G = C7⋊(C42⋊8C4) order 448 = 26·7
The semidirect product of C7 and C42⋊8C4 acting via C42⋊8C4/C2.C42=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊1(C42⋊8C4), (C4×Dic7)⋊12C4, C22.53(D4×D7), (C22×C4).7D14, C22.12(Q8×D7), Dic7.1(C4⋊C4), (C2×Dic7).83D4, (C2×Dic7).10Q8, C2.5(C42⋊D7), C2.2(Dic7.Q8), (C22×C28).3C22, C14.6(C42.C2), C2.C42.1D7, C14.1(C42⋊C2), C14.15(C4.4D4), C22.27(C4○D28), C14.C42.1C2, C23.247(C22×D7), C22.29(D4⋊2D7), (C22×C14).275C23, C2.2(Dic7.D4), (C22×Dic7).3C22, C2.5(C23.11D14), C2.6(D7×C4⋊C4), C14.3(C2×C4⋊C4), C22.82(C2×C4×D7), (C2×C4).122(C4×D7), (C2×C14).59(C2×Q8), (C2×C4×Dic7).23C2, (C2×C28).139(C2×C4), (C2×C14).190(C2×D4), (C2×Dic7⋊C4).3C2, (C2×C14).54(C4○D4), (C2×C14).41(C22×C4), (C2×Dic7).75(C2×C4), (C7×C2.C42).21C2, SmallGroup(448,184)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊(C42⋊8C4)
G = < a,b,c,d | a7=b4=c4=d4=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >
Subgroups: 604 in 154 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic7, Dic7, C28, C2×C14, C2×C14, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C42⋊8C4, C4×Dic7, Dic7⋊C4, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C14.C42, C14.C42, C7×C2.C42, C2×C4×Dic7, C2×Dic7⋊C4, C7⋊(C42⋊8C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D14, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C4×D7, C22×D7, C42⋊8C4, C2×C4×D7, C4○D28, D4×D7, D4⋊2D7, Q8×D7, C42⋊D7, C23.11D14, Dic7.D4, Dic7.Q8, D7×C4⋊C4, C7⋊(C42⋊8C4)
►Regular action on 448 points
Generators in S448
(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)
(1 260 13 253)(2 266 14 259)(3 265 8 258)(4 264 9 257)(5 263 10 256)(6 262 11 255)(7 261 12 254)(15 279 22 272)(16 278 23 271)(17 277 24 270)(18 276 25 269)(19 275 26 268)(20 274 27 267)(21 280 28 273)(29 237 36 230)(30 236 37 229)(31 235 38 228)(32 234 39 227)(33 233 40 226)(34 232 41 225)(35 238 42 231)(43 251 50 244)(44 250 51 243)(45 249 52 242)(46 248 53 241)(47 247 54 240)(48 246 55 239)(49 252 56 245)(57 321 64 314)(58 320 65 313)(59 319 66 312)(60 318 67 311)(61 317 68 310)(62 316 69 309)(63 322 70 315)(71 335 78 328)(72 334 79 327)(73 333 80 326)(74 332 81 325)(75 331 82 324)(76 330 83 323)(77 336 84 329)(85 293 92 286)(86 292 93 285)(87 291 94 284)(88 290 95 283)(89 289 96 282)(90 288 97 281)(91 294 98 287)(99 307 106 300)(100 306 107 299)(101 305 108 298)(102 304 109 297)(103 303 110 296)(104 302 111 295)(105 308 112 301)(113 377 120 370)(114 376 121 369)(115 375 122 368)(116 374 123 367)(117 373 124 366)(118 372 125 365)(119 378 126 371)(127 391 134 384)(128 390 135 383)(129 389 136 382)(130 388 137 381)(131 387 138 380)(132 386 139 379)(133 392 140 385)(141 349 148 342)(142 348 149 341)(143 347 150 340)(144 346 151 339)(145 345 152 338)(146 344 153 337)(147 350 154 343)(155 363 162 356)(156 362 163 355)(157 361 164 354)(158 360 165 353)(159 359 166 352)(160 358 167 351)(161 364 168 357)(169 433 176 426)(170 432 177 425)(171 431 178 424)(172 430 179 423)(173 429 180 422)(174 428 181 421)(175 434 182 427)(183 447 190 440)(184 446 191 439)(185 445 192 438)(186 444 193 437)(187 443 194 436)(188 442 195 435)(189 448 196 441)(197 405 204 398)(198 404 205 397)(199 403 206 396)(200 402 207 395)(201 401 208 394)(202 400 209 393)(203 406 210 399)(211 419 218 412)(212 418 219 411)(213 417 220 410)(214 416 221 409)(215 415 222 408)(216 414 223 407)(217 420 224 413)
