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## G = C7×C4⋊C16order 448 = 26·7

### Direct product of C7 and C4⋊C16

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×C4⋊C16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C56 — C2×C112 — C7×C4⋊C16
 Lower central C1 — C2 — C7×C4⋊C16
 Upper central C1 — C2×C56 — C7×C4⋊C16

Generators and relations for C7×C4⋊C16
G = < a,b,c | a7=b4=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

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

280 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 7A ··· 7F 8A ··· 8H 8I 8J 8K 8L 14A ··· 14R 16A ··· 16P 28A ··· 28X 28Y ··· 28AV 56A ··· 56AV 56AW ··· 56BT 112A ··· 112CR order 1 2 2 2 4 4 4 4 4 4 4 4 7 ··· 7 8 ··· 8 8 8 8 8 14 ··· 14 16 ··· 16 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 1 1 1 1 1 1 2 2 2 2 1 ··· 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

280 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C4 C4 C7 C8 C14 C14 C16 C28 C28 C56 C112 D4 Q8 M4(2) M5(2) C7×D4 C7×Q8 C7×M4(2) C7×M5(2) kernel C7×C4⋊C16 C4×C56 C2×C112 C4×C28 C2×C56 C4⋊C16 C2×C28 C4×C8 C2×C16 C28 C42 C2×C8 C2×C4 C4 C56 C56 C28 C14 C8 C8 C4 C2 # reps 1 1 2 2 2 6 8 6 12 16 12 12 48 96 1 1 2 4 6 6 12 24

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

 1 0 0 0 28 0 0 0 28
,
 112 0 0 0 98 0 0 11 15
,
 73 0 0 0 85 88 0 7 28
G:=sub<GL(3,GF(113))| [1,0,0,0,28,0,0,0,28],[112,0,0,0,98,11,0,0,15],[73,0,0,0,85,7,0,88,28] >;

C7×C4⋊C16 in GAP, Magma, Sage, TeX

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

G:=Group("C7xC4:C16");
// GroupNames label

G:=SmallGroup(448,167);
// by ID

G=gap.SmallGroup(448,167);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,204,102,124]);
// Polycyclic

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

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