)(12 185 377 296)(13 186 378 297)(14 187 379 298)(15 188 380 299)(16 189 381 300)(17 153 161 447)(18 154 162 448)(19 155 163 433)(20 156 164 434)(21 157 165 435)(22 158 166 436)(23 159 167 437)(24 160 168 438)(25 145 169 439)(26 146 170 440)(27 147 171 441)(28 148 172 442)(29 149 173 443)(30 150 174 444)(31 151 175 445)(32 152 176 446)(33 118 364 425)(34 119 365 426)(35 120 366 427)(36 121 367 428)(37 122 368 429)(38 123 353 430)(39 124 354 431)(40 125 355 432)(41 126 356 417)(42 127 357 418)(43 128 358 419)(44 113 359 420)(45 114 360 421)(46 115 361 422)(47 116 362 423)(48 117 363 424)(49 257 342 206)(50 258 343 207)(51 259 344 208)(52 260 345 193)(53 261 346 194)(54 262 347 195)(55 263 348 196)(56 264 349 197)(57 265 350 198)(58 266 351 199)(59 267 352 200)(60 268 337 201)(61 269 338 202)(62 270 339 203)(63 271 340 204)(64 272 341 205)(65 111 328 130)(66 112 329 131)(67 97 330 132)(68 98 331 133)(69 99 332 134)(70 100 333 135)(71 101 334 136)(72 102 335 137)(73 103 336 138)(74 104 321 139)(75 105 322 140)(76 106 323 141)(77 107 324 142)(78 108 325 143)(79 109 326 144)(80 110 327 129)(81 275 313 235)(82 276 314 236)(83 277 315 237)(84 278 316 238)(85 279 317 239)(86 280 318 240)(87 281 319 225)(88 282 320 226)(89 283 305 227)(90 284 306 228)(91 285 307 229)(92 286 308 230)(93 287 309 231)(94 288 310 232)(95 273 311 233)(96 274 312 234)(209 397 242 411)(210 398 243 412)(211 399 244 413)(212 400 245 414)(213 385 246 415)(214 386 247 416)(215 387 248 401)(216 388 249 402)(217 389 250 403)(218 390 251 404)(219 391 252 405)(220 392 253 406)(221 393 254 407)(222 394 255 408)(223 395 256 409)(224 396 241 410)
(1 385 34 71 234 31 270)(2 271 32 235 72 35 386)(3 387 36 73 236 17 272)(4 257 18 237 74 37 388)(5 389 38 75 238 19 258)(6 259 20 239 76 39 390)(7 391 40 77 240 21 260)(8 261 22 225 78 41 392)(9 393 42 79 226 23 262)(10 263 24 227 80 43 394)(11 395 44 65 228 25 264)(12 265 26 229 66 45 396)(13 397 46 67 230 27 266)(14 267 28 231 68 47 398)(15 399 48 69 232 29 268)(16 269 30 233 70 33 400)(49 448 315 139 429 216 304)(50 289 217 430 140 316 433)(51 434 317 141 431 218 290)(52 291 219 432 142 318 435)(53 436 319 143 417 220 292)(54 293 221 418 144 320 437)(55 438 305 129 419 222 294)(56 295 223 420 130 306 439)(57 440 307 131 421 224 296)(58 297 209 422 132 308 441)(59 442 309 133 423 210 298)(60 299 211 424 134 310 443)(61 444 311 135 425 212 300)(62 301 213 426 136 312 445)(63 446 313 137 427 214 302)(64 303 215 428 138 314 447)(81 102 120 247 191 340 152)(82 153 341 192 248 121 103)(83 104 122 249 177 342 154)(84 155 343 178 250 123 105)(85 106 124 251 179 344 156)(86 157 345 180 252 125 107)(87 108 126 253 181 346 158)(88 159 347 182 254 127 109)(89 110 128 255 183 348 160)(90 145 349 184 256 113 111)(91 112 114 241 185 350 146)(92 147 351 186 242 115 97)(93 98 116 243 187 352 148)(94 149 337 188 244 117 99)(95 100 118 245 189 338 150)(96 151 339 190 246 119 101)(161 205 384 401 367 336 276)(162 277 321 368 402 369 206)(163 207 370 403 353 322 278)(164 279 323 354 404 371 208)(165 193 372 405 355 324 280)(166 281 325 356 406 373 194)(167 195 374 407 357 326 282)(168 283 327 358 408 375 196)(169 197 376 409 359 328 284)(170 285 329 360 410 377 198)(171 199 378 411 361 330 286)(172 287 331 362 412 379 200)(173 201 380 413 363 332 288)(174 273 333 364 414 381 202)(175 203 382 415 365 334 274)(176 275 335 366 416 383 204)
(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)

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

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

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

160 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4L 7A 7B 7C 8A ··· 8P 14A ··· 14I 16A ··· 16AF 28A ··· 28AJ 56A ··· 56AV order 1 2 2 2 4 ··· 4 7 7 7 8 ··· 8 14 ··· 14 16 ··· 16 28 ··· 28 56 ··· 56 size 1 1 1 1 1 ··· 1 2 2 2 1 ··· 1 2 ··· 2 7 ··· 7 2 ··· 2 2 ··· 2

160 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - - + image C1 C2 C2 C4 C4 C4 C8 C8 C16 D7 Dic7 Dic7 D14 C7⋊C8 C4×D7 C7⋊C8 C7⋊C16 kernel C4×C7⋊C16 C2×C7⋊C16 C4×C56 C7⋊C16 C4×C28 C2×C56 C56 C2×C28 C28 C4×C8 C42 C2×C8 C2×C8 C8 C8 C2×C4 C4 # reps 1 2 1 8 2 2 8 8 32 3 3 3 3 12 12 12 48

Matrix representation of C4×C7⋊C16 in GL3(𝔽113) generated by

 15 0 0 0 15 0 0 0 15
,
 1 0 0 0 109 0 0 0 28
,
 1 0 0 0 0 98 0 69 0
G:=sub<GL(3,GF(113))| [15,0,0,0,15,0,0,0,15],[1,0,0,0,109,0,0,0,28],[1,0,0,0,0,69,0,98,0] >;

C4×C7⋊C16 in GAP, Magma, Sage, TeX

C_4\times C_7\rtimes C_{16}
% in TeX

G:=Group("C4xC7:C16");
// GroupNames label

G:=SmallGroup(448,17);
// by ID

G=gap.SmallGroup(448,17);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,64,100,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^4=b^7=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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