(1 160 20 146)(2 161 21 147)(3 155 15 141)(4 156 16 142)(5 157 17 143)(6 158 18 144)(7 159 19 145)(8 162 22 148)(9 163 23 149)(10 164 24 150)(11 165 25 151)(12 166 26 152)(13 167 27 153)(14 168 28 154)(29 127 43 113)(30 128 44 114)(31 129 45 115)(32 130 46 116)(33 131 47 117)(34 132 48 118)(35 133 49 119)(36 134 50 120)(37 135 51 121)(38 136 52 122)(39 137 53 123)(40 138 54 124)(41 139 55 125)(42 140 56 126)(57 211 71 197)(58 212 72 198)(59 213 73 199)(60 214 74 200)(61 215 75 201)(62 216 76 202)(63 217 77 203)(64 218 78 204)(65 219 79 205)(66 220 80 206)(67 221 81 207)(68 222 82 208)(69 223 83 209)(70 224 84 210)(85 183 99 169)(86 184 100 170)(87 185 101 171)(88 186 102 172)(89 187 103 173)(90 188 104 174)(91 189 105 175)(92 190 106 176)(93 191 107 177)(94 192 108 178)(95 193 109 179)(96 194 110 180)(97 195 111 181)(98 196 112 182)(225 379 239 365)(226 380 240 366)(227 381 241 367)(228 382 242 368)(229 383 243 369)(230 384 244 370)(231 385 245 371)(232 386 246 372)(233 387 247 373)(234 388 248 374)(235 389 249 375)(236 390 250 376)(237 391 251 377)(238 392 252 378)(253 351 267 337)(254 352 268 338)(255 353 269 339)(256 354 270 340)(257 355 271 341)(258 356 272 342)(259 357 273 343)(260 358 274 344)(261 359 275 345)(262 360 276 346)(263 361 277 347)(264 362 278 348)(265 363 279 349)(266 364 280 350)(281 435 295 421)(282 436 296 422)(283 437 297 423)(284 438 298 424)(285 439 299 425)(286 440 300 426)(287 441 301 427)(288 442 302 428)(289 443 303 429)(290 444 304 430)(291 445 305 431)(292 446 306 432)(293 447 307 433)(294 448 308 434)(309 407 323 393)(310 408 324 394)(311 409 325 395)(312 410 326 396)(313 411 327 397)(314 412 328 398)(315 413 329 399)(316 414 330 400)(317 415 331 401)(318 416 332 402)(319 417 333 403)(320 418 334 404)(321 419 335 405)(322 420 336 406)
(1 90 34 62)(2 91 35 63)(3 85 29 57)(4 86 30 58)(5 87 31 59)(6 88 32 60)(7 89 33 61)(8 92 36 64)(9 93 37 65)(10 94 38 66)(11 95 39 67)(12 96 40 68)(13 97 41 69)(14 98 42 70)(15 99 43 71)(16 100 44 72)(17 101 45 73)(18 102 46 74)(19 103 47 75)(20 104 48 76)(21 105 49 77)(22 106 50 78)(23 107 51 79)(24 108 52 80)(25 109 53 81)(26 110 54 82)(27 111 55 83)(28 112 56 84)(113 204 141 176)(114 205 142 177)(115 206 143 178)(116 207 144 179)(117 208 145 180)(118 209 146 181)(119 210 147 182)(120 197 148 169)(121 198 149 170)(122 199 150 171)(123 200 151 172)(124 201 152 173)(125 202 153 174)(126 203 154 175)(127 218 155 190)(128 219 156 191)(129 220 157 192)(130 221 158 193)(131 222 159 194)(132 223 160 195)(133 224 161 196)(134 211 162 183)(135 212 163 184)(136 213 164 185)(137 214 165 186)(138 215 166 187)(139 216 167 188)(140 217 168 189)(225 323 253 295)(226 324 254 296)(227 325 255 297)(228 326 256 298)(229 327 257 299)(230 328 258 300)(231 329 259 301)(232 330 260 302)(233 331 261 303)(234 332 262 304)(235 333 263 305)(236 334 264 306)(237 335 265 307)(238 336 266 308)(239 309 267 281)(240 310 268 282)(241 311 269 283)(242 312 270 284)(243 313 271 285)(244 