direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C42×Dic7, C7⋊C43, C28⋊3C42, (C4×C28)⋊14C4, C2.2(D7×C42), (C2×C42).23D7, C14.16(C2×C42), (C22×C4).456D14, C23.263(C22×D7), (C22×C14).305C23, (C22×C28).555C22, C22.16(C22×Dic7), (C22×Dic7).241C22, (C2×C4×C28).27C2, C2.2(C2×C4×Dic7), C22.49(C2×C4×D7), (C2×C4).179(C4×D7), (C2×C4×Dic7).42C2, (C2×C28).250(C2×C4), (C2×C4).99(C2×Dic7), (C2×C14).95(C22×C4), (C2×Dic7).109(C2×C4), SmallGroup(448,464)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C42×Dic7 |
Generators and relations for C42×Dic7
G = < a,b,c,d | a4=b4=c14=1, d2=c7, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 708 in 258 conjugacy classes, 183 normal (9 characteristic)
C1, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C42, C42, C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2×C42, C2×C42, C2×Dic7, C2×C28, C22×C14, C43, C4×Dic7, C4×C28, C22×Dic7, C22×C28, C2×C4×Dic7, C2×C4×C28, C42×Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, C22×C4, Dic7, D14, C2×C42, C4×D7, C2×Dic7, C22×D7, C43, C4×Dic7, C2×C4×D7, C22×Dic7, D7×C42, C2×C4×Dic7, C42×Dic7
►Regular action on 448 points
Generators in S448
(1 224 143 77)(2 211 144 78)(3 212 145 79)(4 213 146 80)(5 214 147 81)(6 215 148 82)(7 216 149 83)(8 217 150 84)(9 218 151 71)(10 219 152 72)(11 220 153 73)(12 221 154 74)(13 222 141 75)(14 223 142 76)(15 168 119 205)(16 155 120 206)(17 156 121 207)(18 157 122 208)(19 158 123 209)(20 159 124 210)(21 160 125 197)(22 161 126 198)(23 162 113 199)(24 163 114 200)(25 164 115 201)(26 165 116 202)(27 166 117 203)(28 167 118 204)(29 107 283 408)(30 108 284 409)(31 109 285 410)(32 110 286 411)(33 111 287 412)(34 112 288 413)(35 99 289 414)(36 100 290 415)(37 101 291 416)(38 102 292 417)(39 103 293 418)(40 104 294 419)(41 105 281 420)(42 106 282 407)(43 403 432 233)(44 404 433 234)(45 405 434 235)(46 406 421 236)(47 393 422 237)(48 394 423 238)(49 395 424 225)(50 396 425 226)(51 397 426 227)(52 398 427 228)(53 399 428 229)(54 400 429 230)(55 401 430 231)(56 402 431 232)(57 256 347 250)(58 257 348 251)(59 258 349 252)(60 259 350 239)(61 260 337 240)(62 261 338 241)(63 262 339 242)(64 263 340 243)(65 264 341 244)(66 265 342 245)(67 266 343 246)(68 253 344 247)(69 254 345 248)(70 255 346 249)(85 297 315 365)(86 298 316 366)(87 299 317 367)(88 300 318 368)(89 301 319 369)(90 302 320 370)(91 303 321 371)(92 304 322 372)(93 305 309 373)(94 306 310 374)(95 307 311 375)(96 308 312 376)(97 295 313 377)(98 296 314 378)(127 387 325 351)(128 388 326 352)(129 389 327 353)(130 390 328 354)(131 391 329 355)(132 392 330 356)(133 379 331 357)(134 380 332 358)(135 381 333 359)(136 382 334 360)(137 383 335 361)(138 384 336 362)(139 385 323 363)(140 386 324 364)(169 271 436 193)(170 272 437 194)(171 273 438 195)(172 274 439 196)(173 275 440 183)(174 276 441 184)(175 277 442 185)(176 278 443 186)(177 279 444 187)(178 280 445 188)(179 267 446 189)(180 268 447 190)(181 269 448 191)(182 270 435 192)
