direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C16⋊3C4, C112⋊7C4, C16⋊3C28, C14.7Q32, C56.18Q8, C14.14D16, C28.18Q16, C8.2(C7×Q8), C2.2(C7×D16), C2.2(C7×Q32), C4.1(C7×Q16), (C2×C16).3C14, (C2×C112).9C2, C56.81(C2×C4), C8.13(C2×C28), (C2×C14).51D8, C28.56(C4⋊C4), C2.D8.2C14, (C2×C28).408D4, C22.10(C7×D8), C14.13(C2.D8), (C2×C56).418C22, C4.7(C7×C4⋊C4), C2.3(C7×C2.D8), (C2×C4).62(C7×D4), (C7×C2.D8).9C2, (C2×C8).73(C2×C14), SmallGroup(448,170)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C16⋊3C4
G = < a,b,c | a7=b16=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×C16⋊3C4
►Regular action on 448 points
Generators in S448
(1 279 144 326 440 306 85)(2 280 129 327 441 307 86)(3 281 130 328 442 308 87)(4 282 131 329 443 309 88)(5 283 132 330 444 310 89)(6 284 133 331 445 311 90)(7 285 134 332 446 312 91)(8 286 135 333 447 313 92)(9 287 136 334 448 314 93)(10 288 137 335 433 315 94)(11 273 138 336 434 316 95)(12 274 139 321 435 317 96)(13 275 140 322 436 318 81)(14 276 141 323 437 319 82)(15 277 142 324 438 320 83)(16 278 143 325 439 305 84)(17 111 186 372 38 402 61)(18 112 187 373 39 403 62)(19 97 188 374 40 404 63)(20 98 189 375 41 405 64)(21 99 190 376 42 406 49)(22 100 191 377 43 407 50)(23 101 192 378 44 408 51)(24 102 177 379 45 409 52)(25 103 178 380 46 410 53)(26 104 179 381 47 411 54)(27 105 180 382 48 412 55)(28 106 181 383 33 413 56)(29 107 182 384 34 414 57)(30 108 183 369 35 415 58)(31 109 184 370 36 416 59)(32 110 185 371 37 401 60)(65 299 387 252 231 349 424)(66 300 388 253 232 350 425)(67 301 389 254 233 351 426)(68 302 390 255 234 352 427)(69 303 391 256 235 337 428)(70 304 392 241 236 338 429)(71 289 393 242 237 339 430)(72 290 394 243 238 340 431)(73 291 395 244 239 341 432)(74 292 396 245 240 342 417)(75 293 397 246 225 343 418)(76 294 398 247 226 344 419)(77 295 399 248 227 345 420)(78 296 400 249 228 346 421)(79 297 385 250 229 347 422)(80 298 386 251 230 348 423)(113 169 271 358 158 210 208)(114 170 272 359 159 211 193)(115 171 257 360 160 212 194)(116 172 258 361 145 213 195)(117 173 259 362 146 214 196)(118 174 260 363 147 215 197)(119 175 261 364 148 216 198)(120 176 262 365 149 217 199)(121 161 263 366 150 218 200)(122 162 264 367 151 219 201)(123 163 265 368 152 220 202)(124 164 266 353 153 221 203)(125 165 267 354 154 222 204)(126 166 268 355 155 223 205)(127 167 269 356 156 224 206)(128 168 270 357 157 209 207)
(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)
(1 371 193 229)(2 370 194 228)(3 369 195 227)(4 384 196 226)(5 383 197 225)(6 382 198 240)(7 381 199 239)(8 380 200 238)(9 379 201 237)(10 378 202 236)(11 377 203 235)(12 376 204 234)(13 375 205 233)(14 374 206 232)(15 373 207 231)(16 372 208 230)(17 358 298 439)(18 357 299 438)(19 356 300 437)(20 355 301 436)(21 354 302 435)(22 353 303 434)(23 368 304 433)(24 367 289 448)(25 366 290 447)(26 365 291 446)(27 364 292 445)(28 363 293 444)(29 362 294 443)(30 361 295 442)(31 360 296 441)(32 359 297 440)(33 118 343 283)(34 117 344 282)(35 116 345 281)(36 115 346 280)(37 114 347 279)(38 113 348 278)(39 128 349 277)(40 127 350 276)(41 126 351 275)(42 125 352 274)(43 124 337 273)(44 123 338 288)(45 122 339 287)(46 121 340 286)(47 120 341 285)(48 119 342 284)(49 267 68 321)(50 266 69 336)(51 265 70 335)(52 264 71 334)(53 263 72 333)(54 262 73 332)(55 261 74 331)(56 260 75 330)(57 