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## G = C7×C42⋊5C4order 448 = 26·7

### Direct product of C7 and C42⋊5C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C42⋊5C4
 Chief series C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C7×C2.C42 — C7×C42⋊5C4
 Lower central C1 — C22 — C7×C42⋊5C4
 Upper central C1 — C22×C14 — C7×C42⋊5C4

Generators and relations for C7×C425C4
G = < a,b,c,d | a7=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c-1 >

Subgroups: 210 in 138 conjugacy classes, 82 normal (10 characteristic)
C1, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C42, C22×C4, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C28, C2×C28, C22×C14, C425C4, C4×C28, C22×C28, C7×C2.C42, C2×C4×C28, C7×C425C4
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C22×C4, C4○D4, C28, C2×C14, C42⋊C2, C422C2, C2×C28, C22×C14, C425C4, C22×C28, C7×C4○D4, C7×C42⋊C2, C7×C422C2, C7×C425C4

Smallest permutation representation of C7×C425C4
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)
(1 183 95 181)(2 184 96 182)(3 185 97 176)(4 186 98 177)(5 187 92 178)(6 188 93 179)(7 189 94 180)(8 353 22 344)(9 354 23 345)(10 355 24 346)(11 356 25 347)(12 357 26 348)(13 351 27 349)(14 352 28 350)(15 336 439 362)(16 330 440 363)(17 331 441 364)(18 332 435 358)(19 333 436 359)(20 334 437 360)(21 335 438 361)(29 328 442 375)(30 329 443 376)(31 323 444 377)(32 324 445 378)(33 325 446 372)(34 326 447 373)(35 327 448 374)(36 343 45 369)(37 337 46 370)(38 338 47 371)(39 339 48 365)(40 340 49 366)(41 341 43 367)(42 342 44 368)(50 197 83 171)(51 198 84 172)(52 199 78 173)(53 200 79 174)(54 201 80 175)(55 202 81 169)(56 203 82 170)(57 193 90 167)(58 194 91 168)(59 195 85 162)(60 196 86 163)(61 190 87 164)(62 191 88 165)(63 192 89 166)(64 155 73 209)(65 156 74 210)(66 157 75 204)(67 158 76 205)(68 159 77 206)(69 160 71 207)(70 161 72 208)(99 232 153 241)(100 233 154 242)(101 234 148 243)(102 235 149 244)(103 236 150 245)(104 237 151 239)(105 238 152 240)(106 253 139 227)(107 254 140 228)(108 255 134 229)(109 256 135 230)(110 257 136 231)(111 258 137 225)(112 259 138 226)(113 249 146 223)(114 250 147 224)(115 251 141 218)(116 252 142 219)(117 246 143 220)(118 247 144 221)(119 248 145 222)(120 211 129 265)(121 212 130 266)(122 213 131 260)(123 214 132 261)(124 215 133 262)(125 216 127 263)(126 217 128 264)(267 388 321 414)(268 389 322 415)(269 390 316 416)(270 391 317 417)(271 392 318 418)(272 386 319 419)(273 387 320 420)(274 384 307 431)(275 385 308 432)(276 379 302 433)(277 380 303 434)(278 381 304 428)(279 382 305 429)(280 383 306 430)(281 412 314 403)(282 413 315 404)(283 407 309 405)(284 408 310 406)(285 409 311 400)(286 410 312 401)(287 411 313 402)(288 423 297 397)(289 424 298 398)(290 425 299 399)(291 426 300 393)(292 427 301 394)(293 421 295 395)(294 422 296 396)
(1 106 59 125)(2 107 60 126)(3 108 61 120)(4 109 62 121)(5 110 63 122)(6 111 57 123)(7 112 58 124)(8 400 435 388)(9 401 436 389)(10 402 437 390)(11 403 438 391)(12 404 439 392)(13 405 440 386)(14 406 441 387)(15 418 26 413)(16 419 27 407)(17 420 28 408)(18 414 22 409)(19 415 23 410)(20 416 24 411)(21 417 25 412)(29 431 48 395)(30 432 49 396)(31 433 43 397)(32 434 44 398)(33 428 45 399)(34 429 46 393)(35 430 47 394)(36 425 446 381)(37 426 447 382)(38 427 448 383)(39 421 442 384)(40 422 443 385)(41 423 444 379)(42 424 445 380)(50 151 69 115)(51 152 70 116)(52 153 64 117)(53 154 65 118)(54 148 66 119)(55 149 67 113)(56 150 68 114)(71 141 83 104)(72 142 84 105)(73 143 78 99)(74 144 79 100)(75 145 80 101)(76 146 81 102)(77 147 82 103)(85 127 95 139)(86 128 96 140)(87 129 97 134)(88 130 98 135)(89 131 92 136)(90 132 93 137)(91 133 94 138)(155 246 199 241)(156 247 200 242)(157 248 201 243)(158 249 202 244)(159 250 203 245)(160 251 197 239)(161 252 198 240)(162 263 181 227)(163 264 182 228)(164 265 176 229)(165 266 177 230)(166 260 178 231)(167 261 179 225)(168 262 180 226)(169 235 205 223)(170 236 206 224)(171 237 207 218)(172 238 208 219)(173 232 209 220)(174 233 210 221)(175 234 204 222)(183 253 195 216)(184 254 196 217)(185 255 190 211)(186 256 191 212)(187 257 192 213)(188 258 193 214)(189 259 194 215)(267 344 311 332)(268 345 312 333)(269 346 313 334)(270 347 314 335)(271 348 315 336)(272 349 309 330)(273 350 310 331)(274 365 293 328)(275 366 294 329)(276 367 288 323)(277 368 289 324)(278 369 290 325)(279 370 291 326)(280 371 292 327)(281 361 317 356)(282 362 318 357)(283 363 319 351)(284 364 320 352)(285 358 321 353)(286 359 322 354)(287 360 316 355)(295 375 307 339)(296 376 308 340)(297 377 302 341)(298 378 303 342)(299 372 304 343)(300 373 305 337)(301 374 306 338)
(1 293 69 319)(2 294 70 320)(3 288 64 321)(4 289 65 322)(5 290 66 316)(6 291 67 317)(7 292 68 318)(8 255 31 246)(9 256 32 247)(10 257 33 248)(11 258 34 249)(12 259 35 250)(13 253 29 251)(14 254 30 252)(15 262 38 236)(16 263 39 237)(17 264 40 238)(18 265 41 232)(19 266 42 233)(20 260 36 234)(21 261 37 235)(22 229 444 220)(23 230 445 221)(24 231 446 222)(25 225 447 223)(26 226 448 224)(27 227 442 218)(28 228 443 219)(43 241 435 211)(44 242 436 212)(45 243 437 213)(46 244 438 214)(47 245 439 215)(48 239 440 216)(49 240 441 217)(50 283 59 274)(51 284 60 275)(52 285 61 276)(53 286 62 277)(54 287 63 278)(55 281 57 279)(56 282 58 280)(71 272 95 295)(72 273 96 296)(73 267 97 297)(74 268 98 298)(75 269 92 299)(76 270 93 300)(77 271 94 301)(78 311 87 302)(79 312 88 303)(80 313 89 304)(81 314 90 305)(82 315 91 306)(83 309 85 307)(84 310 86 308)(99 353 129 323)(100 354 130 324)(101 355 131 325)(102 356 132 326)(103 357 133 327)(104 351 127 328)(105 352 128 329)(106 339 115 330)(107 340 116 331)(108 341 117 332)(109 342 118 333)(110 343 119 334)(111 337 113 335)(112 338 114 336)(120 377 153 344)(121 378 154 345)(122 372 148 346)(123 373 149 347)(124 374 150 348)(125 375 151 349)(126 376 152 350)(134 367 143 358)(135 368 144 359)(136 369 145 360)(137 370 146 361)(138 371 147 362)(139 365 141 363)(140 366 142 364)(155 409 185 379)(156 410 186 380)(157 411 187 381)(158 412 188 382)(159 413 189 383)(160 407 183 384)(161 408 184 385)(162 395 171 386)(163 396 172 387)(164 397 173 388)(165 398 174 389)(166 399 175 390)(167 393 169 391)(168 394 170 392)(176 433 209 400)(177 434 210 401)(178 428 204 402)(179 429 205 403)(180 430 206 404)(181 431 207 405)(182 432 208 406)(190 423 199 414)(191 424 200 415)(192 425 201 416)(193 426 202 417)(194 427 203 418)(195 421 197 419)(196 422 198 420)