(1 248 235 33)(2 249 236 34)(3 250 237 35)(4 251 238 36)(5 252 225 37)(6 239 226 38)(7 240 227 39)(8 241 228 40)(9 242 229 41)(10 243 230 42)(11 244 231 29)(12 245 232 30)(13 246 233 31)(14 247 234 32)(15 173 303 359)(16 174 304 360)(17 175 305 361)(18 176 306 362)(19 177 307 363)(20 178 308 364)(21 179 295 351)(22 180 296 352)(23 181 297 353)(24 182 298 354)(25 169 299 355)(26 170 300 356)(27 171 301 357)(28 172 302 358)(43 109 222 67)(44 110 223 68)(45 111 224 69)(46 112 211 70)(47 99 212 57)(48 100 213 58)(49 101 214 59)(50 102 215 60)(51 103 216 61)(52 104 217 62)(53 105 218 63)(54 106 219 64)(55 107 220 65)(56 108 221 66)(71 339 428 420)(72 340 429 407)(73 341 430 408)(74 342 431 409)(75 343 432 410)(76 344 433 411)(77 345 434 412)(78 346 421 413)(79 347 422 414)(80 348 423 415)(81 349 424 416)(82 350 425 417)(83 337 426 418)(84 338 427 419)(85 327 199 191)(86 328 200 192)(87 329 201 193)(88 330 202 194)(89 331 203 195)(90 332 204 196)(91 333 205 183)(92 334 206 184)(93 335 207 185)(94 336 208 186)(95 323 209 187)(96 324 210 188)(97 325 197 189)(98 326 198 190)(113 448 365 389)(114 435 366 390)(115 436 367 391)(116 437 368 392)(117 438 369 379)(118 439 370 380)(119 440 371 381)(120 441 372 382)(121 442 373 383)(122 443 374 384)(123 444 375 385)(124 445 376 386)(125 446 377 387)(126 447 378 388)(127 160 267 313)(128 161 268 314)(129 162 269 315)(130 163 270 316)(131 164 271 317)(132 165 272 318)(133 166 273 319)(134 167 274 320)(135 168 275 321)(136 155 276 322)(137 156 277 309)(138 157 278 310)(139 158 279 311)(140 159 280 312)(141 266 403 285)(142 253 404 286)(143 254 405 287)(144 255 406 288)(145 256 393 289)(146 257 394 290)(147 258 395 291)(148 259 396 292)(149 260 397 293)(150 261 398 294)(151 262 399 281)(152 263 400 282)(153 264 401 283)(154 265 402 284)
(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)
(1 303 8 296)(2 302 9 295)(3 301 10 308)(4 300 11 307)(5 299 12 306)(6 298 13 305)(7 297 14 304)(15 228 22 235)(16 227 23 234)(17 226 24 233)(18 225 25 232)(19 238 26 231)(20 237 27 230)(21 236 28 229)(29 177 36 170)(30 176 37 169)(31 175 38 182)(32 174 39 181)(33 173 40 180)(34 172 41 179)(35 171 42 178)(43 156 50 163)(44 155 51 162)(45 168 52 161)(46 167 53 160)(47 166 54 159)(48 165 55 158)(49 164 56 157)(57 133 64 140)(58 132 65 139)(59 131 66 138)(60 130 67 137)(61 129 68 136)(62 128 69 135)(63 127 70 134)(71 97 78 90)(72 96 79 89)(73 95 80 88)(74 94 81 87)(75 93 82 86)(76 92 83 85)(77 91 84 98)(99 273 106 280)(100 272 107 279)(101 271 108 278)(102 270 109 277)(103 269 110 276)(104 268 111 275)(105 267 112 274)(113 404 120 397)(114 403 121 396)(115 402 122 395)(116 401 123 394)(117 400 124 393)(118 399 125 406)(119 398 126 405)(141 373 148 366)(142 372 149 365)(143 371 150 378)(144 370 151 377)(145 369 152 376)(146 368 153 375)(147 367 154 374)(183 419 190 412)(184 418 191 411)(185 417 192 410)(186 416 193 409)(187 415 194 408)(188 414 195 407)(189 413 196 420)(197 421 204 428)(198 434 205 427)(199 433 206 426)(200 432 207 425)(201 431 208 424)(202 430 209 423)(203 429 210 422)(211 320 218 313)(212 319 219 312)(213 318 220 311)(214 317 221 310)(215 316 222 309)(216 315 223 322)(217 