259 76 329)(58 258 77 328)(59 257 78 327)(60 272 79 326)(61 271 80 325)(62 270 65 324)(63 269 66 323)(64 268 67 322)(81 189 223 254)(82 188 224 253)(83 187 209 252)(84 186 210 251)(85 185 211 250)(86 184 212 249)(87 183 213 248)(88 182 214 247)(89 181 215 246)(90 180 216 245)(91 179 217 244)(92 178 218 243)(93 177 219 242)(94 192 220 241)(95 191 221 256)(96 190 222 255)(97 156 388 319)(98 155 389 318)(99 154 390 317)(100 153 391 316)(101 152 392 315)(102 151 393 314)(103 150 394 313)(104 149 395 312)(105 148 396 311)(106 147 397 310)(107 146 398 309)(108 145 399 308)(109 160 400 307)(110 159 385 306)(111 158 386 305)(112 157 387 320)(129 416 171 421)(130 415 172 420)(131 414 173 419)(132 413 174 418)(133 412 175 417)(134 411 176 432)(135 410 161 431)(136 409 162 430)(137 408 163 429)(138 407 164 428)(139 406 165 427)(140 405 166 426)(141 404 167 425)(142 403 168 424)(143 402 169 423)(144 401 170 422)
G:=sub<Sym(448)| (1,279,144,326,440,306,85)(2,280,129,327,441,307,86)(3,281,130,328,442,308,87)(4,282,131,329,443,309,88)(5,283,132,330,444,310,89)(6,284,133,331,445,311,90)(7,285,134,332,446,312,91)(8,286,135,333,447,313,92)(9,287,136,334,448,314,93)(10,288,137,335,433,315,94)(11,273,138,336,434,316,95)(12,274,139,321,435,317,96)(13,275,140,322,436,318,81)(14,276,141,323,437,319,82)(15,277,142,324,438,320,83)(16,278,143,325,439,305,84)(17,111,186,372,38,402,61)(18,112,187,373,39,403,62)(19,97,188,374,40,404,63)(20,98,189,375,41,405,64)(21,99,190,376,42,406,49)(22,100,191,377,43,407,50)(23,101,192,378,44,408,51)(24,102,177,379,45,409,52)(25,103,178,380,46,410,53)(26,104,179,381,47,411,54)(27,105,180,382,48,412,55)(28,106,181,383,33,413,56)(29,107,182,384,34,414,57)(30,108,183,369,35,415,58)(31,109,184,370,36,416,59)(32,110,185,371,37,401,60)(65,299,387,252,231,349,424)(66,300,388,253,232,350,425)(67,301,389,254,233,351,426)(68,302,390,255,234,352,427)(69,303,391,256,235,337,428)(70,304,392,241,236,338,429)(71,289,393,242,237,339,430)(72,290,394,243,238,340,431)(73,291,395,244,239,341,432)(74,292,396,245,240,342,417)(75,293,397,246,225,343,418)(76,294,398,247,226,344,419)(77,295,399,248,227,345,420)(78,296,400,249,228,346,421)(79,297,385,250,229,347,422)(80,298,386,251,230,348,423)(113,169,271,358,158,210,208)(114,170,272,359,159,211,193)(115,171,257,360,160,212,194)(116,172,258,361,145,213,195)(117,173,259,362,146,214,196)(118,174,260,363,147,215,197)(119,175,261,364,148,216,198)(120,176,262,365,149,217,199)(121,161,263,366,150,218,200)(122,162,264,367,151,219,201)(123,163,265,368,152,220,202)(124,164,266,353,153,221,203)(125,165,267,354,154,222,204)(126,166,268,355,155,223,205)(127,167,269,356,156,224,206)(128,168,270,357,157,209,207), (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), (1,371,193,229)(2,370,194,228)(3,369,195,227)(4,384,196,226)(5,383,197,225)(6,382,198,240)(7,381,199,239)(8,380,200,238)(9,379,201,237)(10,378,202,236)(11,377,203,235)(12,376,204,234)(13,375,205,233)(14,374,206,232)(15,373,207,231)(16,372,208,230)(17,358,298,439)(18,357,299,438)(19,356,300,437)(20,355,301,436)(21,354,302,435)(22,353,303,434)(23,368,304,433)(24,367,289,448)(25,366,290,447)(26,365,291,446)(27,364,292,445)(28,363,293,444)(29,362,294,443)(30,361,295,442)(31,360,296,441)(32,359,297,440)(33,118,343,283)(34,117,344,282)(35,116,345,281)(36,115,346,280)(37,114,347,279)(38,113,348,278)