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## G = C7×C42⋊4C4order 448 = 26·7

### Direct product of C7 and C42⋊4C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×C42⋊4C4
 Chief series C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C7×C2.C42 — C7×C42⋊4C4
 Lower central C1 — C2 — C7×C42⋊4C4
 Upper central C1 — C22×C28 — C7×C42⋊4C4

Generators and relations for C7×C424C4
G = < a,b,c,d | a7=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, cd=dc >

Subgroups: 226 in 178 conjugacy classes, 130 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C22×C4, C22×C4, C28, C28, C2×C14, C2.C42, C2×C42, C2×C28, C2×C28, C22×C14, C424C4, C4×C28, C22×C28, C22×C28, C7×C2.C42, C2×C4×C28, C7×C424C4
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C42, C22×C4, C4○D4, C28, C2×C14, C2×C42, C42⋊C2, C2×C28, C22×C14, C424C4, C4×C28, C22×C28, C7×C4○D4, C2×C4×C28, C7×C42⋊C2, C7×C424C4

Smallest permutation representation of C7×C424C4
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)
(1 384 70 413)(2 385 64 407)(3 379 65 408)(4 380 66 409)(5 381 67 410)(6 382 68 411)(7 383 69 412)(8 133 48 104)(9 127 49 105)(10 128 43 99)(11 129 44 100)(12 130 45 101)(13 131 46 102)(14 132 47 103)(15 140 443 116)(16 134 444 117)(17 135 445 118)(18 136 446 119)(19 137 447 113)(20 138 448 114)(21 139 442 115)(22 108 31 143)(23 109 32 144)(24 110 33 145)(25 111 34 146)(26 112 35 147)(27 106 29 141)(28 107 30 142)(36 148 437 123)(37 149 438 124)(38 150 439 125)(39 151 440 126)(40 152 441 120)(41 153 435 121)(42 154 436 122)(50 419 85 395)(51 420 86 396)(52 414 87 397)(53 415 88 398)(54 416 89 399)(55 417 90 393)(56 418 91 394)(57 426 81 391)(58 427 82 392)(59 421 83 386)(60 422 84 387)(61 423 78 388)(62 424 79 389)(63 425 80 390)(71 400 96 432)(72 401 97 433)(73 402 98 434)(74 403 92 428)(75 404 93 429)(76 405 94 430)(77 406 95 431)(155 296 184 267)(156 297 185 268)(157 298 186 269)(158 299 187 270)(159 300 188 271)(160 301 189 272)(161 295 183 273)(162 309 197 274)(163 310 198 275)(164 311 199 276)(165 312 200 277)(166 313 201 278)(167 314 202 279)(168 315 203 280)(169 305 193 281)(170 306 194 282)(171 307 195 283)(172 308 196 284)(173 302 190 285)(174 303 191 286)(175 304 192 287)(176 320 208 288)(177 321 209 289)(178 322 210 290)(179 316 204 291)(180 317 205 292)(181 318 206 293)(182 319 207 294)(211 352 240 323)(212 353 241 324)(213 354 242 325)(214 355 243 326)(215 356 244 327)(216 357 245 328)(217 351 239 329)(218 365 253 330)(219 366 254 331)(220 367 255 332)(221 368 256 333)(222 369 257 334)(223 370 258 335)(224 371 259 336)(225 361 249 337)(226 362 250 338)(227 363 251 339)(228 364 252 340)(229 358 246 341)(230 359 247 342)(231 360 248 343)(232 376 264 344)(233 377 265 345)(234 378 266 346)(235 372 260 347)(236 373 261 348)(237 374 262 349)(238 375 263 350)
(1 106 59 133)(2 107 60 127)(3 108 61 128)(4 109 62 129)(5 110 63 130)(6 111 57 131)(7 112 58 132)(8 413 27 386)(9 407 28 387)(10 408 22 388)(11 409 23 389)(12 410 24 390)(13 411 25 391)(14 412 26 392)(15 420 441 400)(16 414 435 401)(17 415 436 402)(18 416 437 403)(19 417 438 404)(20 418 439 405)(21 419 440 406)(29 421 48 384)(30 422 49 385)(31 423 43 379)(32 424 44 380)(33 425 45 381)(34 426 46 382)(35 427 47 383)(36 428 446 399)(37 429 447 393)(38 430 448 394)(39 431 442 395)(40 432 443 396)(41 433 444 397)(42 434 445 398)(50 151 77 115)(51 152 71 116)(52 153 72 117)(53 154 73 118)(54 148 74 119)(55 149 75 113)(56 150 76 114)(64 142 84 105)(65 143 78 99)(66 144 79 100)(67 145 80 101)(68 146 81 102)(69 147 82 103)(70 141 83 104)(85 126 95 139)(86 120 96 140)(87 121 97 134)(88 122 98 135)(89 123 92 136)(90 124 93 137)(91 125 94 138)(155 246 199 233)(156 247 200 234)(157 248 201 235)(158 249 202 236)(159 250 203 237)(160 251 197 238)(161 252 198 232)(162 263 189 227)(163 264 183 228)(164 265 184 229)(165 266 185 230)(166 260 186 231)(167 261 187 225)(168 262 188 226)(169 243 205 223)(170 244 206 224)(171 245 207 218)(172 239 208 219)(173 240 209 220)(174 241 210 221)(175 242 204 222)(176 254 196 217)(177 255 190 211)(178 256 191 212)(179 257 192 213)(180 258 193 214)(181 259 194 215)(182 253 195 216)(267 358 311 345)(268 359 312 346)(269 360 313 347)(270 361 314 348)(271 362 315 349)(272 363 309 350)(273 364 310 344)(274 375 301 339)(275 376 295 340)(276 377 296 341)(277 378 297 342)(278 372 298 343)(279 373 299 337)(280 374 300 338)(281 355 317 335)(282 356 318 336)(283 357 319 330)(284 351 320 331)(285 352 321 332)(286 353 322 333)(287 354 316 334)(288 366 308 329)(289 367 302 323)(290 368 303 324)(291 369 304 325)(292 370 305 326)(293 371 306 327)(294 365 307 328)
(1 189 77 207)(2 183 71 208)(3 184 72 209)(4 185 73 210)(5 186 74 204)(6 187 75 205)(7 188 76 206)(8 339 39 330)(9 340 40 331)(10 341 41 332)(11 342 42 333)(12 343 36 334)(13 337 37 335)(14 338 38 336)(15 329 30 344)(16 323 31 345)(17 324 32 346)(18 325 33 347)(19 326 34 348)(20 327 35 349)(21 328 29 350)(22 377 444 352)(23 378 445 353)(24 372 446 354)(25 373 447 355)(26 374 448 356)(27 375 442 357)(28 376 443 351)(43 358 435 367)(44 359 436 368)(45 360 437 369)(46 361 438 370)(47 362 439 371)(48 363 440 365)(49 364 441 366)(50 171 59 162)(51 172 60 163)(52 173 61 164)(53 174 62 165)(54 175 63 166)(55 169 57 167)(56 170 58 168)(64 161 96 176)(65 155 97 177)(66 156 98 178)(67 157 92 179)(68 158 93 180)(69 159 94 181)(70 160 95 182)(78 199 87 190)(79 200 88 191)(80 201 89 192)(81 202 90 193)(82 203 91 194)(83 197 85 195)(84 198 86 196)(99 233 121 211)(100 234 122 212)(101 235 123 213)(102 236 124 214)(103 237 125 215)(104 238 126 216)(105 232 120 217)(106 227 115 218)(107 228 116 219)(108 229 117 220)(109 230 118 221)(110 231 119 222)(111 225 113 223)(112 226 114 224)(127 264 152 239)(128 265 153 240)(129 266 154 241)(130 260 148 242)(131 261 149 243)(132 262 150 244)(133 263 151 245)(134 255 143 246)(135 256 144 247)(136 257 145 248)(137 258 146 249)(138 259 147 250)(139 253 141 251)(140 254 142 252)(267 414 289 423)(268 415 290 424)(269 416 291 425)(270 417 292 426)(271 418 293 427)(272 419 294 421)(273 420 288 422)(274 431 283 413)(275 432 284 407)(276 433 285 408)(277 434 286 409)(278 428 287 410)(279 429 281 411)(280 430 282 412)(295 396 320 387)(296 397 321 388)(297 398 322 389)(298 399 316 390)(299 393 317 391)(300 394 318 392)(301 395 319 386)(302 379 311 401)(303 380 312 402)(304 381 313 403)(305 382 314 404)(306 383 315 405)(307 384 309 406)(308 385 310 400)

