direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C16⋊4C4, C16⋊4C28, C112⋊10C4, C56.19Q8, C28.19Q16, C14.11SD32, C8.3(C7×Q8), C4.2(C7×Q16), C8.14(C2×C28), C56.82(C2×C4), (C2×C16).6C14, (C2×C14).52D8, C28.57(C4⋊C4), C2.D8.3C14, C2.3(C7×SD32), (C2×C112).16C2, (C2×C28).409D4, C22.11(C7×D8), C14.14(C2.D8), (C2×C56).419C22, C4.8(C7×C4⋊C4), C2.4(C7×C2.D8), (C2×C4).63(C7×D4), (C2×C8).74(C2×C14), (C7×C2.D8).10C2, SmallGroup(448,171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C16⋊4C4
G = < a,b,c | a7=b16=c4=1, ab=ba, ac=ca, cbc-1=b7 >
Smallest permutation representation of C7×C16⋊4C4
►Regular action on 448 points
Generators in S448
(1 287 241 384 31 169 193)(2 288 242 369 32 170 194)(3 273 243 370 17 171 195)(4 274 244 371 18 172 196)(5 275 245 372 19 173 197)(6 276 246 373 20 174 198)(7 277 247 374 21 175 199)(8 278 248 375 22 176 200)(9 279 249 376 23 161 201)(10 280 250 377 24 162 202)(11 281 251 378 25 163 203)(12 282 252 379 26 164 204)(13 283 253 380 27 165 205)(14 284 254 381 28 166 206)(15 285 255 382 29 167 207)(16 286 256 383 30 168 208)(33 109 319 437 85 297 224)(34 110 320 438 86 298 209)(35 111 305 439 87 299 210)(36 112 306 440 88 300 211)(37 97 307 441 89 301 212)(38 98 308 442 90 302 213)(39 99 309 443 91 303 214)(40 100 310 444 92 304 215)(41 101 311 445 93 289 216)(42 102 312 446 94 290 217)(43 103 313 447 95 291 218)(44 104 314 448 96 292 219)(45 105 315 433 81 293 220)(46 106 316 434 82 294 221)(47 107 317 435 83 295 222)(48 108 318 436 84 296 223)(49 336 403 361 76 155 261)(50 321 404 362 77 156 262)(51 322 405 363 78 157 263)(52 323 406 364 79 158 264)(53 324 407 365 80 159 265)(54 325 408 366 65 160 266)(55 326 409 367 66 145 267)(56 327 410 368 67 146 268)(57 328 411 353 68 147 269)(58 329 412 354 69 148 270)(59 330 413 355 70 149 271)(60 331 414 356 71 150 272)(61 332 415 357 72 151 257)(62 333 416 358 73 152 258)(63 334 401 359 74 153 259)(64 335 402 360 75 154 260)(113 184 425 388 236 141 349)(114 185 426 389 237 142 350)(115 186 427 390 238 143 351)(116 187 428 391 239 144 352)(117 188 429 392 240 129 337)(118 189 430 393 225 130 338)(119 190 431 394 226 131 339)(120 191 432 395 227 132 340)(121 192 417 396 228 133 341)(122 177 418 397 229 134 342)(123 178 419 398 230 135 343)(124 179 420 399 231 136 344)(125 180 421 400 232 137 345)(126 181 422 385 233 138 346)(127 182 423 386 234 139 347)(128 183 424 387 235 140 348)
(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)
(1 238 91 66)(2 229 92 73)(3 236 93 80)(4 227 94 71)(5 234 95 78)(6 225 96 69)(7 232 81 76)(8 239 82 67)(9 230 83 74)(10 237 84 65)(11 228 85 72)(12 235 86 79)(13 226 87 70)(14 233 88 77)(15 240 89 68)(16 231 90 75)(17 184 101 324)(18 191 102 331)(19 182 103 322)(20 189 104 329)(21 180 105 336)(22 187 106 327)(23 178 107 334)(24 185 108 325)(25 192 109 332)(26 183 110 323)(27 190 111 330)(28 181 112 321)(29 188 97 328)(30 179 98 335)(31 186 99 326)(32 177 100 333)(33 61 378 121)(34 52 379 128)(35 59 380 119)(36 50 381 126)(37 57 382 117)(38 64 383 124)(39 55 384 115)(40 62 369 122)(41 53 370 113)(42 60 371 120)(43 51 372 127)(44 58 373 118)(45 49 374 125)(46 56 375 116)(47 63 376 123)(48 54 377 114)(129 301 147 285)(130 292 148 276)(131 299 149 283)(132 290 150 274)(133 297 151 281)(134 304 152 288)(135 295 153 279)(136 302 154 286)(137 293 155 277)(138 300 156 284)(139 