(39,128,349,277)(40,127,350,276)(41,126,351,275)(42,125,352,274)(43,124,337,273)(44,123,338,288)(45,122,339,287)(46,121,340,286)(47,120,341,285)(48,119,342,284)(49,267,68,321)(50,266,69,336)(51,265,70,335)(52,264,71,334)(53,263,72,333)(54,262,73,332)(55,261,74,331)(56,260,75,330)(57,259,76,329)(58,258,77,328)(59,257,78,327)(60,272,79,326)(61,271,80,325)(62,270,65,324)(63,269,66,323)(64,268,67,322)(81,189,223,254)(82,188,224,253)(83,187,209,252)(84,186,210,251)(85,185,211,250)(86,184,212,249)(87,183,213,248)(88,182,214,247)(89,181,215,246)(90,180,216,245)(91,179,217,244)(92,178,218,243)(93,177,219,242)(94,192,220,241)(95,191,221,256)(96,190,222,255)(97,156,388,319)(98,155,389,318)(99,154,390,317)(100,153,391,316)(101,152,392,315)(102,151,393,314)(103,150,394,313)(104,149,395,312)(105,148,396,311)(106,147,397,310)(107,146,398,309)(108,145,399,308)(109,160,400,307)(110,159,385,306)(111,158,386,305)(112,157,387,320)(129,416,171,421)(130,415,172,420)(131,414,173,419)(132,413,174,418)(133,412,175,417)(134,411,176,432)(135,410,161,431)(136,409,162,430)(137,408,163,429)(138,407,164,428)(139,406,165,427)(140,405,166,426)(141,404,167,425)(142,403,168,424)(143,402,169,423)(144,401,170,422)>;
G:=Group( (1,279,144,326,440,306,85)(2,280,129,327,441,307,86)(3,281,130,328,442,308,87)(4,282,131,329,443,309,88)(5,283,132,330,444,310,89)(6,284,133,331,445,311,90)(7,285,134,332,446,312,91)(8,286,135,333,447,313,92)(9,287,136,334,448,314,93)(10,288,137,335,433,315,94)(11,273,138,336,434,316,95)(12,274,139,321,435,317,96)(13,275,140,322,436,318,81)(14,276,141,323,437,319,82)(15,277,142,324,438,320,83)(16,278,143,325,439,305,84)(17,111,186,372,38,402,61)(18,112,187,373,39,403,62)(19,97,188,374,40,404,63)(20,98,189,375,41,405,64)(21,99,190,376,42,406,49)(22,100,191,377,43,407,50)(23,101,192,378,44,408,51)(24,102,177,379,45,409,52)(25,103,178,380,46,410,53)(26,104,179,381,47,411,54)(27,105,180,382,48,412,55)(28,106,181,383,33,413,56)(29,107,182,384,34,414,57)(30,108,183,369,35,415,58)(31,109,184,370,36,416,59)(32,110,185,371,37,401,60)(65,299,387,252,231,349,424)(66,300,388,253,232,350,425)(67,301,389,254,233,351,426)(68,302,390,255,234,352,427)(69,303,391,256,235,337,428)(70,304,392,241,236,338,429)(71,289,393,242,237,339,430)(72,290,394,243,238,340,431)(73,291,395,244,239,341,432)(74,292,396,245,240,342,417)(75,293,397,246,225,343,418)(76,294,398,247,226,344,419)(77,295,399,248,227,345,420)(78,296,400,249,228,346,421)(79,297,385,250,229,347,422)(80,298,386,251,230,348,423)(113,169,271,358,158,210,208)(114,170,272,359,159,211,193)(115,171,257,360,160,212,194)(116,172,258,361,145,213,195)(117,173,259,362,146,214,196)(118,174,260,363,147,215,197)(119,175,261,364,148,216,198)(120,176,262,365,149,217,199)(121,161,263,366,150,218,200)(122,162,264,367,151,219,201)(123,163,265,368,152,220,202)(124,164,266,353,153,221,203)(125,165,267,354,154,222,204)(126,166,268,355,155,223,205)(127,167,269,356,156,224,206)(128,168,270,357,157,209,207), (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), (1,371,193,229)(2,370,194,228)(3,369,195,227)(4,384,196,226)(5,383,197,225)(6,382,198,240)(7,381,199,239)(8,380,200,238)(9,379,201,237)(10,378,202,236)(11,377,203,235)(12,376,204,234)(13,375,205,233)(14,374,206,232)(15,373,207,231)(16,372,208,230)(17,358,298,439)(18,357,299,438)(19,356,300,437)(20,355,301,436)(21,354,