314 272 286)(245 315 273 287)(246 316 274 288)(247 317 275 289)(248 318 276 290)(249 319 277 291)(250 320 278 292)(251 321 279 293)(252 322 280 294)(337 442 365 414)(338 443 366 415)(339 444 367 416)(340 445 368 417)(341 446 369 418)(342 447 370 419)(343 448 371 420)(344 435 372 407)(345 436 373 408)(346 437 374 409)(347 438 375 410)(348 439 376 411)(349 440 377 412)(350 441 378 413)(351 428 379 400)(352 429 380 401)(353 430 381 402)(354 431 382 403)(355 432 383 404)(356 433 384 405)(357 434 385 406)(358 421 386 393)(359 422 387 394)(360 423 388 395)(361 424 389 396)(362 425 390 397)(363 426 391 398)(364 427 392 399)
G:=sub<Sym(448)| (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), (1,260,13,253)(2,266,14,259)(3,265,8,258)(4,264,9,257)(5,263,10,256)(6,262,11,255)(7,261,12,254)(15,279,22,272)(16,278,23,271)(17,277,24,270)(18,276,25,269)(19,275,26,268)(20,274,27,267)(21,280,28,273)(29,237,36,230)(30,236,37,229)(31,235,38,228)(32,234,39,227)(33,233,40,226)(34,232,41,225)(35,238,42,231)(43,251,50,244)(44,250,51,243)(45,249,52,242)(46,248,53,241)(47,247,54,240)(48,246,55,239)(49,252,56,245)(57,321,64,314)(58,320,65,313)(59,319,66,312)(60,318,67,311)(61,317,68,310)(62,316,69,309)(63,322,70,315)(71,335,78,328)(72,334,79,327)(73,333,80,326)(74,332,81,325)(75,331,82,324)(76,330,83,323)(77,336,84,329)(85,293,92,286)(86,292,93,285)(87,291,94,284)(88,290,95,283)(89,289,96,282)(90,288,97,281)(91,294,98,287)(99,307,106,300)(100,306,107,299)(101,305,108,298)(102,304,109,297)(103,303,110,296)(104,302,111,295)(105,308,112,301)(113,377,120,370)(114,376,121,369)(115,375,122,368)(116,374,123,367)(117,373,124,366)(118,372,125,365)(119,378,126,371)(127,391,134,384)(128,390,135,383)(129,389,136,382)(130,388,137,381)(131,387,138,380)(132,386,139,379)(133,392,140,385)(141,349,148,342)(142,348,149,341)(143,347,150,340)(144,346,151,339)(145,345,152,338)(146,344,153,337)(147,350,154,343)(155,363,162,356)(156,362,163,355)(157,361,164,354)(158,360,165,353)(159,359,166,352)(160,358,167,351)(161,364,168,357)(169,433,176,426)(170,432,177,425)(171,431,178,424)(172,430,179,423)(173,429,180,422)(174,428,181,421)(175,434,182,427)(183,447,190,440)(184,446,191,439)(185,445,192,438)(186,444,193,437)(187,443,194,436)(188,442,195,435)(189,448,196,441)(197,405,204,398)(198,404,205,397)(199,403,206,396)(200,402,207,395)(201,401,208,394)(202,400,209,393)(203,406,210,399)(211,419,218,412)(212,418,219,411)(213,417,220,410)(214,416,221,409)(215,415,222,408)(216,414,223,407)(217,420,224,413), (1,160,20,146)(2,161,21,147)(3,155,15,141)(4,156,16,142)(5,157,17,143)(6,158,18,144)(7,159,19,145)(8,162,22,148)(9,163,23,149)(10,164,24,150)(11,165,25,151)(12,166,26,152)(13,167,27,153)(14,168,28,154)(29,127,43,113)(30,128,44,114)(31,129,45,115)(32,130,46,116)(33,131,47,117)(34,132,48,118)(35,133,49,119)(36,134,50,120)(37,135,51,121)(38,136,52,122)(39,137,53,123)(40,138,54,124)(41,139,55,125)(42,140,56,126)(57,211,71,197)(58,212,72,198)(59,213,73,199)(60,214,74,200)(61,215,75,201)(62,216,76,202)(63,217,77,203)(64,218,78,204)(65,219,79,205)(66,220,80,206)(67,221,81,207)(68,222,82,208)(69,223,83,209)(70,224,84,210)(85,183,99,169)(86,184,100,170)(87,185,101,171