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

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

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

196 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 7A ··· 7F 14A ··· 14AP 28A ··· 28BT 28BU ··· 28DP order 1 2 ··· 2 4 ··· 4 4 ··· 4 7 ··· 7 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C4 C7 C14 C14 C28 C4○D4 C7×C4○D4 kernel C7×C42⋊5C4 C7×C2.C42 C2×C4×C28 C4×C28 C42⋊5C4 C2.C42 C2×C42 C42 C2×C14 C22 # reps 1 6 1 8 6 36 6 48 12 72

Matrix representation of C7×C425C4 in GL6(𝔽29)

 25 0 0 0 0 0 0 25 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 23 0 0 0 0 0 0 23
,
 17 0 0 0 0 0 0 17 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 27 0 0 0 0 0 17
,
 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 2 0 0 0 0 0 1 0 0 0 0 0 0 17 0 0 0 0 0 0 17
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 19 13 0 0 0 0 19 10 0 0 0 0 0 0 16 26 0 0 0 0 18 13

G:=sub<GL(6,GF(29))| [25,0,0,0,0,0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,0,0,0,0,0,0,23],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,27,17],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,2,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,19,19,0,0,0,0,13,10,0,0,0,0,0,0,16,18,0,0,0,0,26,13] >;

C7×C425C4 in GAP, Magma, Sage, TeX

C_7\times C_4^2\rtimes_5C_4
% in TeX

G:=Group("C7xC4^2:5C4");
// GroupNames label

G:=SmallGroup(448,791);
// by ID

G=gap.SmallGroup(448,791);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1184,2438,310]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^4=c^4=d^4=1,a*b=b*a,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^-1>;
// generators/relations

׿
×
𝔽