direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C16⋊5C4, C16⋊5C28, C56.8C8, C8.2C56, C112⋊11C4, C42.4C28, C28.34C42, C14.7M5(2), (C2×C28).6C8, (C2×C4).2C56, (C4×C28).7C4, C2.3(C4×C56), C4.6(C4×C28), (C4×C56).32C2, (C2×C16).7C14, (C4×C8).14C14, C8.21(C2×C28), C28.52(C2×C8), C56.90(C2×C4), (C2×C56).31C4, C14.12(C4×C8), (C2×C8).12C28, C4.12(C2×C56), (C2×C112).17C2, C22.8(C2×C56), C2.1(C7×M5(2)), (C2×C56).450C22, (C2×C14).39(C2×C8), (C2×C4).82(C2×C28), (C2×C28).344(C2×C4), (C2×C8).104(C2×C14), SmallGroup(448,150)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C16⋊5C4
G = < a,b,c | a7=b16=c4=1, ab=ba, ac=ca, cbc-1=b9 >
Smallest permutation representation of C7×C16⋊5C4
►Regular action on 448 points
Generators in S448
(1 73 295 237 188 36 273)(2 74 296 238 189 37 274)(3 75 297 239 190 38 275)(4 76 298 240 191 39 276)(5 77 299 225 192 40 277)(6 78 300 226 177 41 278)(7 79 301 227 178 42 279)(8 80 302 228 179 43 280)(9 65 303 229 180 44 281)(10 66 304 230 181 45 282)(11 67 289 231 182 46 283)(12 68 290 232 183 47 284)(13 69 291 233 184 48 285)(14 70 292 234 185 33 286)(15 71 293 235 186 34 287)(16 72 294 236 187 35 288)(17 342 223 329 107 134 361)(18 343 224 330 108 135 362)(19 344 209 331 109 136 363)(20 345 210 332 110 137 364)(21 346 211 333 111 138 365)(22 347 212 334 112 139 366)(23 348 213 335 97 140 367)(24 349 214 336 98 141 368)(25 350 215 321 99 142 353)(26 351 216 322 100 143 354)(27 352 217 323 101 144 355)(28 337 218 324 102 129 356)(29 338 219 325 103 130 357)(30 339 220 326 104 131 358)(31 340 221 327 105 132 359)(32 341 222 328 106 133 360)(49 203 370 254 261 156 167)(50 204 371 255 262 157 168)(51 205 372 256 263 158 169)(52 206 373 241 264 159 170)(53 207 374 242 265 160 171)(54 208 375 243 266 145 172)(55 193 376 244 267 146 173)(56 194 377 245 268 147 174)(57 195 378 246 269 148 175)(58 196 379 247 270 149 176)(59 197 380 248 271 150 161)(60 198 381 249 272 151 162)(61 199 382 250 257 152 163)(62 200 383 251 258 153 164)(63 201 384 252 259 154 165)(64 202 369 253 260 155 166)(81 442 125 388 424 414 310)(82 443 126 389 425 415 311)(83 444 127 390 426 416 312)(84 445 128 391 427 401 313)(85 446 113 392 428 402 314)(86 447 114 393 429 403 315)(87 448 115 394 430 404 316)(88 433 116 395 431 405 317)(89 434 117 396 432 406 318)(90 435 118 397 417 407 319)(91 436 119 398 418 408 320)(92 437 120 399 419 409 305)(93 438 121 400 420 410 306)(94 439 122 385 421 411 307)(95 440 123 386 422 412 308)(96 441 124 387 423 413 309)
(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)
(1 373 441 211)(2 382 442 220)(3 375 443 213)(4 384 444 222)(5 377 445 215)(6 370 446 224)(7 379 447 217)(8 372 448 210)(9 381 433 219)(10 374 434 212)(11 383 435 221)(12 376 436 214)(13 369 437 223)(14 378 438 216)(15 371 439 209)(16 380 440 218)(17 48 64 305)(18 41 49 314)(19 34 50 307)(20 43 51 316)(21 36 52 309)(22 45 53 318)(23 38 54 311)(24 47 55 320)(25 40 56 313)(26 33 57 306)(27 42 58 315)(28 35 59 308)(29 44 60 317)(30 37 61 310)(31 46 62 319)(32 39 63 312)(65 249 116 325)(66 242 117 334)(67 251 118 327)(68 244 119 336)(69 253 120 329)(70 246 121 322)(71 255 122 331)(72 248 123 324)(73 241 124 333)(74 250 125 326)(75 243 126 335)(76 252 127 328)(77 245 128 321)(78 254 113 330)(79 247 114 323)(80 256 115 332)(81 339 274 199)(82 348 275 208)(83 341 