291 157 275)(140 298 158 282)(141 289 159 273)(142 296 160 280)(143 303 145 287)(144 294 146 278)(161 419 317 401)(162 426 318 408)(163 417 319 415)(164 424 320 406)(165 431 305 413)(166 422 306 404)(167 429 307 411)(168 420 308 402)(169 427 309 409)(170 418 310 416)(171 425 311 407)(172 432 312 414)(173 423 313 405)(174 430 314 412)(175 421 315 403)(176 428 316 410)(193 390 443 367)(194 397 444 358)(195 388 445 365)(196 395 446 356)(197 386 447 363)(198 393 448 354)(199 400 433 361)(200 391 434 368)(201 398 435 359)(202 389 436 366)(203 396 437 357)(204 387 438 364)(205 394 439 355)(206 385 440 362)(207 392 441 353)(208 399 442 360)(209 264 252 348)(210 271 253 339)(211 262 254 346)(212 269 255 337)(213 260 256 344)(214 267 241 351)(215 258 242 342)(216 265 243 349)(217 272 244 340)(218 263 245 347)(219 270 246 338)(220 261 247 345)(221 268 248 352)(222 259 249 343)(223 266 250 350)(224 257 251 341)
G:=sub<Sym(448)| (1,287,241,384,31,169,193)(2,288,242,369,32,170,194)(3,273,243,370,17,171,195)(4,274,244,371,18,172,196)(5,275,245,372,19,173,197)(6,276,246,373,20,174,198)(7,277,247,374,21,175,199)(8,278,248,375,22,176,200)(9,279,249,376,23,161,201)(10,280,250,377,24,162,202)(11,281,251,378,25,163,203)(12,282,252,379,26,164,204)(13,283,253,380,27,165,205)(14,284,254,381,28,166,206)(15,285,255,382,29,167,207)(16,286,256,383,30,168,208)(33,109,319,437,85,297,224)(34,110,320,438,86,298,209)(35,111,305,439,87,299,210)(36,112,306,440,88,300,211)(37,97,307,441,89,301,212)(38,98,308,442,90,302,213)(39,99,309,443,91,303,214)(40,100,310,444,92,304,215)(41,101,311,445,93,289,216)(42,102,312,446,94,290,217)(43,103,313,447,95,291,218)(44,104,314,448,96,292,219)(45,105,315,433,81,293,220)(46,106,316,434,82,294,221)(47,107,317,435,83,295,222)(48,108,318,436,84,296,223)(49,336,403,361,76,155,261)(50,321,404,362,77,156,262)(51,322,405,363,78,157,263)(52,323,406,364,79,158,264)(53,324,407,365,80,159,265)(54,325,408,366,65,160,266)(55,326,409,367,66,145,267)(56,327,410,368,67,146,268)(57,328,411,353,68,147,269)(58,329,412,354,69,148,270)(59,330,413,355,70,149,271)(60,331,414,356,71,150,272)(61,332,415,357,72,151,257)(62,333,416,358,73,152,258)(63,334,401,359,74,153,259)(64,335,402,360,75,154,260)(113,184,425,388,236,141,349)(114,185,426,389,237,142,350)(115,186,427,390,238,143,351)(116,187,428,391,239,144,352)(117,188,429,392,240,129,337)(118,189,430,393,225,130,338)(119,190,431,394,226,131,339)(120,191,432,395,227,132,340)(121,192,417,396,228,133,341)(122,177,418,397,229,134,342)(123,178,419,398,230,135,343)(124,179,420,399,231,136,344)(125,180,421,400,232,137,345)(126,181,422,385,233,138,346)(127,182,423,386,234,139,347)(128,183,424,387,235,140,348), (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), (1,238,91,66)(2,229,92,73)(3,236,93,80)(4,227,94,71)(5,234,95,78)(6,225,96,69)(7,232,81,76)(8,239,82,67)(9,230,83,74)(10,237,84,65)(11,228,85,72)(12,235,86,79)(13,226,87,70)(14,233,88,77)(15,240,89,68)(16,231,90,75)(17,184,101,324)(18,191,102,331)(19,182,103,322)(20,189,104,329)(21,180,105,336)(22,187,106,327)(23,178,107,334)(24,185,108,325)(25,192,109,332)(26,183,110,323)(27,190,111,330)(28,181,112,321)(29,188,97,328)(30,179,98,335)(31,186,99,326)(32,177,100,333)(33,61,378,121)(34,52,379,128)(35,59,380,119)(36,50,381,126)(37,57,382,117)(38,64,383,124)(39,55,384,115)(40,62,369,122)(41,