314 224 321)(239 354 246 361)(240 353 247 360)(241 352 248 359)(242 351 249 358)(243 364 250 357)(244 363 251 356)(245 362 252 355)(253 382 260 389)(254 381 261 388)(255 380 262 387)(256 379 263 386)(257 392 264 385)(258 391 265 384)(259 390 266 383)(281 446 288 439)(282 445 289 438)(283 444 290 437)(284 443 291 436)(285 442 292 435)(286 441 293 448)(287 440 294 447)(323 348 330 341)(324 347 331 340)(325 346 332 339)(326 345 333 338)(327 344 334 337)(328 343 335 350)(329 342 336 349)
G:=sub<Sym(448)| (1,224,143,77)(2,211,144,78)(3,212,145,79)(4,213,146,80)(5,214,147,81)(6,215,148,82)(7,216,149,83)(8,217,150,84)(9,218,151,71)(10,219,152,72)(11,220,153,73)(12,221,154,74)(13,222,141,75)(14,223,142,76)(15,168,119,205)(16,155,120,206)(17,156,121,207)(18,157,122,208)(19,158,123,209)(20,159,124,210)(21,160,125,197)(22,161,126,198)(23,162,113,199)(24,163,114,200)(25,164,115,201)(26,165,116,202)(27,166,117,203)(28,167,118,204)(29,107,283,408)(30,108,284,409)(31,109,285,410)(32,110,286,411)(33,111,287,412)(34,112,288,413)(35,99,289,414)(36,100,290,415)(37,101,291,416)(38,102,292,417)(39,103,293,418)(40,104,294,419)(41,105,281,420)(42,106,282,407)(43,403,432,233)(44,404,433,234)(45,405,434,235)(46,406,421,236)(47,393,422,237)(48,394,423,238)(49,395,424,225)(50,396,425,226)(51,397,426,227)(52,398,427,228)(53,399,428,229)(54,400,429,230)(55,401,430,231)(56,402,431,232)(57,256,347,250)(58,257,348,251)(59,258,349,252)(60,259,350,239)(61,260,337,240)(62,261,338,241)(63,262,339,242)(64,263,340,243)(65,264,341,244)(66,265,342,245)(67,266,343,246)(68,253,344,247)(69,254,345,248)(70,255,346,249)(85,297,315,365)(86,298,316,366)(87,299,317,367)(88,300,318,368)(89,301,319,369)(90,302,320,370)(91,303,321,371)(92,304,322,372)(93,305,309,373)(94,306,310,374)(95,307,311,375)(96,308,312,376)(97,295,313,377)(98,296,314,378)(127,387,325,351)(128,388,326,352)(129,389,327,353)(130,390,328,354)(131,391,329,355)(132,392,330,356)(133,379,331,357)(134,380,332,358)(135,381,333,359)(136,382,334,360)(137,383,335,361)(138,384,336,362)(139,385,323,363)(140,386,324,364)(169,271,436,193)(170,272,437,194)(171,273,438,195)(172,274,439,196)(173,275,440,183)(174,276,441,184)(175,277,442,185)(176,278,443,186)(177,279,444,187)(178,280,445,188)(179,267,446,189)(180,268,447,190)(181,269,448,191)(182,270,435,192), (1,248,235,33)(2,249,236,34)(3,250,237,35)(4,251,238,36)(5,252,225,37)(6,239,226,38)(7,240,227,39)(8,241,228,40)(9,242,229,41)(10,243,230,42)(11,244,231,29)(12,245,232,30)(13,246,233,31)(14,247,234,32)(15,173,303,359)(16,174,304,360)(17,175,305,361)(18,176,306,362)(19,177,307,363)(20,178,308,364)(21,179,295,351)(22,180,296,352)(23,181,297,353)(24,182,298,354)(25,169,299,355)(26,170,300,356)(27,171,301,357)(28,172,302,358)(43,109,222,67)(44,110,223,68)(45,111,224,69)(46,112,211,70)(47,99,212,57)(48,100,213,58)(49,101,214,59)(50,102,215,60)(51,103,216,61)(52,104,217,62)(53,105,218,63)(54,106,219,64)(55,107,220,65)(56,108,221,66)(71,339,428,420)(72,340,429,407)(73,341,430,408)(74,342,431,409)(75,343,432,410)(76,344,433,411)(77,345,434,412