)(88,186,102,172)(89,187,103,173)(90,188,104,174)(91,189,105,175)(92,190,106,176)(93,191,107,177)(94,192,108,178)(95,193,109,179)(96,194,110,180)(97,195,111,181)(98,196,112,182)(225,379,239,365)(226,380,240,366)(227,381,241,367)(228,382,242,368)(229,383,243,369)(230,384,244,370)(231,385,245,371)(232,386,246,372)(233,387,247,373)(234,388,248,374)(235,389,249,375)(236,390,250,376)(237,391,251,377)(238,392,252,378)(253,351,267,337)(254,352,268,338)(255,353,269,339)(256,354,270,340)(257,355,271,341)(258,356,272,342)(259,357,273,343)(260,358,274,344)(261,359,275,345)(262,360,276,346)(263,361,277,347)(264,362,278,348)(265,363,279,349)(266,364,280,350)(281,435,295,421)(282,436,296,422)(283,437,297,423)(284,438,298,424)(285,439,299,425)(286,440,300,426)(287,441,301,427)(288,442,302,428)(289,443,303,429)(290,444,304,430)(291,445,305,431)(292,446,306,432)(293,447,307,433)(294,448,308,434)(309,407,323,393)(310,408,324,394)(311,409,325,395)(312,410,326,396)(313,411,327,397)(314,412,328,398)(315,413,329,399)(316,414,330,400)(317,415,331,401)(318,416,332,402)(319,417,333,403)(320,418,334,404)(321,419,335,405)(322,420,336,406), (1,90,34,62)(2,91,35,63)(3,85,29,57)(4,86,30,58)(5,87,31,59)(6,88,32,60)(7,89,33,61)(8,92,36,64)(9,93,37,65)(10,94,38,66)(11,95,39,67)(12,96,40,68)(13,97,41,69)(14,98,42,70)(15,99,43,71)(16,100,44,72)(17,101,45,73)(18,102,46,74)(19,103,47,75)(20,104,48,76)(21,105,49,77)(22,106,50,78)(23,107,51,79)(24,108,52,80)(25,109,53,81)(26,110,54,82)(27,111,55,83)(28,112,56,84)(113,204,141,176)(114,205,142,177)(115,206,143,178)(116,207,144,179)(117,208,145,180)(118,209,146,181)(119,210,147,182)(120,197,148,169)(121,198,149,170)(122,199,150,171)(123,200,151,172)(124,201,152,173)(125,202,153,174)(126,203,154,175)(127,218,155,190)(128,219,156,191)(129,220,157,192)(130,221,158,193)(131,222,159,194)(132,223,160,195)(133,224,161,196)(134,211,162,183)(135,212,163,184)(136,213,164,185)(137,214,165,186)(138,215,166,187)(139,216,167,188)(140,217,168,189)(225,323,253,295)(226,324,254,296)(227,325,255,297)(228,326,256,298)(229,327,257,299)(230,328,258,300)(231,329,259,301)(232,330,260,302)(233,331,261,303)(234,332,262,304)(235,333,263,305)(236,334,264,306)(237,335,265,307)(238,336,266,308)(239,309,267,281)(240,310,268,282)(241,311,269,283)(242,312,270,284)(243,313,271,285)(244,314,272,286)(245,315,273,287)(246,316,274,288)(247,317,275,289)(248,318,276,290)(249,319,277,291)(250,320,278,292)(251,321,279,293)(252,322,280,294)(337,442,365,414)(338,443,366,415)(339,444,367,416)(340,445,368,417)(341,446,369,418)(342,447,370,419)(343,448,371,420)(344,435,372,407)(345,436,373,408)(346,437,374,409)(347,438,375,410)(348,439,376,411)(349,440,377,412)(350,441,378,413)(351,428,379,400)(352,429,380,401)(353,430,381,402)(354,431,382,403)(355,432,383,404)(356,433,384,405)(357,434,385,406)(358,421,386,393)(359,422,387,394)(360,423,388,395)(361,424,389,396)(362,425,390,397)(363,426,391,398)(364,427,392,399)>;