53,370,113)(42,60,371,120)(43,51,372,127)(44,58,373,118)(45,49,374,125)(46,56,375,116)(47,63,376,123)(48,54,377,114)(129,301,147,285)(130,292,148,276)(131,299,149,283)(132,290,150,274)(133,297,151,281)(134,304,152,288)(135,295,153,279)(136,302,154,286)(137,293,155,277)(138,300,156,284)(139,291,157,275)(140,298,158,282)(141,289,159,273)(142,296,160,280)(143,303,145,287)(144,294,146,278)(161,419,317,401)(162,426,318,408)(163,417,319,415)(164,424,320,406)(165,431,305,413)(166,422,306,404)(167,429,307,411)(168,420,308,402)(169,427,309,409)(170,418,310,416)(171,425,311,407)(172,432,312,414)(173,423,313,405)(174,430,314,412)(175,421,315,403)(176,428,316,410)(193,390,443,367)(194,397,444,358)(195,388,445,365)(196,395,446,356)(197,386,447,363)(198,393,448,354)(199,400,433,361)(200,391,434,368)(201,398,435,359)(202,389,436,366)(203,396,437,357)(204,387,438,364)(205,394,439,355)(206,385,440,362)(207,392,441,353)(208,399,442,360)(209,264,252,348)(210,271,253,339)(211,262,254,346)(212,269,255,337)(213,260,256,344)(214,267,241,351)(215,258,242,342)(216,265,243,349)(217,272,244,340)(218,263,245,347)(219,270,246,338)(220,261,247,345)(221,268,248,352)(222,259,249,343)(223,266,250,350)(224,257,251,341)>;
G:=Group( (1,287,241,384,31,169,193)(2,288,242,369,32,170,194)(3,273,243,370,17,171,195)(4,274,244,371,18,172,196)(5,275,245,372,19,173,197)(6,276,246,373,20,174,198)(7,277,247,374,21,175,199)(8,278,248,375,22,176,200)(9,279,249,376,23,161,201)(10,280,250,377,24,162,202)(11,281,251,378,25,163,203)(12,282,252,379,26,164,204)(13,283,253,380,27,165,205)(14,284,254,381,28,166,206)(15,285,255,382,29,167,207)(16,286,256,383,30,168,208)(33,109,319,437,85,297,224)(34,110,320,438,86,298,209)(35,111,305,439,87,299,210)(36,112,306,440,88,300,211)(37,97,307,441,89,301,212)(38,98,308,442,90,302,213)(39,99,309,443,91,303,214)(40,100,310,444,92,304,215)(41,101,311,445,93,289,216)(42,102,312,446,94,290,217)(43,103,313,447,95,291,218)(44,104,314,448,96,292,219)(45,105,315,433,81,293,220)(46,106,316,434,82,294,221)(47,107,317,435,83,295,222)(48,108,318,436,84,296,223)(49,336,403,361,76,155,261)(50,321,404,362,77,156,262)(51,322,405,363,78,157,263)(52,323,406,364,79,158,264)(53,324,407,365,80,159,265)(54,325,408,366,65,160,266)(55,326,409,367,66,145,267)(56,327,410,368,67,146,268)(57,328,411,353,68,147,269)(58,329,412,354,69,148,270)(59,330,413,355,70,149,271)(60,331,414,356,71,150,272)(61,332,415,357,72,151,257)(62,333,416,358,73,152,258)(63,334,401,359,74,153,259)(64,335,402,360,75,154,260)(113,184,425,388,236,141,349)(114,185,426,389,237,142,350)(115,186,427,390,238,143,351)(116,187,428,391,239,144,352)(117,188,429,392,240,129,337)(118,189,430,393,225,130,338)(119,190,431,394,226,131,339)(120,191,432,395,227,132,340)(121,192,417,396,228,133,341)(122,177,418,397,229,134,342)(123,178,419,398,230,135,343)(124,179,420,399,231,136,344)(125,180,421,400,232,137,345)(126,181,422,385,233,138,346)(127,182,423,386,234,139,347)(128,183,424,387,235,140,348), (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), (1,238,91,66)(2,229,92,73)(3,236,93,80)(4,227,94,71)(5,234,95,78)(6,225,96,69)(7,232,81,76)(8,239,82,67)(9,230,83,74)(10,237,84,65)(11,228,85,72)(12,235,86,79)(13,226,87,70)(14,233,88,77)(15,240,89,68)(16,231,90,75)(17,184,101,324)(18,191,102,331)(19,182,103,322)(20,189,104,329)(21,180,105,336)(22,187,106,327