)(78,346,421,413)(79,347,422,414)(80,348,423,415)(81,349,424,416)(82,350,425,417)(83,337,426,418)(84,338,427,419)(85,327,199,191)(86,328,200,192)(87,329,201,193)(88,330,202,194)(89,331,203,195)(90,332,204,196)(91,333,205,183)(92,334,206,184)(93,335,207,185)(94,336,208,186)(95,323,209,187)(96,324,210,188)(97,325,197,189)(98,326,198,190)(113,448,365,389)(114,435,366,390)(115,436,367,391)(116,437,368,392)(117,438,369,379)(118,439,370,380)(119,440,371,381)(120,441,372,382)(121,442,373,383)(122,443,374,384)(123,444,375,385)(124,445,376,386)(125,446,377,387)(126,447,378,388)(127,160,267,313)(128,161,268,314)(129,162,269,315)(130,163,270,316)(131,164,271,317)(132,165,272,318)(133,166,273,319)(134,167,274,320)(135,168,275,321)(136,155,276,322)(137,156,277,309)(138,157,278,310)(139,158,279,311)(140,159,280,312)(141,266,403,285)(142,253,404,286)(143,254,405,287)(144,255,406,288)(145,256,393,289)(146,257,394,290)(147,258,395,291)(148,259,396,292)(149,260,397,293)(150,261,398,294)(151,262,399,281)(152,263,400,282)(153,264,401,283)(154,265,402,284), (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), (1,303,8,296)(2,302,9,295)(3,301,10,308)(4,300,11,307)(5,299,12,306)(6,298,13,305)(7,297,14,304)(15,228,22,235)(16,227,23,234)(17,226,24,233)(18,225,25,232)(19,238,26,231)(20,237,27,230)(21,236,28,229)(29,177,36,170)(30,176,37,169)(31,175,38,182)(32,174,39,181)(33,173,40,180)(34,172,41,179)(35,171,42,178)(43,156,50,163)(44,155,51,162)(45,168,52,161)(46,167,53,160)(47,166,54,159)(48,165,55,158)(49,164,56,157)(57,133,64,140)(58,132,65,139)(59,131,66,138)(60,130,67,137)(61,129,68,136)(62,128,69,135)(63,127,70,134)(71,97,78,90)(72,96,79,89)(73,95,80,88)(74,94,81,87)(75,93,82,86)(76,92,83,85)(77,91,84,98)(99,273,106,280)(100,272,107,279)(101,271,108,278)(102,270,109,277)(103,269,110,276)(104,268,111,275)(105,267,112,274)(113,404,120,397)(114,403,121,396)(115,402,122,395)(116,401,123,394)(117,400,124,393)(118,399,125,406)(119,398,126,405)(141,373,148,366)(142,372,149,365)(143,371,150,378)(144,370,151,377)(145,369,152,376)(146,368,153,375)(147,367,154,374)(183,419,190,412)(184,418,191,411)(185,417,192,410)(186,416,193,409)(187,415,194,408)(188,414,195,407)(189,413,196,420)(197,421,204,428)(198,434,205,427)(199,433,206,426)(200,432,207,425)(201,431,208,424)(202,430,209,423)(203,429,210,422)(211,320,218,313)(212,319,219,312)(213,318,220,311)(214,317,221,310)(215,316,222,309)(216,315,223,322)(217,314,224,321)(239,354,246,361)(240,353,247,360)(241,352,248,359)(242,351,249,358)(243,364,250,357)(244,363,251,356)(245,362,252,355)(253,382,260,389)(254,381,261,388)(255,380,262,387)(256,379,263,386)(257,392,264,385)(258,391,265,384)(259,390,266,383)(281,446,288,439)(282,445,289,438)(283,444,290,437)(284,443,291,436)(285,442,292,435)(286,441,293,448)(287,440,294,447)(323,348,330,341)(324,347,331,340)(325,346,332,339)(326,345,333,338)(327,344,334,337)(328,343,335,350)(329,342,336,349)>;