G:=Group( (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), (1,260,13,253)(2,266,14,259)(3,265,8,258)(4,264,9,257)(5,263,10,256)(6,262,11,255)(7,261,12,254)(15,279,22,272)(16,278,23,271)(17,277,24,270)(18,276,25,269)(19,275,26,268)(20,274,27,267)(21,280,28,273)(29,237,36,230)(30,236,37,229)(31,235,38,228)(32,234,39,227)(33,233,40,226)(34,232,41,225)(35,238,42,231)(43,251,50,244)(44,250,51,243)(45,249,52,242)(46,248,53,241)(47,247,54,240)(48,246,55,239)(49,252,56,245)(57,321,64,314)(58,320,65,313)(59,319,66,312)(60,318,67,311)(61,317,68,310)(62,316,69,309)(63,322,70,315)(71,335,78,328)(72,334,79,327)(73,333,80,326)(74,332,81,325)(75,331,82,324)(76,330,83,323)(77,336,84,329)(85,293,92,286)(86,292,93,285)(87,291,94,284)(88,290,95,283)(89,289,96,282)(90,288,97,281)(91,294,98,287)(99,307,106,300)(100,306,107,299)(101,305,108,298)(102,304,109,297)(103,303,110,296)(104,302,111,295)(105,308,112,301)(113,377,120,370)(114,376,121,369)(115,375,122,368)(116,374,123,367)(117,373,124,366)(118,372,125,365)(119,378,126,371)(127,391,134,384)(128,390,135,383)(129,389,136,382)(130,388,137,381)(131,387,138,380)(132,386,139,379)(133,392,140,385)(141,349,148,342)(142,348,149,341)(143,347,150,340)(144,346,151,339)(145,345,152,338)(146,344,153,337)(147,350,154,343)(155,363,162,356)(156,362,163,355)(157,361,164,354)(158,360,165,353)(159,359,166,352)(160,358,167,351)(161,364,168,357)(169,433,176,426)(170,432,177,425)(171,431,178,424)(172,430,179,423)(173,429,180,422)(174,428,181,421)(175,434,182,427)(183,447,190,440)(184,446,191,439)(185,445,192,438)(186,444,193,437)(187,443,194,436)(188,442,195,435)(189,448,196,441)(197,405,204,398)(198,404,205,397)(199,403,206,396)(200,402,207,395)(201,401,208,394)(202,400,209,393)(203,406,210,399)(211,419,218,412)(212,418,219,411)(213,417,220,410)(214,416,221,409)(215,415,222,408)(216,414,223,407)(217,420,224,413), (1,160,20,146)(2,161,21,147)(3,155,15,141)(4,156,16,142)(5,157,17,143)(6,158,18,144)(7,159,19,145)(8,162,22,148)(9,163,23,149)(10,164,24,150)(11,165,25,151)(12,166,26,152)(13,167,27,153)(14,168,28,154)(29,127,43,113)(30,128,44,114)(31,129,45,115)(32,130,46,116)(33,131,47,117)(34,132,48,118)(35,133,49,119)(36,134,50,120)(37,135,51,121)(38,136,52,122)(39,137,53,123)(40,138,54,124)(41,139,55,125)(42,140,56,126)(57,211,71,197)(58,212,72,198)(59,213,73,199)(60,214,74,200)(61,215,75,201)(62,216,76,202)(63,217,77,203)(64,218,78,204)(65,219,79,205)(66,220,80,206)(67,221,81,207)(68,222,82,208)(69,223,83,209)(70,224,84,210)(85,183,99,169)(86,184,100,170)(87,185,101,171)(88,186,102,172)(89,187,103,173)(90,188,104,174)(91,189,105,175)(92,190,106,176)(93,191,107,177)(94,192,108,178)(95,193,109,179)(96,194,110,180)(97,195,111,181)(98,196,112,182)(225,379,239,365)(226,380,240,366)(227,381,241,367)(228,382,242,368)(229,383,243,369)(230,384,244,370)(231,385,245,371)(232,386,246,372)(233,387,247,373)(234,388,248,374)(235,389,249,375)(236,390,250,376)(237,391,251,377)(238,392,252,378)(253,351,267,337)(254,352,268,338)(255,353,269,339)(256,354,270,340)(257,355,271,341)(258,356,272,342)(259,357,273,343)(260,358,274,344)(261,359,275,345)(262,360,276,346)(263,361,277,347)(264,362,278,348)(265,363,279,349)(266,364,280,350)(281,435,295,421)(282,436,296,422)(283,437,297,423)(284,438,298,424)(285,439,299,425)(286,440,300,426)(287,441,301,427)(288,442,302,428)(289,443,303,429)(290,444,304,430)(291,445,305,431)(292,446,306,432)(293,447