)(23,178,107,334)(24,185,108,325)(25,192,109,332)(26,183,110,323)(27,190,111,330)(28,181,112,321)(29,188,97,328)(30,179,98,335)(31,186,99,326)(32,177,100,333)(33,61,378,121)(34,52,379,128)(35,59,380,119)(36,50,381,126)(37,57,382,117)(38,64,383,124)(39,55,384,115)(40,62,369,122)(41,53,370,113)(42,60,371,120)(43,51,372,127)(44,58,373,118)(45,49,374,125)(46,56,375,116)(47,63,376,123)(48,54,377,114)(129,301,147,285)(130,292,148,276)(131,299,149,283)(132,290,150,274)(133,297,151,281)(134,304,152,288)(135,295,153,279)(136,302,154,286)(137,293,155,277)(138,300,156,284)(139,291,157,275)(140,298,158,282)(141,289,159,273)(142,296,160,280)(143,303,145,287)(144,294,146,278)(161,419,317,401)(162,426,318,408)(163,417,319,415)(164,424,320,406)(165,431,305,413)(166,422,306,404)(167,429,307,411)(168,420,308,402)(169,427,309,409)(170,418,310,416)(171,425,311,407)(172,432,312,414)(173,423,313,405)(174,430,314,412)(175,421,315,403)(176,428,316,410)(193,390,443,367)(194,397,444,358)(195,388,445,365)(196,395,446,356)(197,386,447,363)(198,393,448,354)(199,400,433,361)(200,391,434,368)(201,398,435,359)(202,389,436,366)(203,396,437,357)(204,387,438,364)(205,394,439,355)(206,385,440,362)(207,392,441,353)(208,399,442,360)(209,264,252,348)(210,271,253,339)(211,262,254,346)(212,269,255,337)(213,260,256,344)(214,267,241,351)(215,258,242,342)(216,265,243,349)(217,272,244,340)(218,263,245,347)(219,270,246,338)(220,261,247,345)(221,268,248,352)(222,259,249,343)(223,266,250,350)(224,257,251,341) );
G=PermutationGroup([[(1,287,241,384,31,169,193),(2,288,242,369,32,170,194),(3,273,243,370,17,171,195),(4,274,244,371,18,172,196),(5,275,245,372,19,173,197),(6,276,246,373,20,174,198),(7,277,247,374,21,175,199),(8,278,248,375,22,176,200),(9,279,249,376,23,161,201),(10,280,250,377,24,162,202),(11,281,251,378,25,163,203),(12,282,252,379,26,164,204),(13,283,253,380,27,165,205),(14,284,254,381,28,166,206),(15,285,255,382,29,167,207),(16,286,256,383,30,168,208),(33,109,319,437,85,297,224),(34,110,320,438,86,298,209),(35,111,305,439,87,299,210),(36,112,306,440,88,300,211),(37,97,307,441,89,301,212),(38,98,308,442,90,302,213),(39,99,309,443,91,303,214),(40,100,310,444,92,304,215),(41,101,311,445,93,289,216),(42,102,312,446,94,290,217),(43,103,313,447,95,291,218),(44,104,314,448,96,292,219),(45,105,315,433,81,293,220),(46,106,316,434,82,294,221),(47,107,317,435,83,295,222),(48,108,318,436,84,296,223),(49,336,403,361,76,155,261),(50,321,404,362,77,156,262),(51,322,405,363,78,157,263),(52,323,406,364,79,158,264),(53,324,407,365,80,159,265),(54,325,408,366,65,160,266),(55,326,409,367,66,145,267),(56,327,410,368,67,146,268),(57,328,411,353,68,147,269),(58,329,412,354,69,148,270),(59,330,413,355,70,149,271),(60,331,414,356,71,150,272),(61,332,415,357,72,151,257),(62,333,416,358,73,152,258),(63,334,401,359,74,153,259),(64,335,402,360,75,154,260),(113,184,425,388,236,141,349),(114,185,426,389,237,142,350),(115,186,427,390,238,143,351),(116,187,428,391,239,144,352),(117,188,429,392,240,129,337),(118,189,430,393,225,130,338),(119,190,431,394,226,131,339),(120,191,432,395,227,132,340),(121,192,417,396,228,133,341),(122,177,418,397,229,134,342),(123,178,419,398,230,135,343),(124,179,420,399,231,136,344),(125,180,421,400,232,137,345),(126,181,422,385,233,138,346),(127,182,423,386,234,139,347),(128,183,424,387,235,140,348)], [(