302,435)(22,353,303,434)(23,368,304,433)(24,367,289,448)(25,366,290,447)(26,365,291,446)(27,364,292,445)(28,363,293,444)(29,362,294,443)(30,361,295,442)(31,360,296,441)(32,359,297,440)(33,118,343,283)(34,117,344,282)(35,116,345,281)(36,115,346,280)(37,114,347,279)(38,113,348,278)(39,128,349,277)(40,127,350,276)(41,126,351,275)(42,125,352,274)(43,124,337,273)(44,123,338,288)(45,122,339,287)(46,121,340,286)(47,120,341,285)(48,119,342,284)(49,267,68,321)(50,266,69,336)(51,265,70,335)(52,264,71,334)(53,263,72,333)(54,262,73,332)(55,261,74,331)(56,260,75,330)(57,259,76,329)(58,258,77,328)(59,257,78,327)(60,272,79,326)(61,271,80,325)(62,270,65,324)(63,269,66,323)(64,268,67,322)(81,189,223,254)(82,188,224,253)(83,187,209,252)(84,186,210,251)(85,185,211,250)(86,184,212,249)(87,183,213,248)(88,182,214,247)(89,181,215,246)(90,180,216,245)(91,179,217,244)(92,178,218,243)(93,177,219,242)(94,192,220,241)(95,191,221,256)(96,190,222,255)(97,156,388,319)(98,155,389,318)(99,154,390,317)(100,153,391,316)(101,152,392,315)(102,151,393,314)(103,150,394,313)(104,149,395,312)(105,148,396,311)(106,147,397,310)(107,146,398,309)(108,145,399,308)(109,160,400,307)(110,159,385,306)(111,158,386,305)(112,157,387,320)(129,416,171,421)(130,415,172,420)(131,414,173,419)(132,413,174,418)(133,412,175,417)(134,411,176,432)(135,410,161,431)(136,409,162,430)(137,408,163,429)(138,407,164,428)(139,406,165,427)(140,405,166,426)(141,404,167,425)(142,403,168,424)(143,402,169,423)(144,401,170,422) );
G=PermutationGroup([[(1,279,144,326,440,306,85),(2,280,129,327,441,307,86),(3,281,130,328,442,308,87),(4,282,131,329,443,309,88),(5,283,132,330,444,310,89),(6,284,133,331,445,311,90),(7,285,134,332,446,312,91),(8,286,135,333,447,313,92),(9,287,136,334,448,314,93),(10,288,137,335,433,315,94),(11,273,138,336,434,316,95),(12,274,139,321,435,317,96),(13,275,140,322,436,318,81),(14,276,141,323,437,319,82),(15,277,142,324,438,320,83),(16,278,143,325,439,305,84),(17,111,186,372,38,402,61),(18,112,187,373,39,403,62),(19,97,188,374,40,404,63),(20,98,189,375,41,405,64),(21,99,190,376,42,406,49),(22,100,191,377,43,407,50),(23,101,192,378,44,408,51),(24,102,177,379,45,409,52),(25,103,178,380,46,410,53),(26,104,179,381,47,411,54),(27,105,180,382,48,412,55),(28,106,181,383,33,413,56),(29,107,182,384,34,414,57),(30,108,183,369,35,415,58),(31,109,184,370,36,416,59),(32,110,185,371,37,401,60),(65,299,387,252,231,349,424),(66,300,388,253,232,350,425),(67,301,389,254,233,351,426),(68,302,390,255,234,352,427),(69,303,391,256,235,337,428),(70,304,392,241,236,338,429),(71,289,393,242,237,339,430),(72,290,394,243,238,340,431),(73,291,395,244,239,341,432),(74,292,396,245,240,342,417),(75,293,397,246,225,343,418),(76,294,398,247,226,344,419),(77,295,399,248,227,345,420),(78,296,400,249,228,346,421),(79,297,385,250,229,347,422),(80,298,386,251,230,348,423),(113,169,271,358,158,210,208),(114,170,272,359,159,211,193),(115,171,257,360,160,212,194),(116,172,258,361,145,213,195),(117,173,259,362,146,214,196),(118,174,260,363,147,215,197),(119,175,261,364,148,216,198),(120,176,262,365,149,217,199),(121,161,263,366,150,218,200),(122,162,264,367,151,219,201),(123,163,265,368,152,220,202),(124,164,266,353,153,221,203),(125,165,267,354,154,222,204),(126,166,268,355,155,223,205),(127,167,269,356,156,224,206),(128,168,270,357,157,209,207)], [(