G:=Group( (1,224,143,77)(2,211,144,78)(3,212,145,79)(4,213,146,80)(5,214,147,81)(6,215,148,82)(7,216,149,83)(8,217,150,84)(9,218,151,71)(10,219,152,72)(11,220,153,73)(12,221,154,74)(13,222,141,75)(14,223,142,76)(15,168,119,205)(16,155,120,206)(17,156,121,207)(18,157,122,208)(19,158,123,209)(20,159,124,210)(21,160,125,197)(22,161,126,198)(23,162,113,199)(24,163,114,200)(25,164,115,201)(26,165,116,202)(27,166,117,203)(28,167,118,204)(29,107,283,408)(30,108,284,409)(31,109,285,410)(32,110,286,411)(33,111,287,412)(34,112,288,413)(35,99,289,414)(36,100,290,415)(37,101,291,416)(38,102,292,417)(39,103,293,418)(40,104,294,419)(41,105,281,420)(42,106,282,407)(43,403,432,233)(44,404,433,234)(45,405,434,235)(46,406,421,236)(47,393,422,237)(48,394,423,238)(49,395,424,225)(50,396,425,226)(51,397,426,227)(52,398,427,228)(53,399,428,229)(54,400,429,230)(55,401,430,231)(56,402,431,232)(57,256,347,250)(58,257,348,251)(59,258,349,252)(60,259,350,239)(61,260,337,240)(62,261,338,241)(63,262,339,242)(64,263,340,243)(65,264,341,244)(66,265,342,245)(67,266,343,246)(68,253,344,247)(69,254,345,248)(70,255,346,249)(85,297,315,365)(86,298,316,366)(87,299,317,367)(88,300,318,368)(89,301,319,369)(90,302,320,370)(91,303,321,371)(92,304,322,372)(93,305,309,373)(94,306,310,374)(95,307,311,375)(96,308,312,376)(97,295,313,377)(98,296,314,378)(127,387,325,351)(128,388,326,352)(129,389,327,353)(130,390,328,354)(131,391,329,355)(132,392,330,356)(133,379,331,357)(134,380,332,358)(135,381,333,359)(136,382,334,360)(137,383,335,361)(138,384,336,362)(139,385,323,363)(140,386,324,364)(169,271,436,193)(170,272,437,194)(171,273,438,195)(172,274,439,196)(173,275,440,183)(174,276,441,184)(175,277,442,185)(176,278,443,186)(177,279,444,187)(178,280,445,188)(179,267,446,189)(180,268,447,190)(181,269,448,191)(182,270,435,192), (1,248,235,33)(2,249,236,34)(3,250,237,35)(4,251,238,36)(5,252,225,37)(6,239,226,38)(7,240,227,39)(8,241,228,40)(9,242,229,41)(10,243,230,42)(11,244,231,29)(12,245,232,30)(13,246,233,31)(14,247,234,32)(15,173,303,359)(16,174,304,360)(17,175,305,361)(18,176,306,362)(19,177,307,363)(20,178,308,364)(21,179,295,351)(22,180,296,352)(23,181,297,353)(24,182,298,354)(25,169,299,355)(26,170,300,356)(27,171,301,357)(28,172,302,358)(43,109,222,67)(44,110,223,68)(45,111,224,69)(46,112,211,70)(47,99,212,57)(48,100,213,58)(49,101,214,59)(50,102,215,60)(51,103,216,61)(52,104,217,62)(53,105,218,63)(54,106,219,64)(55,107,220,65)(56,108,221,66)(71,339,428,420)(72,340,429,407)(73,341,430,408)(74,342,431,409)(75,343,432,410)(76,344,433,411)(77,345,434,412)(78,346,421,413)(79,347,422,414)(80,348,423,415)(81,349,424,416)(82,350,425,417)(83,337,426,418)(84,338,427,419)(85,327,199,191)(86,328,200,192)(87,329,201,193)(88,330,202,194)(89,331,203,195)(90,332,204,196)(91,333,205,183)(92,334,206,184)(93,335,207,185)(94,336,208,186)(95,323,209,187)(96,324,210,188)(97,325,197,189)(98,326,198,190)(113,448,365,389)(114,435,366,390)(115,436,367,391)(116,437,368,392)(117,438,369,379)(118,439,370,380)(119,440,371,381)(120,441,372,382)(121,442,373,383)(122,443,374,384)(123,444,375,385)(124,445,376,386)(125,446,377,387)(126,447,378,388)(127,160,267,313)(128,161,268,314)(129,162,269,315)(130,163,270,316