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

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

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

280 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4AF 7A ··· 7F 14A ··· 14AP 28A ··· 28AV 28AW ··· 28GJ order 1 2 ··· 2 4 ··· 4 4 ··· 4 7 ··· 7 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 1 ··· 1 2 ··· 2

280 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⋊4C4 C7×C2.C42 C2×C4×C28 C4×C28 C42⋊4C4 C2.C42 C2×C42 C42 C2×C14 C22 # reps 1 4 3 24 6 24 18 144 8 48

Matrix representation of C7×C424C4 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 23 0 0 0 0 23
,
 12 0 0 0 0 12 0 0 0 0 9 15 0 0 14 20
,
 1 0 0 0 0 28 0 0 0 0 12 0 0 0 0 12
,
 17 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,23,0,0,0,0,23],[12,0,0,0,0,12,0,0,0,0,9,14,0,0,15,20],[1,0,0,0,0,28,0,0,0,0,12,0,0,0,0,12],[17,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C7×C424C4 in GAP, Magma, Sage, TeX

C_7\times C_4^2\rtimes_4C_4
% in TeX

G:=Group("C7xC4^2:4C4");
// GroupNames label

G:=SmallGroup(448,784);
// by ID

G=gap.SmallGroup(448,784);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1576,310]);
// 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,c*d=d*c>;
// generators/relations

׿
×
𝔽