276 201)(84 350 277 194)(85 343 278 203)(86 352 279 196)(87 345 280 205)(88 338 281 198)(89 347 282 207)(90 340 283 200)(91 349 284 193)(92 342 285 202)(93 351 286 195)(94 344 287 204)(95 337 288 197)(96 346 273 206)(97 297 266 389)(98 290 267 398)(99 299 268 391)(100 292 269 400)(101 301 270 393)(102 294 271 386)(103 303 272 395)(104 296 257 388)(105 289 258 397)(106 298 259 390)(107 291 260 399)(108 300 261 392)(109 293 262 385)(110 302 263 394)(111 295 264 387)(112 304 265 396)(129 236 150 422)(130 229 151 431)(131 238 152 424)(132 231 153 417)(133 240 154 426)(134 233 155 419)(135 226 156 428)(136 235 157 421)(137 228 158 430)(138 237 159 423)(139 230 160 432)(140 239 145 425)(141 232 146 418)(142 225 147 427)(143 234 148 420)(144 227 149 429)(161 412 356 187)(162 405 357 180)(163 414 358 189)(164 407 359 182)(165 416 360 191)(166 409 361 184)(167 402 362 177)(168 411 363 186)(169 404 364 179)(170 413 365 188)(171 406 366 181)(172 415 367 190)(173 408 368 183)(174 401 353 192)(175 410 354 185)(176 403 355 178)
G:=sub<Sym(448)| (1,73,295,237,188,36,273)(2,74,296,238,189,37,274)(3,75,297,239,190,38,275)(4,76,298,240,191,39,276)(5,77,299,225,192,40,277)(6,78,300,226,177,41,278)(7,79,301,227,178,42,279)(8,80,302,228,179,43,280)(9,65,303,229,180,44,281)(10,66,304,230,181,45,282)(11,67,289,231,182,46,283)(12,68,290,232,183,47,284)(13,69,291,233,184,48,285)(14,70,292,234,185,33,286)(15,71,293,235,186,34,287)(16,72,294,236,187,35,288)(17,342,223,329,107,134,361)(18,343,224,330,108,135,362)(19,344,209,331,109,136,363)(20,345,210,332,110,137,364)(21,346,211,333,111,138,365)(22,347,212,334,112,139,366)(23,348,213,335,97,140,367)(24,349,214,336,98,141,368)(25,350,215,321,99,142,353)(26,351,216,322,100,143,354)(27,352,217,323,101,144,355)(28,337,218,324,102,129,356)(29,338,219,325,103,130,357)(30,339,220,326,104,131,358)(31,340,221,327,105,132,359)(32,341,222,328,106,133,360)(49,203,370,254,261,156,167)(50,204,371,255,262,157,168)(51,205,372,256,263,158,169)(52,206,373,241,264,159,170)(53,207,374,242,265,160,171)(54,208,375,243,266,145,172)(55,193,376,244,267,146,173)(56,194,377,245,268,147,174)(57,195,378,246,269,148,175)(58,196,379,247,270,149,176)(59,197,380,248,271,150,161)(60,198,381,249,272,151,162)(61,199,382,250,257,152,163)(62,200,383,251,258,153,164)(63,201,384,252,259,154,165)(64,202,369,253,260,155,166)(81,442,125,388,424,414,310)(82,443,126,389,425,415,311)(83,444,127,390,426,416,312)(84,445,128,391,427,401,313)(85,446,113,392,428,402,314)(86,447,114,393,429,403,315)(87,448,115,394,430,404,316)(88,433,116,395,431,405,317)(89,434,117,396,432,406,318)(90,435,118,397,417,407,319)(91,436,119,398,418,408,320)(92,437,120,399,419,409,305)(93,438,121,400,420,410,306)(94,439,122,385,421,411,307)(95,440,123,386,422,412,308)(96,441,124,387,423,413,309), (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), (1,373,441,211)(2,382,442,220)(3,375,443,213)(4,384,444,222)(5,377,445,215)(6,370,446,224)(7,379,447,217)(8,372,448,210)(9,381,433,219)(10,374,434,212)(11,383,435,221)(12,376,436,214)(13,369,437,223)(14,378,438,216)(15,371,439,209)(16,380,440,218)(17,48,64,305)(18,41,49,314)(19,34,50,307)(20,43,51,316)(21,36,52,309)(22,45,53,318)(23,38,54,311)(24,47,55,320)(25,40,56,313)(26,33,57,306)(27,42,58,315)(28,35,59,308)(29,44,60,317)(30,37,61,310)(31,46,62,319