)(131,164,271,317)(132,165,272,318)(133,166,273,319)(134,167,274,320)(135,168,275,321)(136,155,276,322)(137,156,277,309)(138,157,278,310)(139,158,279,311)(140,159,280,312)(141,266,403,285)(142,253,404,286)(143,254,405,287)(144,255,406,288)(145,256,393,289)(146,257,394,290)(147,258,395,291)(148,259,396,292)(149,260,397,293)(150,261,398,294)(151,262,399,281)(152,263,400,282)(153,264,401,283)(154,265,402,284), (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), (1,303,8,296)(2,302,9,295)(3,301,10,308)(4,300,11,307)(5,299,12,306)(6,298,13,305)(7,297,14,304)(15,228,22,235)(16,227,23,234)(17,226,24,233)(18,225,25,232)(19,238,26,231)(20,237,27,230)(21,236,28,229)(29,177,36,170)(30,176,37,169)(31,175,38,182)(32,174,39,181)(33,173,40,180)(34,172,41,179)(35,171,42,178)(43,156,50,163)(44,155,51,162)(45,168,52,161)(46,167,53,160)(47,166,54,159)(48,165,55,158)(49,164,56,157)(57,133,64,140)(58,132,65,139)(59,131,66,138)(60,130,67,137)(61,129,68,136)(62,128,69,135)(63,127,70,134)(71,97,78,90)(72,96,79,89)(73,95,80,88)(74,94,81,87)(75,93,82,86)(76,92,83,85)(77,91,84,98)(99,273,106,280)(100,272,107,279)(101,271,108,278)(102,270,109,277)(103,269,110,276)(104,268,111,275)(105,267,112,274)(113,404,120,397)(114,403,121,396)(115,402,122,395)(116,401,123,394)(117,400,124,393)(118,399,125,406)(119,398,126,405)(141,373,148,366)(142,372,149,365)(143,371,150,378)(144,370,151,377)(145,369,152,376)(146,368,153,375)(147,367,154,374)(183,419,190,412)(184,418,191,411)(185,417,192,410)(186,416,193,409)(187,415,194,408)(188,414,195,407)(189,413,196,420)(197,421,204,428)(198,434,205,427)(199,433,206,426)(200,432,207,425)(201,431,208,424)(202,430,209,423)(203,429,210,422)(211,320,218,313)(212,319,219,312)(213,318,220,311)(214,317,221,310)(215,316,222,309)(216,315,223,322)(217,314,224,321)(239,354,246,361)(240,353,247,360)(241,352,248,359)(242,351,249,358)(243,364,250,357)(244,363,251,356)(245,362,252,355)(253,382,260,389)(254,381,261,388)(255,380,262,387)(256,379,263,386)(257,392,264,385)(258,391,265,384)(259,390,266,383)(281,446,288,439)(282,445,289,438)(283,444,290,437)(284,443,291,436)(285,442,292,435)(286,441,293,448)(287,440,294,447)(323,348,330,341)(324,347,331,340)(325,346,332,339)(326,345,333,338)(327,344,334,337)(328,343,335,350)(329,342,336,349) );
G=PermutationGroup([[(1,224,143,77),(2,211,144,78),(3,212,145,79),(4,213,146,80),(5,214,147,81),(6,215,148,82),(7,216,149,83),(8,217,150,84),(9,218,151,71),(10,219,152,72),(11,220,153,73),(12,221,154,74),(13,222,141,75),(14,223,142,76),(15,168,119,205),(16,155,120,206),(17,156,121,207),(18,157,122,208),(19,158,123,209),(20,159,124,210),(21,160,125,197),(22,161,126,198),(23,162,113,199),(24,163,114,200),(25,164,115,201),(26,165,116,202),(27,166,117,203),(28,167,118,204),(29,107,283,408),(30,108,284,409),(31,109,285,410),(32,110,286,411),(33,111,287,412),(34,112,288,413),(35,99,289,414),(36,100,290,415),(37,101,291,416),(38,102,292,417),(39,103,293,418),(40,104,294,419),(41,105,281,420),(42,106,282,407),(43,403,432,233),(44,404,433,234),(45,405,434,235),(46,406,421,236),(47,