,307,433)(294,448,308,434)(309,407,323,393)(310,408,324,394)(311,409,325,395)(312,410,326,396)(313,411,327,397)(314,412,328,398)(315,413,329,399)(316,414,330,400)(317,415,331,401)(318,416,332,402)(319,417,333,403)(320,418,334,404)(321,419,335,405)(322,420,336,406), (1,90,34,62)(2,91,35,63)(3,85,29,57)(4,86,30,58)(5,87,31,59)(6,88,32,60)(7,89,33,61)(8,92,36,64)(9,93,37,65)(10,94,38,66)(11,95,39,67)(12,96,40,68)(13,97,41,69)(14,98,42,70)(15,99,43,71)(16,100,44,72)(17,101,45,73)(18,102,46,74)(19,103,47,75)(20,104,48,76)(21,105,49,77)(22,106,50,78)(23,107,51,79)(24,108,52,80)(25,109,53,81)(26,110,54,82)(27,111,55,83)(28,112,56,84)(113,204,141,176)(114,205,142,177)(115,206,143,178)(116,207,144,179)(117,208,145,180)(118,209,146,181)(119,210,147,182)(120,197,148,169)(121,198,149,170)(122,199,150,171)(123,200,151,172)(124,201,152,173)(125,202,153,174)(126,203,154,175)(127,218,155,190)(128,219,156,191)(129,220,157,192)(130,221,158,193)(131,222,159,194)(132,223,160,195)(133,224,161,196)(134,211,162,183)(135,212,163,184)(136,213,164,185)(137,214,165,186)(138,215,166,187)(139,216,167,188)(140,217,168,189)(225,323,253,295)(226,324,254,296)(227,325,255,297)(228,326,256,298)(229,327,257,299)(230,328,258,300)(231,329,259,301)(232,330,260,302)(233,331,261,303)(234,332,262,304)(235,333,263,305)(236,334,264,306)(237,335,265,307)(238,336,266,308)(239,309,267,281)(240,310,268,282)(241,311,269,283)(242,312,270,284)(243,313,271,285)(244,314,272,286)(245,315,273,287)(246,316,274,288)(247,317,275,289)(248,318,276,290)(249,319,277,291)(250,320,278,292)(251,321,279,293)(252,322,280,294)(337,442,365,414)(338,443,366,415)(339,444,367,416)(340,445,368,417)(341,446,369,418)(342,447,370,419)(343,448,371,420)(344,435,372,407)(345,436,373,408)(346,437,374,409)(347,438,375,410)(348,439,376,411)(349,440,377,412)(350,441,378,413)(351,428,379,400)(352,429,380,401)(353,430,381,402)(354,431,382,403)(355,432,383,404)(356,433,384,405)(357,434,385,406)(358,421,386,393)(359,422,387,394)(360,423,388,395)(361,424,389,396)(362,425,390,397)(363,426,391,398)(364,427,392,399) );
G=PermutationGroup([[(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)], [(1,260,13,253),(2,266,14,259),(3,265,8,258),(4,264,9,257),(5,263,10,256),(6,262,11,255),(7,261,12,254),(15,279,22,272),(16,278,23,271),(17,277,24,270),(18,276,25,269),(19,275,26,268),(20,274,27,267),(21,280,28,273),(29,237,36,230),(30,236,37,229),(31,235,38,228),(32,234,39,227),(33,233,40,226),(34,232,41,225),(35,238,42,231),(43,251,50,244),(44,250,51,243),(45,249,52,242),(46,248,53,241),(47,247,54,240),(48,246,55,239),(49,252,56,245),(57,321,64,314),(58,320,65,313),(59,319,66,312),(60,318,67,311),(61,317,68,310),(62,316,69,309),(63,322,70,315),(71,335,78,328),(72,334,79,327),(73,333,80,326),(74,332,81,325),(75,331,82,324),(76,330,83,323),(77,336,84,329),(85,293,92,286),(86,292,93,285),(87,291,94,284),(88,290,95,283),(89,289,96,282),(90,288,97,281),(91,294,98,287),(99,307,106,300),(100,306,107,299),(101,305,108,298),(102,304,109,297),(103,303,110,296),(104,302,111,295),(105,308,112,301),(113,377,120,370),(114,376,121,369),(115,375,122,368),(116,374,123,367),(117,373,124,366),(118,372,125,365),(119,378,