)(32,39,63,312)(65,249,116,325)(66,242,117,334)(67,251,118,327)(68,244,119,336)(69,253,120,329)(70,246,121,322)(71,255,122,331)(72,248,123,324)(73,241,124,333)(74,250,125,326)(75,243,126,335)(76,252,127,328)(77,245,128,321)(78,254,113,330)(79,247,114,323)(80,256,115,332)(81,339,274,199)(82,348,275,208)(83,341,276,201)(84,350,277,194)(85,343,278,203)(86,352,279,196)(87,345,280,205)(88,338,281,198)(89,347,282,207)(90,340,283,200)(91,349,284,193)(92,342,285,202)(93,351,286,195)(94,344,287,204)(95,337,288,197)(96,346,273,206)(97,297,266,389)(98,290,267,398)(99,299,268,391)(100,292,269,400)(101,301,270,393)(102,294,271,386)(103,303,272,395)(104,296,257,388)(105,289,258,397)(106,298,259,390)(107,291,260,399)(108,300,261,392)(109,293,262,385)(110,302,263,394)(111,295,264,387)(112,304,265,396)(129,236,150,422)(130,229,151,431)(131,238,152,424)(132,231,153,417)(133,240,154,426)(134,233,155,419)(135,226,156,428)(136,235,157,421)(137,228,158,430)(138,237,159,423)(139,230,160,432)(140,239,145,425)(141,232,146,418)(142,225,147,427)(143,234,148,420)(144,227,149,429)(161,412,356,187)(162,405,357,180)(163,414,358,189)(164,407,359,182)(165,416,360,191)(166,409,361,184)(167,402,362,177)(168,411,363,186)(169,404,364,179)(170,413,365,188)(171,406,366,181)(172,415,367,190)(173,408,368,183)(174,401,353,192)(175,410,354,185)(176,403,355,178)>;
G:=Group( (1,73,295,237,188,36,273)(2,74,296,238,189,37,274)(3,75,297,239,190,38,275)(4,76,298,240,191,39,276)(5,77,299,225,192,40,277)(6,78,300,226,177,41,278)(7,79,301,227,178,42,279)(8,80,302,228,179,43,280)(9,65,303,229,180,44,281)(10,66,304,230,181,45,282)(11,67,289,231,182,46,283)(12,68,290,232,183,47,284)(13,69,291,233,184,48,285)(14,70,292,234,185,33,286)(15,71,293,235,186,34,287)(16,72,294,236,187,35,288)(17,342,223,329,107,134,361)(18,343,224,330,108,135,362)(19,344,209,331,109,136,363)(20,345,210,332,110,137,364)(21,346,211,333,111,138,365)(22,347,212,334,112,139,366)(23,348,213,335,97,140,367)(24,349,214,336,98,141,368)(25,350,215,321,99,142,353)(26,351,216,322,100,143,354)(27,352,217,323,101,144,355)(28,337,218,324,102,129,356)(29,338,219,325,103,130,357)(30,339,220,326,104,131,358)(31,340,221,327,105,132,359)(32,341,222,328,106,133,360)(49,203,370,254,261,156,167)(50,204,371,255,262,157,168)(51,205,372,256,263,158,169)(52,206,373,241,264,159,170)(53,207,374,242,265,160,171)(54,208,375,243,266,145,172)(55,193,376,244,267,146,173)(56,194,377,245,268,147,174)(57,195,378,246,269,148,175)(58,196,379,247,270,149,176)(59,197,380,248,271,150,161)(60,198,381,249,272,151,162)(61,199,382,250,257,152,163)(62,200,383,251,258,153,164)(63,201,384,252,259,154,165)(64,202,369,253,260,155,166)(81,442,125,388,424,414,310)(82,443,126,389,425,415,311)(83,444,127,390,426,416,312)(84,445,128,391,427,401,313)(85,446,113,392,428,402,314)(86,447,114,393,429,403,315)(87,448,115,394,430,404,316)(88,433,116,395,431,405,317)(89,434,117,396,432,406,318)(90,435,118,397,417,407,319)(91,436,119,398,418,408,320)(92,437,120,399,419,409,305)(93,438,121,400,420,410,306)(94,439,122,385,421,411,307)(95,440,123,386,422,412,308)(96,441,124,387,423,413,309), (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), (1,373,441,211)(2,382,442,220)(3,375,443,213)(4,384,444,222)(5,377,445,215)(6,370,446,224)(7,379,447,217)(8,372,448,210)(9,381,433,219)(10,374,434,