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)], [(1,371,193,229),(2,370,194,228),(3,369,195,227),(4,384,196,226),(5,383,197,225),(6,382,198,240),(7,381,199,239),(8,380,200,238),(9,379,201,237),(10,378,202,236),(11,377,203,235),(12,376,204,234),(13,375,205,233),(14,374,206,232),(15,373,207,231),(16,372,208,230),(17,358,298,439),(18,357,299,438),(19,356,300,437),(20,355,301,436),(21,354,302,435),(22,353,303,434),(23,368,304,433),(24,367,289,448),(25,366,290,447),(26,365,291,446),(27,364,292,445),(28,363,293,444),(29,362,294,443),(30,361,295,442),(31,360,296,441),(32,359,297,440),(33,118,343,283),(34,117,344,282),(35,116,345,281),(36,115,346,280),(37,114,347,279),(38,113,348,278),(39,128,349,277),(40,127,350,276),(41,126,351,275),(42,125,352,274),(43,124,337,273),(44,123,338,288),(45,122,339,287),(46,121,340,286),(47,120,341,285),(48,119,342,284),(49,267,68,321),(50,266,69,336),(51,265,70,335),(52,264,71,334),(53,263,72,333),(54,262,73,332),(55,261,74,331),(56,260,75,330),(57,259,76,329),(58,258,77,328),(59,257,78,327),(60,272,79,326),(61,271,80,325),(62,270,65,324),(63,269,66,323),(64,268,67,322),(81,189,223,254),(82,188,224,253),(83,187,209,252),(84,186,210,251),(85,185,211,250),(86,184,212,249),(87,183,213,248),(88,182,214,247),(89,181,215,246),(90,180,216,245),(91,179,217,244),(92,178,218,243),(93,177,219,242),(94,192,220,241),(95,191,221,256),(96,190,222,255),(97,156,388,319),(98,155,389,318),(99,154,390,317),(100,153,391,316),(101,152,392,315),(102,151,393,314),(103,150,394,313),(104,149,395,312),(105,148,396,311),(106,147,397,310),(107,146,398,309),(108,145,399,308),(109,160,400,307),(110,159,385,306),(111,158,386,305),(112,157,387,320),(129,416,171,421),(130,415,172,420),(131,414,173,419),(132,413,174,418),(133,412,175,417),(134,411,176,432),(135,410,161,431),(136,409,162,430),(137,408,163,429),(138,407,164,428),(139,406,165,427),(140,405,166,426),(141,404,167,425),(142,403,168,424),(143,402,169,423),(144,401,170,422)]])
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 | 2 | 2 |
| type | + | + | + | - | + | - | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | Q8 | D4 | Q16 | D8 | D16 | Q32 | C7×Q8 | C7×D4 | C7×Q16 | C7×D8 | C7×D16 | C7×Q32 |
| kernel | C7×C16⋊3C4 | C7×C2.D8 | C2×C112 | C112 | C16⋊3C4 | C2.D8 | C2×C16 | C16 | C56 | C2×C28 | C28 | C2×C14 | C14 | C14 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 1 | 1 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 24 | 24 |
Matrix representation of C7×C16⋊3C4 ►in GL4(𝔽113) generated by
| 106 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 |
| 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 106 |
| 31 | 82 | 0 | 0 |
| 31 | 31 | 0 | 0 |
| 0 | 0 | 18 | 109 |
| 0 | 0 | 4 | 18 |
| 76 | 72 | 0 | 0 |
| 72 | 37 | 0 | 0 |
| 0 | 0 | 72 | 76 |
| 0 | 0 | 76 | 41 |
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,106,0,0,0,0,106],[31,31,0,0,82,31,0,0,0,0,18,4,0,0,109,18],[76,72,0,0,72,37,0,0,0,0,72,76,0,0,76,41] >;
C7×C16⋊3C4 in GAP, Magma, Sage, TeX
C_7\times C_{16}\rtimes_3C_4 % in TeX
G:=Group("C7xC16:3C4"); // GroupNames label
G:=SmallGroup(448,170);
// by ID
G=gap.SmallGroup(448,170);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,988,3923,360,14117,124]);
// 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^-1>;
// generators/relations
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