126,371),(127,391,134,384),(128,390,135,383),(129,389,136,382),(130,388,137,381),(131,387,138,380),(132,386,139,379),(133,392,140,385),(141,349,148,342),(142,348,149,341),(143,347,150,340),(144,346,151,339),(145,345,152,338),(146,344,153,337),(147,350,154,343),(155,363,162,356),(156,362,163,355),(157,361,164,354),(158,360,165,353),(159,359,166,352),(160,358,167,351),(161,364,168,357),(169,433,176,426),(170,432,177,425),(171,431,178,424),(172,430,179,423),(173,429,180,422),(174,428,181,421),(175,434,182,427),(183,447,190,440),(184,446,191,439),(185,445,192,438),(186,444,193,437),(187,443,194,436),(188,442,195,435),(189,448,196,441),(197,405,204,398),(198,404,205,397),(199,403,206,396),(200,402,207,395),(201,401,208,394),(202,400,209,393),(203,406,210,399),(211,419,218,412),(212,418,219,411),(213,417,220,410),(214,416,221,409),(215,415,222,408),(216,414,223,407),(217,420,224,413)], [(1,160,20,146),(2,161,21,147),(3,155,15,141),(4,156,16,142),(5,157,17,143),(6,158,18,144),(7,159,19,145),(8,162,22,148),(9,163,23,149),(10,164,24,150),(11,165,25,151),(12,166,26,152),(13,167,27,153),(14,168,28,154),(29,127,43,113),(30,128,44,114),(31,129,45,115),(32,130,46,116),(33,131,47,117),(34,132,48,118),(35,133,49,119),(36,134,50,120),(37,135,51,121),(38,136,52,122),(39,137,53,123),(40,138,54,124),(41,139,55,125),(42,140,56,126),(57,211,71,197),(58,212,72,198),(59,213,73,199),(60,214,74,200),(61,215,75,201),(62,216,76,202),(63,217,77,203),(64,218,78,204),(65,219,79,205),(66,220,80,206),(67,221,81,207),(68,222,82,208),(69,223,83,209),(70,224,84,210),(85,183,99,169),(86,184,100,170),(87,185,101,171),(88,186,102,172),(89,187,103,173),(90,188,104,174),(91,189,105,175),(92,190,106,176),(93,191,107,177),(94,192,108,178),(95,193,109,179),(96,194,110,180),(97,195,111,181),(98,196,112,182),(225,379,239,365),(226,380,240,366),(227,381,241,367),(228,382,242,368),(229,383,243,369),(230,384,244,370),(231,385,245,371),(232,386,246,372),(233,387,247,373),(234,388,248,374),(235,389,249,375),(236,390,250,376),(237,391,251,377),(238,392,252,378),(253,351,267,337),(254,352,268,338),(255,353,269,339),(256,354,270,340),(257,355,271,341),(258,356,272,342),(259,357,273,343),(260,358,274,344),(261,359,275,345),(262,360,276,346),(263,361,277,347),(264,362,278,348),(265,363,279,349),(266,364,280,350),(281,435,295,421),(282,436,296,422),(283,437,297,423),(284,438,298,424),(285,439,299,425),(286,440,300,426),(287,441,301,427),(288,442,302,428),(289,443,303,429),(290,444,304,430),(291,445,305,431),(292,446,306,432),(293,447,307,433),(294,448,308,434),(309,407,323,393),(310,408,324,394),(311,409,325,395),(312,410,326,396),(313,411,327,397),(314,412,328,398),(315,413,329,399),(316,414,330,400),(317,415,331,401),(318,416,332,402),(319,417,333,403),(320,418,334,404),(321,419,335,405),(322,420,336,406)], [(1,90,34,62),(2,91,35,63),(3,85,29,57),(4,86,30,58),(5,87,31,59),(6,88,32,60),(7,89,33,61),(8,92,36,64),(9,93,37,65),(10,94,38,66),(11,95,39,67),(12,96,40,68),(13,97,41,69),(14,98,42,70),(15,99,43,71),(16,100,44,72),(17,101,45,73),(18,102,46,74),(19,103,47,75),(20,104,48,76),(21,105,49,77),(22,106,50,78),(23,107,51,79),(24,108,52,80),(25,109,53,81),(26,110,