212)(11,383,435,221)(12,376,436,214)(13,369,437,223)(14,378,438,216)(15,371,439,209)(16,380,440,218)(17,48,64,305)(18,41,49,314)(19,34,50,307)(20,43,51,316)(21,36,52,309)(22,45,53,318)(23,38,54,311)(24,47,55,320)(25,40,56,313)(26,33,57,306)(27,42,58,315)(28,35,59,308)(29,44,60,317)(30,37,61,310)(31,46,62,319)(32,39,63,312)(65,249,116,325)(66,242,117,334)(67,251,118,327)(68,244,119,336)(69,253,120,329)(70,246,121,322)(71,255,122,331)(72,248,123,324)(73,241,124,333)(74,250,125,326)(75,243,126,335)(76,252,127,328)(77,245,128,321)(78,254,113,330)(79,247,114,323)(80,256,115,332)(81,339,274,199)(82,348,275,208)(83,341,276,201)(84,350,277,194)(85,343,278,203)(86,352,279,196)(87,345,280,205)(88,338,281,198)(89,347,282,207)(90,340,283,200)(91,349,284,193)(92,342,285,202)(93,351,286,195)(94,344,287,204)(95,337,288,197)(96,346,273,206)(97,297,266,389)(98,290,267,398)(99,299,268,391)(100,292,269,400)(101,301,270,393)(102,294,271,386)(103,303,272,395)(104,296,257,388)(105,289,258,397)(106,298,259,390)(107,291,260,399)(108,300,261,392)(109,293,262,385)(110,302,263,394)(111,295,264,387)(112,304,265,396)(129,236,150,422)(130,229,151,431)(131,238,152,424)(132,231,153,417)(133,240,154,426)(134,233,155,419)(135,226,156,428)(136,235,157,421)(137,228,158,430)(138,237,159,423)(139,230,160,432)(140,239,145,425)(141,232,146,418)(142,225,147,427)(143,234,148,420)(144,227,149,429)(161,412,356,187)(162,405,357,180)(163,414,358,189)(164,407,359,182)(165,416,360,191)(166,409,361,184)(167,402,362,177)(168,411,363,186)(169,404,364,179)(170,413,365,188)(171,406,366,181)(172,415,367,190)(173,408,368,183)(174,401,353,192)(175,410,354,185)(176,403,355,178) );
G=PermutationGroup([[(1,73,295,237,188,36,273),(2,74,296,238,189,37,274),(3,75,297,239,190,38,275),(4,76,298,240,191,39,276),(5,77,299,225,192,40,277),(6,78,300,226,177,41,278),(7,79,301,227,178,42,279),(8,80,302,228,179,43,280),(9,65,303,229,180,44,281),(10,66,304,230,181,45,282),(11,67,289,231,182,46,283),(12,68,290,232,183,47,284),(13,69,291,233,184,48,285),(14,70,292,234,185,33,286),(15,71,293,235,186,34,287),(16,72,294,236,187,35,288),(17,342,223,329,107,134,361),(18,343,224,330,108,135,362),(19,344,209,331,109,136,363),(20,345,210,332,110,137,364),(21,346,211,333,111,138,365),(22,347,212,334,112,139,366),(23,348,213,335,97,140,367),(24,349,214,336,98,141,368),(25,350,215,321,99,142,353),(26,351,216,322,100,143,354),(27,352,217,323,101,144,355),(28,337,218,324,102,129,356),(29,338,219,325,103,130,357),(30,339,220,326,104,131,358),(31,340,221,327,105,132,359),(32,341,222,328,106,133,360),(49,203,370,254,261,156,167),(50,204,371,255,262,157,168),(51,205,372,256,263,158,169),(52,206,373,241,264,159,170),(53,207,374,242,265,160,171),(54,208,375,243,266,145,172),(55,193,376,244,267,146,173),(56,194,377,245,268,147,174),(57,195,378,246,269,148,175),(58,196,379,247,270,149,176),(59,197,380,248,271,150,161),(60,198,381,249,272,151,162),(61,199,382,250,257,152,163),(62,200,383,251,258,153,164),(63,201,384,252,259,154,165),(64,202,369,253,260,155,166),(81,442,125,388,424,414,310),(82,443,126,389,425,415,311),(83,444,127,390,426,416,312),(84,445,128,391,427,401,313),(85,446,113,392,428,402,314),(86,447,114,393,429,403,315),(87,448,115,394,430,404,316),(88,433,116,395,431,405,317),(89,434,117,396,432,406,318),(90,435,118,397,417,407,319