393,422,237),(48,394,423,238),(49,395,424,225),(50,396,425,226),(51,397,426,227),(52,398,427,228),(53,399,428,229),(54,400,429,230),(55,401,430,231),(56,402,431,232),(57,256,347,250),(58,257,348,251),(59,258,349,252),(60,259,350,239),(61,260,337,240),(62,261,338,241),(63,262,339,242),(64,263,340,243),(65,264,341,244),(66,265,342,245),(67,266,343,246),(68,253,344,247),(69,254,345,248),(70,255,346,249),(85,297,315,365),(86,298,316,366),(87,299,317,367),(88,300,318,368),(89,301,319,369),(90,302,320,370),(91,303,321,371),(92,304,322,372),(93,305,309,373),(94,306,310,374),(95,307,311,375),(96,308,312,376),(97,295,313,377),(98,296,314,378),(127,387,325,351),(128,388,326,352),(129,389,327,353),(130,390,328,354),(131,391,329,355),(132,392,330,356),(133,379,331,357),(134,380,332,358),(135,381,333,359),(136,382,334,360),(137,383,335,361),(138,384,336,362),(139,385,323,363),(140,386,324,364),(169,271,436,193),(170,272,437,194),(171,273,438,195),(172,274,439,196),(173,275,440,183),(174,276,441,184),(175,277,442,185),(176,278,443,186),(177,279,444,187),(178,280,445,188),(179,267,446,189),(180,268,447,190),(181,269,448,191),(182,270,435,192)], [(1,248,235,33),(2,249,236,34),(3,250,237,35),(4,251,238,36),(5,252,225,37),(6,239,226,38),(7,240,227,39),(8,241,228,40),(9,242,229,41),(10,243,230,42),(11,244,231,29),(12,245,232,30),(13,246,233,31),(14,247,234,32),(15,173,303,359),(16,174,304,360),(17,175,305,361),(18,176,306,362),(19,177,307,363),(20,178,308,364),(21,179,295,351),(22,180,296,352),(23,181,297,353),(24,182,298,354),(25,169,299,355),(26,170,300,356),(27,171,301,357),(28,172,302,358),(43,109,222,67),(44,110,223,68),(45,111,224,69),(46,112,211,70),(47,99,212,57),(48,100,213,58),(49,101,214,59),(50,102,215,60),(51,103,216,61),(52,104,217,62),(53,105,218,63),(54,106,219,64),(55,107,220,65),(56,108,221,66),(71,339,428,420),(72,340,429,407),(73,341,430,408),(74,342,431,409),(75,343,432,410),(76,344,433,411),(77,345,434,412),(78,346,421,413),(79,347,422,414),(80,348,423,415),(81,349,424,416),(82,350,425,417),(83,337,426,418),(84,338,427,419),(85,327,199,191),(86,328,200,192),(87,329,201,193),(88,330,202,194),(89,331,203,195),(90,332,204,196),(91,333,205,183),(92,334,206,184),(93,335,207,185),(94,336,208,186),(95,323,209,187),(96,324,210,188),(97,325,197,189),(98,326,198,190),(113,448,365,389),(114,435,366,390),(115,436,367,391),(116,437,368,392),(117,438,369,379),(118,439,370,380),(119,440,371,381),(120,441,372,382),(121,442,373,383),(122,443,374,384),(123,444,375,385),(124,445,376,386),(125,446,377,387),(126,447,378,388),(127,160,267,313),(128,161,268,314),(129,162,269,315),(130,163,270,316),(131,164,271,317),(132,165,272,318),(133,166,273,319),(134,167,274,320),(135,168,275,321),(136,155,276,322),(137,156,277,309),(138,157,278,310),(139,158,279,311),(140,159,280,312),(141,266,403,285),(142,253,404,286),(143,254,405,287),(144,255,406,288),(145,256,393,289),(146,257,394,290),(147,258,395,291),(148,259,396,292),(149,260,397,293),(150,261,398,294),(151,262,399,281),(152,263,400,282),(153,264,401,283),(154,265,402