54,82),(27,111,55,83),(28,112,56,84),(113,204,141,176),(114,205,142,177),(115,206,143,178),(116,207,144,179),(117,208,145,180),(118,209,146,181),(119,210,147,182),(120,197,148,169),(121,198,149,170),(122,199,150,171),(123,200,151,172),(124,201,152,173),(125,202,153,174),(126,203,154,175),(127,218,155,190),(128,219,156,191),(129,220,157,192),(130,221,158,193),(131,222,159,194),(132,223,160,195),(133,224,161,196),(134,211,162,183),(135,212,163,184),(136,213,164,185),(137,214,165,186),(138,215,166,187),(139,216,167,188),(140,217,168,189),(225,323,253,295),(226,324,254,296),(227,325,255,297),(228,326,256,298),(229,327,257,299),(230,328,258,300),(231,329,259,301),(232,330,260,302),(233,331,261,303),(234,332,262,304),(235,333,263,305),(236,334,264,306),(237,335,265,307),(238,336,266,308),(239,309,267,281),(240,310,268,282),(241,311,269,283),(242,312,270,284),(243,313,271,285),(244,314,272,286),(245,315,273,287),(246,316,274,288),(247,317,275,289),(248,318,276,290),(249,319,277,291),(250,320,278,292),(251,321,279,293),(252,322,280,294),(337,442,365,414),(338,443,366,415),(339,444,367,416),(340,445,368,417),(341,446,369,418),(342,447,370,419),(343,448,371,420),(344,435,372,407),(345,436,373,408),(346,437,374,409),(347,438,375,410),(348,439,376,411),(349,440,377,412),(350,441,378,413),(351,428,379,400),(352,429,380,401),(353,430,381,402),(354,431,382,403),(355,432,383,404),(356,433,384,405),(357,434,385,406),(358,421,386,393),(359,422,387,394),(360,423,388,395),(361,424,389,396),(362,425,390,397),(363,426,391,398),(364,427,392,399)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D7 | C4○D4 | D14 | C4×D7 | C4○D28 | D4×D7 | D4⋊2D7 | Q8×D7 |
| kernel | C7⋊(C42⋊8C4) | C14.C42 | C7×C2.C42 | C2×C4×Dic7 | C2×Dic7⋊C4 | C4×Dic7 | C2×Dic7 | C2×Dic7 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 1 | 1 | 2 | 8 | 2 | 2 | 3 | 8 | 9 | 12 | 24 | 3 | 6 | 3 |
Matrix representation of C7⋊(C42⋊8C4) ►in GL6(𝔽29)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 28 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 3 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 3 |
| 0 | 0 | 0 | 0 | 27 | 11 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 26 |
| 0 | 0 | 0 | 0 | 2 | 18 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 16 | 0 | 0 |
| 0 | 0 | 13 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 16 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(29))| [0,28,0,0,0,0,1,3,0,0,0,0,0,0,0,28,0,0,0,0,1,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,3,0,0,0,0,0,28,0,0,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,0,0,18,27,0,0,0,0,3,11],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,11,2,0,0,0,0,26,18],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,5,13,0,0,0,0,16,24,0,0,0,0,0,0,17,0,0,0,0,0,16,12] >;
C7⋊(C42⋊8C4) in GAP, Magma, Sage, TeX
C_7\rtimes (C_4^2\rtimes_8C_4)
% in TeX
G:=Group("C7:(C4^2:8C4)"); // GroupNames label
G:=SmallGroup(448,184);
// by ID
G=gap.SmallGroup(448,184);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,422,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^4=d^4=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c>;
// generators/relations