),(91,436,119,398,418,408,320),(92,437,120,399,419,409,305),(93,438,121,400,420,410,306),(94,439,122,385,421,411,307),(95,440,123,386,422,412,308),(96,441,124,387,423,413,309)], [(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)], [(1,373,441,211),(2,382,442,220),(3,375,443,213),(4,384,444,222),(5,377,445,215),(6,370,446,224),(7,379,447,217),(8,372,448,210),(9,381,433,219),(10,374,434,212),(11,383,435,221),(12,376,436,214),(13,369,437,223),(14,378,438,216),(15,371,439,209),(16,380,440,218),(17,48,64,305),(18,41,49,314),(19,34,50,307),(20,43,51,316),(21,36,52,309),(22,45,53,318),(23,38,54,311),(24,47,55,320),(25,40,56,313),(26,33,57,306),(27,42,58,315),(28,35,59,308),(29,44,60,317),(30,37,61,310),(31,46,62,319),(32,39,63,312),(65,249,116,325),(66,242,117,334),(67,251,118,327),(68,244,119,336),(69,253,120,329),(70,246,121,322),(71,255,122,331),(72,248,123,324),(73,241,124,333),(74,250,125,326),(75,243,126,335),(76,252,127,328),(77,245,128,321),(78,254,113,330),(79,247,114,323),(80,256,115,332),(81,339,274,199),(82,348,275,208),(83,341,276,201),(84,350,277,194),(85,343,278,203),(86,352,279,196),(87,345,280,205),(88,338,281,198),(89,347,282,207),(90,340,283,200),(91,349,284,193),(92,342,285,202),(93,351,286,195),(94,344,287,204),(95,337,288,197),(96,346,273,206),(97,297,266,389),(98,290,267,398),(99,299,268,391),(100,292,269,400),(101,301,270,393),(102,294,271,386),(103,303,272,395),(104,296,257,388),(105,289,258,397),(106,298,259,390),(107,291,260,399),(108,300,261,392),(109,293,262,385),(110,302,263,394),(111,295,264,387),(112,304,265,396),(129,236,150,422),(130,229,151,431),(131,238,152,424),(132,231,153,417),(133,240,154,426),(134,233,155,419),(135,226,156,428),(136,235,157,421),(137,228,158,430),(138,237,159,423),(139,230,160,432),(140,239,145,425),(141,232,146,418),(142,225,147,427),(143,234,148,420),(144,227,149,429),(161,412,356,187),(162,405,357,180),(163,414,358,189),(164,407,359,182),(165,416,360,191),(166,409,361,184),(167,402,362,177),(168,411,363,186),(169,404,364,179),(170,413,365,188),(171,406,366,181),(172,415,367,190),(173,408,368,183),(174,401,353,192),(175,410,354,185),(176,403,355,178)]])
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 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C7 | C8 | C8 | C14 | C14 | C28 | C28 | C28 | C56 | C56 | M5(2) | C7×M5(2) |
| kernel | C7×C16⋊5C4 | C4×C56 | C2×C112 | C112 | C4×C28 | C2×C56 | C16⋊5C4 | C56 | C2×C28 | C4×C8 | C2×C16 | C16 | C42 | C2×C8 | C8 | C2×C4 | C14 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 6 | 8 | 8 | 6 | 12 | 48 | 12 | 12 | 48 | 48 | 8 | 48 |
Matrix representation of C7×C16⋊5C4 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 49 | 0 |
| 0 | 0 | 49 |
| 15 | 0 | 0 |
| 0 | 56 | 33 |
| 0 | 33 | 57 |
| 15 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 112 | 0 |
G:=sub<GL(3,GF(113))| [1,0,0,0,49,0,0,0,49],[15,0,0,0,56,33,0,33,57],[15,0,0,0,0,112,0,1,0] >;
C7×C16⋊5C4 in GAP, Magma, Sage, TeX
C_7\times C_{16}\rtimes_5C_4 % in TeX
G:=Group("C7xC16:5C4"); // GroupNames label
G:=SmallGroup(448,150);
// by ID
G=gap.SmallGroup(448,150);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,196,3165,400,136,124]);
// 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^9>;
// generators/relations
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