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)], [(1,238,91,66),(2,229,92,73),(3,236,93,80),(4,227,94,71),(5,234,95,78),(6,225,96,69),(7,232,81,76),(8,239,82,67),(9,230,83,74),(10,237,84,65),(11,228,85,72),(12,235,86,79),(13,226,87,70),(14,233,88,77),(15,240,89,68),(16,231,90,75),(17,184,101,324),(18,191,102,331),(19,182,103,322),(20,189,104,329),(21,180,105,336),(22,187,106,327),(23,178,107,334),(24,185,108,325),(25,192,109,332),(26,183,110,323),(27,190,111,330),(28,181,112,321),(29,188,97,328),(30,179,98,335),(31,186,99,326),(32,177,100,333),(33,61,378,121),(34,52,379,128),(35,59,380,119),(36,50,381,126),(37,57,382,117),(38,64,383,124),(39,55,384,115),(40,62,369,122),(41,53,370,113),(42,60,371,120),(43,51,372,127),(44,58,373,118),(45,49,374,125),(46,56,375,116),(47,63,376,123),(48,54,377,114),(129,301,147,285),(130,292,148,276),(131,299,149,283),(132,290,150,274),(133,297,151,281),(134,304,152,288),(135,295,153,279),(136,302,154,286),(137,293,155,277),(138,300,156,284),(139,291,157,275),(140,298,158,282),(141,289,159,273),(142,296,160,280),(143,303,145,287),(144,294,146,278),(161,419,317,401),(162,426,318,408),(163,417,319,415),(164,424,320,406),(165,431,305,413),(166,422,306,404),(167,429,307,411),(168,420,308,402),(169,427,309,409),(170,418,310,416),(171,425,311,407),(172,432,312,414),(173,423,313,405),(174,430,314,412),(175,421,315,403),(176,428,316,410),(193,390,443,367),(194,397,444,358),(195,388,445,365),(196,395,446,356),(197,386,447,363),(198,393,448,354),(199,400,433,361),(200,391,434,368),(201,398,435,359),(202,389,436,366),(203,396,437,357),(204,387,438,364),(205,394,439,355),(206,385,440,362),(207,392,441,353),(208,399,442,360),(209,264,252,348),(210,271,253,339),(211,262,254,346),(212,269,255,337),(213,260,256,344),(214,267,241,351),(215,258,242,342),(216,265,243,349),(217,272,244,340),(218,263,245,347),(219,270,246,338),(220,261,247,345),(221,268,248,352),(222,259,249,343),(223,266,250,350),(224,257,251,341)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 16A | ··· | 16H | 28A | ··· | 28L | 28M | ··· | 28AJ | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | Q8 | D4 | Q16 | D8 | SD32 | C7×Q8 | C7×D4 | C7×Q16 | C7×D8 | C7×SD32 |
| kernel | C7×C16⋊4C4 | C7×C2.D8 | C2×C112 | C112 | C16⋊4C4 | C2.D8 | C2×C16 | C16 | C56 | C2×C28 | C28 | C2×C14 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 1 | 1 | 2 | 2 | 8 | 6 | 6 | 12 | 12 | 48 |
Matrix representation of C7×C16⋊4C4 ►in GL4(𝔽113) generated by
| 106 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 |
| 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 106 |
| 18 | 0 | 0 | 0 |
| 0 | 44 | 0 | 0 |
| 0 | 0 | 69 | 60 |
| 0 | 0 | 53 | 69 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 22 | 46 |
| 0 | 0 | 46 | 91 |
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,106,0,0,0,0,106],[18,0,0,0,0,44,0,0,0,0,69,53,0,0,60,69],[0,112,0,0,1,0,0,0,0,0,22,46,0,0,46,91] >;
C7×C16⋊4C4 in GAP, Magma, Sage, TeX
C_7\times C_{16}\rtimes_4C_4 % in TeX
G:=Group("C7xC16:4C4"); // GroupNames label
G:=SmallGroup(448,171);
// by ID
G=gap.SmallGroup(448,171);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,2556,3923,360,14117,124]);
// Polycyclic
G:=Group<a,b,c|a^7=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^7>;
// generators/relations
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