,284)], [(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)], [(1,303,8,296),(2,302,9,295),(3,301,10,308),(4,300,11,307),(5,299,12,306),(6,298,13,305),(7,297,14,304),(15,228,22,235),(16,227,23,234),(17,226,24,233),(18,225,25,232),(19,238,26,231),(20,237,27,230),(21,236,28,229),(29,177,36,170),(30,176,37,169),(31,175,38,182),(32,174,39,181),(33,173,40,180),(34,172,41,179),(35,171,42,178),(43,156,50,163),(44,155,51,162),(45,168,52,161),(46,167,53,160),(47,166,54,159),(48,165,55,158),(49,164,56,157),(57,133,64,140),(58,132,65,139),(59,131,66,138),(60,130,67,137),(61,129,68,136),(62,128,69,135),(63,127,70,134),(71,97,78,90),(72,96,79,89),(73,95,80,88),(74,94,81,87),(75,93,82,86),(76,92,83,85),(77,91,84,98),(99,273,106,280),(100,272,107,279),(101,271,108,278),(102,270,109,277),(103,269,110,276),(104,268,111,275),(105,267,112,274),(113,404,120,397),(114,403,121,396),(115,402,122,395),(116,401,123,394),(117,400,124,393),(118,399,125,406),(119,398,126,405),(141,373,148,366),(142,372,149,365),(143,371,150,378),(144,370,151,377),(145,369,152,376),(146,368,153,375),(147,367,154,374),(183,419,190,412),(184,418,191,411),(185,417,192,410),(186,416,193,409),(187,415,194,408),(188,414,195,407),(189,413,196,420),(197,421,204,428),(198,434,205,427),(199,433,206,426),(200,432,207,425),(201,431,208,424),(202,430,209,423),(203,429,210,422),(211,320,218,313),(212,319,219,312),(213,318,220,311),(214,317,221,310),(215,316,222,309),(216,315,223,322),(217,314,224,321),(239,354,246,361),(240,353,247,360),(241,352,248,359),(242,351,249,358),(243,364,250,357),(244,363,251,356),(245,362,252,355),(253,382,260,389),(254,381,261,388),(255,380,262,387),(256,379,263,386),(257,392,264,385),(258,391,265,384),(259,390,266,383),(281,446,288,439),(282,445,289,438),(283,444,290,437),(284,443,291,436),(285,442,292,435),(286,441,293,448),(287,440,294,447),(323,348,330,341),(324,347,331,340),(325,346,332,339),(326,345,333,338),(327,344,334,337),(328,343,335,350),(329,342,336,349)]])
160 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 4Y | ··· | 4BD | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | C4 | D7 | Dic7 | D14 | C4×D7 |
| kernel | C42×Dic7 | C2×C4×Dic7 | C2×C4×C28 | C4×Dic7 | C4×C28 | C2×C42 | C42 | C22×C4 | C2×C4 |
| # reps | 1 | 6 | 1 | 48 | 8 | 3 | 12 | 9 | 72 |
Matrix representation of C42×Dic7 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 1 |
| 0 | 0 | 24 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 18 | 8 |
| 0 | 0 | 14 | 11 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,17,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,17,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,28,0,0,0,0,28,24,0,0,1,4],[1,0,0,0,0,17,0,0,0,0,18,14,0,0,8,11] >;
C42×Dic7 in GAP, Magma, Sage, TeX
C_4^2\times {\rm Dic}_7 % in TeX
G:=Group("C4^2xDic7"); // GroupNames label
G:=SmallGroup(448,464);
// by ID
G=gap.SmallGroup(448,464);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,100,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=c^7,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations