metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊Dic7⋊7C4, C14.1(C4×D4), C2.4(C4×D28), C14.15(C4×Q8), (C2×C28).233D4, (C2×C4).111D28, (C22×C4).9D14, C22.14(Q8×D7), (C2×Dic7).12Q8, C22.23(C2×D28), C2.1(D14⋊2Q8), C2.4(Dic7.Q8), C14.19(C22⋊Q8), C14.8(C42.C2), C2.C42.9D7, C14.8(C42⋊2C2), C2.6(Dic7⋊3Q8), C22.30(C4○D28), (C22×C28).42C22, C23.250(C22×D7), C14.18(C42⋊C2), C22.32(D4⋊2D7), C14.C42.22C2, (C22×C14).278C23, C7⋊2(C23.63C23), C14.2(C22.D4), C2.3(C23.D14), C2.1(C22.D28), (C22×Dic7).4C22, C2.7(C23.11D14), (C2×C4).24(C4×D7), C22.85(C2×C4×D7), (C2×C28).32(C2×C4), (C2×C14).92(C2×D4), (C2×C4⋊Dic7).3C2, (C2×C14).61(C2×Q8), (C2×C4×Dic7).26C2, (C2×Dic7⋊C4).26C2, (C2×C14).44(C22×C4), (C2×Dic7).42(C2×C4), (C2×C14).125(C4○D4), (C7×C2.C42).16C2, SmallGroup(448,187)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊Dic7⋊7C4
G = < a,b,c,d | a4=b14=d4=1, c2=b7, ab=ba, cac-1=a-1, dad-1=ab7, cbc-1=b-1, bd=db, cd=dc >
Subgroups: 604 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.63C23, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C22×Dic7, C22×C28, C14.C42, C7×C2.C42, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C4⋊Dic7⋊7C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C22×C4, C2×D4, C2×Q8, C4○D4, D14, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×D7, D28, C22×D7, C23.63C23, C2×C4×D7, C2×D28, C4○D28, D4⋊2D7, Q8×D7, C4×D28, C23.11D14, C23.D14, C22.D28, Dic7⋊3Q8, Dic7.Q8, D14⋊2Q8, C4⋊Dic7⋊7C4
►Regular action on 448 points
Generators in S448
(1 134 15 120)(2 135 16 121)(3 136 17 122)(4 137 18 123)(5 138 19 124)(6 139 20 125)(7 140 21 126)(8 127 22 113)(9 128 23 114)(10 129 24 115)(11 130 25 116)(12 131 26 117)(13 132 27 118)(14 133 28 119)(29 155 50 148)(30 156 51 149)(31 157 52 150)(32 158 53 151)(33 159 54 152)(34 160 55 153)(35 161 56 154)(36 162 43 141)(37 163 44 142)(38 164 45 143)(39 165 46 144)(40 166 47 145)(41 167 48 146)(42 168 49 147)(57 190 78 176)(58 191 79 177)(59 192 80 178)(60 193 81 179)(61 194 82 180)(62 195 83 181)(63 196 84 182)(64 183 71 169)(65 184 72 170)(66 185 73 171)(67 186 74 172)(68 187 75 173)(69 188 76 174)(70 189 77 175)(85 218 106 204)(86 219 107 205)(87 220 108 206)(88 221 109 207)(89 222 110 208)(90 223 111 209)(91 224 112 210)(92 211 99 197)(93 212 100 198)(94 213 101 199)(95 214 102 200)(96 215 103 201)(97 216 104 202)(98 217 105 203)(225 340 239 364)(226 341 240 351)(227 342 241 352)(228 343 242 353)(229 344 243 354)(230 345 244 355)(231 346 245 356)(232 347 246 357)(233 348 247 358)(234 349 248 359)(235 350 249 360)(236 337 250 361)(237 338 251 362)(238 339 252 363)(253 376 267 385)(254 377 268 386)(255 378 269 387)(256 365 270 388)(257 366 271 389)(258 367 272 390)(259 368 273 391)(260 369 274 392)(261 370 275 379)(262 371 276 380)(263 372 277 381)(264 373 278 382)(265 374 279 383)(266 375 280 384)(281 393 305 412)(282 394 306 413)(283 395 307 414)(284 396 308 415)(285 397 295 416)(286 398 296 417)(287 399 297 418)(288 400 298 419)(289 401 299 420)(290 402 300 407)(291 403 301 408)(292 404 302 409)(293 405 303 410)(294 406 304 411)(309 429 333 440)(310 430 334 441)(311 431 335 442)(312 432 336 443)(313 433 323 444)(314 434 324 445)(315 421 325 446)(316 422 326 447)(317 423 327 448)(318 424 328 435)(319 425 329 436)(320 426 330 437)(321 427 331 438)(322 428 332 439)
(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)
(1 227 8 234)(2 226 9 233)(3 225 10 232)(4 238 11 231)(5 237 12 230)(6 236 13 229)(7 235 14 228)(15 241 22 248)(16 240 23 247)(17 239 24 246)(18 252 25 245)(19 251 26 244)(20 250 27 243)(21 249 28 242)(29 263 36 256)(30 262 37 255)(31 261 38 254)(32 260 39 253)(33 259 40 266)(34 258 41 265)(35 257 42 264)(43 270 50 277)(44 269 51 276)(45 268 52 275)(46 267 53 274)(47 280 54 273)(48 279 55 272)(49 278 56 271)(57 293 64 286)(58 292 65 285)(59 291 66 284)(60 290 67 283)(61 289 68 282)(62 288 69 281)(63 287 70 294)(71 296 78 303)(72 295 79 302)(73 308 80 301)(74 307 81 300)(75 306 82 299)(76 305 83 298)(77 304 84 297)(85 320 92 313)(86 319 93 312)(87 318 94 311)(88 317 95 310)(89 316 96 309)(90 315 97 322)(91 314 98 321)(99 323 106 330)(100 336 107 329)(101 335 108 328)(102 334 109 327)(103 333 110 326)(104 332 111 325)(105 331 112 324)(113 349 120 342)(114 348 121 341)(115 347 122 340)(116 346 123 339)(117 345 124 338)(118 344 125 337)(119 343 126 350)(127 359 134 352)(128 358 135 351)(129 357 136 364)(130 356 137 363)(131 355 138 362)(132 354 139 361)(133 353 140 360)(141 365 148 372)(142 378 149 371)(143 377 150 370)(144 376 151 369)(145 375 152 368)(146 374 153 367)(147 373 154 366)(155 381 162 388)(156 380 163 387)(157 379 164 386)(158 392 165 385)(159 391 166 384)(160 390 167 383)(161 389 168 382)(169 398 176 405)(170 397 177 404)(171 396 178 403)(172 395 179 402)(173 394 180 401)(174 393 181 400)(175 406 182 399)(183 417 190 410)(184 416 191 409)(185 415 192 408)(186 414 193 407)(187 413 194 420)(188 412 195 419)(189 411 196 418)(197 433 204 426)(198 432 205 425)(199 431 206 424)(200 430 207 423)(201 429 208 422)(202 428 209 421)(203 427 210 434)(211 444 218 437)(212 443 219 436)(213 442 220 435)(214 441 221 448)(215 440 222 447)(216 439 223 446)(217 438 224 445)
(1 85 29 64)(2 86 30 65)(3 87 31 66)(4 88 32 67)(5 89 33 68)(6 90 34 69)(7 91 35 70)(8 92 36 57)(9 93 37 58)(10 94 38 59)(11 95 39 60)(12 96 40 61)(13 97 41 62)(14 98 42 63)(15 106 50 71)(16 107 51 72)(17 108 52 73)(18 109 53 74)(19 110 54 75)(20 111 55 76)(21 112 56 77)(22 99 43 78)(23 100 44 79)(24 101 45 80)(25 102 46 81)(26 103 47 82)(27 104 48 83)(28 105 49 84)(113 204 141 169)(114 205 142 170)(115 206 143 171)(116 207 144 172)(117 208 145 173)(118 209 146 174)(119 210 147 175)(120 197 148 176)(121 198 149 177)(122 199 150 178)(123 200 151 179)(124 201 152 180)(125 202 153 181)(126 203 154 182)(127 218 162 183)(128 219 163 184)(129 220 164 185)(130 221 165 186)(131 222 166 187)(132 223 167 188)(133 224 168 189)(134 211 155 190)(135 212 156 191)(136 213 157 192)(137 214 158 193)(138 215 159 194)(139 216 160 195)(140 217 161 196)(225 318 261 284)(226 319 262 285)(227 320 263 286)(228 321 264 287)(229 322 265 288)(230 309 266 289)(231 310 253 290)(232 311 254 291)(233 312 255 292)(234 313 256 293)(235 314 257 294)(236 315 258 281)(237 316 259 282)(238 317 260 283)(239 328 275 308)(240 329 276 295)(241 330 277 296)(242 331 278 297)(243 332 279 298)(244 333 280 299)(245 334 267 300)(246 335 268 301)(247 336 269 302)(248 323 270 303)(249 324 271 304)(250 325 272 305)(251 326 273 306)(252 327 274 307)(337 428 367 400)(338 429 368 401)(339 430 369 402)(340 431 370 403)(341 432 371 404)(342 433 372 405)(343 434 373 406)(344 421 374 393)(345 422 375 394)(346 423 376 395)(347 424 377 396)(348 425 378 397)(349 426 365 398)(350 427 366 399)(351 443 380 409)(352 444 381 410)(353 445 382 411)(354 446 383 412)(355 447 384 413)(356 448 385 414)(357 435 386 415)(358 436 387 416)(359 437 388 417)(360 438 389 418)(361 439 390 419)(362 440 391 420)(363 441 392 407)(364 442 379 408)
G:=sub<Sym(448)| (1,134,15,120)(2,135,16,121)(3,136,17,122)(4,137,18,123)(5,138,19,124)(6,139,20,125)(7,140,21,126)(8,127,22,113)(9,128,23,114)(10,129,24,115)(11,130,25,116)(12,131,26,117)(13,132,27,118)(14,133,28,119)(29,155,50,148)(30,156,51,149)(31,157,52,150)(32,158,53,151)(33,159,54,152)(34,160,55,153)(35,161,56,154)(36,162,43,141)(37,163,44,142)(38,164,45,143)(39,165,46,144)(40,166,47,145)(41,167,48,146)(42,168,49,147)(57,190,78,176)(58,191,79,177)(59,192,80,178)(60,193,81,179)(61,194,82,180)(62,195,83,181)(63,196,84,182)(64,183,71,169)(65,184,72,170)(66,185,73,171)(67,186,74,172)(68,187,75,173)(69,188,76,174)(70,189,77,175)(85,218,106,204)(86,219,107,205)(87,220,108,206)(88,221,109,207)(89,222,110,208)(90,223,111,209)(91,224,112,210)(92,211,99,197)(93,212,100,198)(94,213,101,199)(95,214,102,200)(96,215,103,201)(97,216,104,202)(98,217,105,203)(225,340,239,364)(226,341,240,351)(227,342,241,352)(228,343,242,353)(229,344,243,354)(230,345,244,355)(231,346,245,356)(232,347,246,357)(233,348,247,358)(234,349,248,359)(235,350,249,360)(236,337,250,361)(237,338,251,362)(238,339,252,363)(253,376,267,385)(254,377,268,386)(255,378,269,387)(256,365,270,388)(257,366,271,389)(258,367,272,390)(259,368,273,391)(260,369,274,392)(261,370,275,379)(262,371,276,380)(263,372,277,381)(264,373,278,382)(265,374,279,383)(266,375,280,384)(281,393,305,412)(282,394,306,413)(283,395,307,414)(284,396,308,415)(285,397,295,416)(286,398,296,417)(287,399,297,418)(288,400,298,419)(289,401,299,420)(290,402,300,407)(291,403,301,408)(292,404,302,409)(293,405,303,410)(294,406,304,411)(309,429,333,440)(310,430,334,441)(311,431,335,442)(312,432,336,443)(313,433,323,444)(314,434,324,445)(315,421,325,446)(316,422,326,447)(317,423,327,448)(318,424,328,435)(319,425,329,436)(320,426,330,437)(321,427,331,438)(322,428,332,439), (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), (1,227,8,234)(2,226,9,233)(3,225,10,232)(4,238,11,231)(5,237,12,230)(6,236,13,229)(7,235,14,228)(15,241,22,248)(16,240,23,247)(17,239,24,246)(18,252,25,245)(19,251,26,244)(20,250,27,243)(21,249,28,242)(29,263,36,256)(30,262,37,255)(31,261,38,254)(32,260,39,253)(33,259,40,266)(34,258,41,265)(35,257,42,264)(43,270,50,277)(44,269,51,276)(45,268,52,275)(46,267,53,274)(47,280,54,273)(48,279,55,272)(49,278,56,271)(57,293,64,286)(58,292,65,285)(59,291,66,284)(60,290,67,283)(61,289,68,282)(62,288,69,281)(63,287,70,294)(71,296,78,303)(72,295,79,302)(73,308,80,301)(74,307,81,300)(75,306,82,299)(76,305,83,298)(77,304,84,297)(85,320,92,313)(86,319,93,312)(87,318,94,311)(88,317,95,310)(89,316,96,309)(90,315,97,322)(91,314,98,321)(99,323,106,330)(100,336,107,329)(101,335,108,328)(102,334,109,327)(103,333,110,326)(104,332,111,325)(105,331,112,324)(113,349,120,342)(114,348,121,341)(115,347,122,340)(116,346,123,339)(117,345,124,338)(118,344,125,337)(119,343,126,350)(127,359,134,352)(128,358,135,351)(129,357,136,364)(130,356,137,363)(131,355,138,362)(132,354,139,361)(133,353,140,360)(141,365,148,372)(142,378,149,371)(143,377,150,370)(144,376,151,369)(145,375,152,368)(146,374,153,367)(147,373,154,366)(155,381,162,388)(156,380,163,387)(157,379,164,386)(158,392,165,385)(159,391,166,384)(160,390,167,383)(161,389,168,382)(169,398,176,405)(170,397,177,404)(171,396,178,403)(172,395,179,402)(173,394,180,401)(174,393,181,400)(175,406,182,399)(183,417,190,410)(184,416,191,409)(185,415,192,408)(186,414,193,407)(187,413,194,420)(188,412,195,419)(189,411,196,418)(197,433,204,426)(198,432,205,425)(199,431,206,424)(200,430,207,423)(201,429,208,422)(202,428,209,421)(203,427,210,434)(211,444,218,437)(212,443,219,436)(213,442,220,435)(214,441,221,448)(215,440,222,447)(216,439,223,446)(217,438,224,445), (1,85,29,64)(2,86,30,65)(3,87,31,66)(4,88,32,67)(5,89,33,68)(6,90,34,69)(7,91,35,70)(8,92,36,57)(9,93,37,58)(10,94,38,59)(11,95,39,60)(12,96,40,61)(13,97,41,62)(14,98,42,63)(15,106,50,71)(16,107,51,72)(17,108,52,73)(18,109,53,74)(19,110,54,75)(20,111,55,76)(21,112,56,77)(22,99,43,78)(23,100,44,79)(24,101,45,80)(25,102,46,81)(26,103,47,82)(27,104,48,83)(28,105,49,84)(113,204,141,169)(114,205,142,170)(115,206,143,171)(116,207,144,172)(117,208,145,173)(118,209,146,174)(119,210,147,175)(120,197,148,176)(121,198,149,177)(122,199,150,178)(123,200,151,179)(124,201,152,180)(125,202,153,181)(126,203,154,182)(127,218,162,183)(128,219,163,184)(129,220,164,185)(130,221,165,186)(131,222,166,187)(132,223,167,188)(133,224,168,189)(134,211,155,190)(135,212,156,191)(136,213,157,192)(137,214,158,193)(138,215,159,194)(139,216,160,195)(140,217,161,196)(225,318,261,284)(226,319,262,285)(227,320,263,286)(228,321,264,287)(229,322,265,288)(230,309,266,289)(231,310,253,290)(232,311,254,291)(233,312,255,292)(234,313,256,293)(235,314,257,294)(236,315,258,281)(237,316,259,282)(238,317,260,283)(239,328,275,308)(240,329,276,295)(241,330,277,296)(242,331,278,297)(243,332,279,298)(244,333,280,299)(245,334,267,300)(246,335,268,301)(247,336,269,302)(248,323,270,303)(249,324,271,304)(250,325,272,305)(251,326,273,306)(252,327,274,307)(337,428,367,400)(338,429,368,401)(339,430,369,402)(340,431,370,403)(341,432,371,404)(342,433,372,405)(343,434,373,406)(344,421,374,393)(345,422,375,394)(346,423,376,395)(347,424,377,396)(348,425,378,397)(349,426,365,398)(350,427,366,399)(351,443,380,409)(352,444,381,410)(353,445,382,411)(354,446,383,412)(355,447,384,413)(356,448,385,414)(357,435,386,415)(358,436,387,416)(359,437,388,417)(360,438,389,418)(361,439,390,419)(362,440,391,420)(363,441,392,407)(364,442,379,408)>;
G:=Group( (1,134,15,120)(2,135,16,121)(3,136,17,122)(4,137,18,123)(5,138,19,124)(6,139,20,125)(7,140,21,126)(8,127,22,113)(9,128,23,114)(10,129,24,115)(11,130,25,116)(12,131,26,117)(13,132,27,118)(14,133,28,119)(29,155,50,148)(30,156,51,149)(31,157,52,150)(32,158,53,151)(33,159,54,152)(34,160,55,153)(35,161,56,154)(36,162,43,141)(37,163,44,142)(38,164,45,143)(39,165,46,144)(40,166,47,145)(41,167,48,146)(42,168,49,147)(57,190,78,176)(58,191,79,177)(59,192,80,178)(60,193,81,179)(61,194,82,180)(62,195,83,181)(63,196,84,182)(64,183,71,169)(65,184,72,170)(66,185,73,171)(67,186,74,172)(68,187,75,173)(69,188,76,174)(70,189,77,175)(85,218,106,204)(86,219,107,205)(87,220,108,206)(88,221,109,207)(89,222,110,208)(90,223,111,209)(91,224,112,210)(92,211,99,197)(93,212,100,198)(94,213,101,199)(95,214,102,200)(96,215,103,201)(97,216,104,202)(98,217,105,203)(225,340,239,364)(226,341,240,351)(227,342,241,352)(228,343,242,353)(229,344,243,354)(230,345,244,355)(231,346,245,356)(232,347,246,357)(233,348,247,358)(234,349,248,359)(235,350,249,360)(236,337,250,361)(237,338,251,362)(238,339,252,363)(253,376,267,385)(254,377,268,386)(255,378,269,387)(256,365,270,388)(257,366,271,389)(258,367,272,390)(259,368,273,391)(260,369,274,392)(261,370,275,379)(262,371,276,380)(263,372,277,381)(264,373,278,382)(265,374,279,383)(266,375,280,384)(281,393,305,412)(282,394,306,413)(283,395,307,414)(284,396,308,415)(285,397,295,416)(286,398,296,417)(287,399,297,418)(288,400,298,419)(289,401,299,420)(290,402,300,407)(291,403,301,408)(292,404,302,409)(293,405,303,410)(294,406,304,411)(309,429,333,440)(310,430,334,441)(311,431,335,442)(312,432,336,443)(313,433,323,444)(314,434,324,445)(315,421,325,446)(316,422,326,447)(317,423,327,448)(318,424,328,435)(319,425,329,436)(320,426,330,437)(321,427,331,438)(322,428,332,439), (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), (1,227,8,234)(2,226,9,233)(3,225,10,232)(4,238,11,231)(5,237,12,230)(6,236,13,229)(7,235,14,228)(15,241,22,248)(16,240,23,247)(17,239,24,246)(18,252,25,245)(19,251,26,244)(20,250,27,243)(21,249,28,242)(29,263,36,256)(30,262,37,255)(31,261,38,254)(32,260,39,253)(33,259,40,266)(34,258,41,265)(35,257,42,264)(43,270,50,277)(44,269,51,276)(45,268,52,275)(46,267,53,274)(47,280,54,273)(48,279,55,272)(49,278,56,271)(57,293,64,286)(58,292,65,285)(59,291,66,284)(60,290,67,283)(61,289,68,282)(62,288,69,281)(63,287,70,294)(71,296,78,303)(72,295,79,302)(73,308,80,301)(74,307,81,300)(75,306,82,299)(76,305,83,298)(77,304,84,297)(85,320,92,313)(86,319,93,312)(87,318,94,311)(88,317,95,310)(89,316,96,309)(90,315,97,322)(91,314,98,321)(99,323,106,330)(100,336,107,329)(101,335,108,328)(102,334,109,327)(103,333,110,326)(104,332,111,325)(105,331,112,324)(113,349,120,342)(114,348,121,341)(115,347,122,340)(116,346,123,339)(117,345,124,338)(118,344,125,337)(119,343,126,350)(127,359,134,352)(128,358,135,351)(129,357,136,364)(130,356,137,363)(131,355,138,362)(132,354,139,361)(133,353,140,360)(141,365,148,372)(142,378,149,371)(143,377,150,370)(144,376,151,369)(145,375,152,368)(146,374,153,367)(147,373,154,366)(155,381,162,388)(156,380,163,387)(157,379,164,386)(158,392,165,385)(159,391,166,384)(160,390,167,383)(161,389,168,382)(169,398,176,405)(170,397,177,404)(171,396,178,403)(172,395,179,402)(173,394,180,401)(174,393,181,400)(175,406,182,399)(183,417,190,410)(184,416,191,409)(185,415,192,408)(186,414,193,407)(187,413,194,420)(188,412,195,419)(189,411,196,418)(197,433,204,426)(198,432,205,425)(199,431,206,424)(200,430,207,423)(201,429,208,422)(202,428,209,421)(203,427,210,434)(211,444,218,437)(212,443,219,436)(213,442,220,435)(214,441,221,448)(215,440,222,447)(216,439,223,446)(217,438,224,445), (1,85,29,64)(2,86,30,65)(3,87,31,66)(4,88,32,67)(5,89,33,68)(6,90,34,69)(7,91,35,70)(8,92,36,57)(9,93,37,58)(10,94,38,59)(11,95,39,60)(12,96,40,61)(13,97,41,62)(14,98,42,63)(15,106,50,71)(16,107,51,72)(17,108,52,73)(18,109,53,74)(19,110,54,75)(20,111,55,76)(21,112,56,77)(22,99,43,78)(23,100,44,79)(24,101,45,80)(25,102,46,81)(26,103,47,82)(27,104,48,83)(28,105,49,84)(113,204,141,169)(114,205,142,170)(115,206,143,171)(116,207,144,172)(117,208,145,173)(118,209,146,174)(119,210,147,175)(120,197,148,176)(121,198,149,177)(122,199,150,178)(123,200,151,179)(124,201,152,180)(125,202,153,181)(126,203,154,182)(127,218,162,183)(128,219,163,184)(129,220,164,185)(130,221,165,186)(131,222,166,187)(132,223,167,188)(133,224,168,189)(134,211,155,190)(135,212,156,191)(136,213,157,192)(137,214,158,193)(138,215,159,194)(139,216,160,195)(140,217,161,196)(225,318,261,284)(226,319,262,285)(227,320,263,286)(228,321,264,287)(229,322,265,288)(230,309,266,289)(231,310,253,290)(232,311,254,291)(233,312,255,292)(234,313,256,293)(235,314,257,294)(236,315,258,281)(237,316,259,282)(238,317,260,283)(239,328,275,308)(240,329,276,295)(241,330,277,296)(242,331,278,297)(243,332,279,298)(244,333,280,299)(245,334,267,300)(246,335,268,301)(247,336,269,302)(248,323,270,303)(249,324,271,304)(250,325,272,305)(251,326,273,306)(252,327,274,307)(337,428,367,400)(338,429,368,401)(339,430,369,402)(340,431,370,403)(341,432,371,404)(342,433,372,405)(343,434,373,406)(344,421,374,393)(345,422,375,394)(346,423,376,395)(347,424,377,396)(348,425,378,397)(349,426,365,398)(350,427,366,399)(351,443,380,409)(352,444,381,410)(353,445,382,411)(354,446,383,412)(355,447,384,413)(356,448,385,414)(357,435,386,415)(358,436,387,416)(359,437,388,417)(360,438,389,418)(361,439,390,419)(362,440,391,420)(363,441,392,407)(364,442,379,408) );
G=PermutationGroup([[(1,134,15,120),(2,135,16,121),(3,136,17,122),(4,137,18,123),(5,138,19,124),(6,139,20,125),(7,140,21,126),(8,127,22,113),(9,128,23,114),(10,129,24,115),(11,130,25,116),(12,131,26,117),(13,132,27,118),(14,133,28,119),(29,155,50,148),(30,156,51,149),(31,157,52,150),(32,158,53,151),(33,159,54,152),(34,160,55,153),(35,161,56,154),(36,162,43,141),(37,163,44,142),(38,164,45,143),(39,165,46,144),(40,166,47,145),(41,167,48,146),(42,168,49,147),(57,190,78,176),(58,191,79,177),(59,192,80,178),(60,193,81,179),(61,194,82,180),(62,195,83,181),(63,196,84,182),(64,183,71,169),(65,184,72,170),(66,185,73,171),(67,186,74,172),(68,187,75,173),(69,188,76,174),(70,189,77,175),(85,218,106,204),(86,219,107,205),(87,220,108,206),(88,221,109,207),(89,222,110,208),(90,223,111,209),(91,224,112,210),(92,211,99,197),(93,212,100,198),(94,213,101,199),(95,214,102,200),(96,215,103,201),(97,216,104,202),(98,217,105,203),(225,340,239,364),(226,341,240,351),(227,342,241,352),(228,343,242,353),(229,344,243,354),(230,345,244,355),(231,346,245,356),(232,347,246,357),(233,348,247,358),(234,349,248,359),(235,350,249,360),(236,337,250,361),(237,338,251,362),(238,339,252,363),(253,376,267,385),(254,377,268,386),(255,378,269,387),(256,365,270,388),(257,366,271,389),(258,367,272,390),(259,368,273,391),(260,369,274,392),(261,370,275,379),(262,371,276,380),(263,372,277,381),(264,373,278,382),(265,374,279,383),(266,375,280,384),(281,393,305,412),(282,394,306,413),(283,395,307,414),(284,396,308,415),(285,397,295,416),(286,398,296,417),(287,399,297,418),(288,400,298,419),(289,401,299,420),(290,402,300,407),(291,403,301,408),(292,404,302,409),(293,405,303,410),(294,406,304,411),(309,429,333,440),(310,430,334,441),(311,431,335,442),(312,432,336,443),(313,433,323,444),(314,434,324,445),(315,421,325,446),(316,422,326,447),(317,423,327,448),(318,424,328,435),(319,425,329,436),(320,426,330,437),(321,427,331,438),(322,428,332,439)], [(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)], [(1,227,8,234),(2,226,9,233),(3,225,10,232),(4,238,11,231),(5,237,12,230),(6,236,13,229),(7,235,14,228),(15,241,22,248),(16,240,23,247),(17,239,24,246),(18,252,25,245),(19,251,26,244),(20,250,27,243),(21,249,28,242),(29,263,36,256),(30,262,37,255),(31,261,38,254),(32,260,39,253),(33,259,40,266),(34,258,41,265),(35,257,42,264),(43,270,50,277),(44,269,51,276),(45,268,52,275),(46,267,53,274),(47,280,54,273),(48,279,55,272),(49,278,56,271),(57,293,64,286),(58,292,65,285),(59,291,66,284),(60,290,67,283),(61,289,68,282),(62,288,69,281),(63,287,70,294),(71,296,78,303),(72,295,79,302),(73,308,80,301),(74,307,81,300),(75,306,82,299),(76,305,83,298),(77,304,84,297),(85,320,92,313),(86,319,93,312),(87,318,94,311),(88,317,95,310),(89,316,96,309),(90,315,97,322),(91,314,98,321),(99,323,106,330),(100,336,107,329),(101,335,108,328),(102,334,109,327),(103,333,110,326),(104,332,111,325),(105,331,112,324),(113,349,120,342),(114,348,121,341),(115,347,122,340),(116,346,123,339),(117,345,124,338),(118,344,125,337),(119,343,126,350),(127,359,134,352),(128,358,135,351),(129,357,136,364),(130,356,137,363),(131,355,138,362),(132,354,139,361),(133,353,140,360),(141,365,148,372),(142,378,149,371),(143,377,150,370),(144,376,151,369),(145,375,152,368),(146,374,153,367),(147,373,154,366),(155,381,162,388),(156,380,163,387),(157,379,164,386),(158,392,165,385),(159,391,166,384),(160,390,167,383),(161,389,168,382),(169,398,176,405),(170,397,177,404),(171,396,178,403),(172,395,179,402),(173,394,180,401),(174,393,181,400),(175,406,182,399),(183,417,190,410),(184,416,191,409),(185,415,192,408),(186,414,193,407),(187,413,194,420),(188,412,195,419),(189,411,196,418),(197,433,204,426),(198,432,205,425),(199,431,206,424),(200,430,207,423),(201,429,208,422),(202,428,209,421),(203,427,210,434),(211,444,218,437),(212,443,219,436),(213,442,220,435),(214,441,221,448),(215,440,222,447),(216,439,223,446),(217,438,224,445)], [(1,85,29,64),(2,86,30,65),(3,87,31,66),(4,88,32,67),(5,89,33,68),(6,90,34,69),(7,91,35,70),(8,92,36,57),(9,93,37,58),(10,94,38,59),(11,95,39,60),(12,96,40,61),(13,97,41,62),(14,98,42,63),(15,106,50,71),(16,107,51,72),(17,108,52,73),(18,109,53,74),(19,110,54,75),(20,111,55,76),(21,112,56,77),(22,99,43,78),(23,100,44,79),(24,101,45,80),(25,102,46,81),(26,103,47,82),(27,104,48,83),(28,105,49,84),(113,204,141,169),(114,205,142,170),(115,206,143,171),(116,207,144,172),(117,208,145,173),(118,209,146,174),(119,210,147,175),(120,197,148,176),(121,198,149,177),(122,199,150,178),(123,200,151,179),(124,201,152,180),(125,202,153,181),(126,203,154,182),(127,218,162,183),(128,219,163,184),(129,220,164,185),(130,221,165,186),(131,222,166,187),(132,223,167,188),(133,224,168,189),(134,211,155,190),(135,212,156,191),(136,213,157,192),(137,214,158,193),(138,215,159,194),(139,216,160,195),(140,217,161,196),(225,318,261,284),(226,319,262,285),(227,320,263,286),(228,321,264,287),(229,322,265,288),(230,309,266,289),(231,310,253,290),(232,311,254,291),(233,312,255,292),(234,313,256,293),(235,314,257,294),(236,315,258,281),(237,316,259,282),(238,317,260,283),(239,328,275,308),(240,329,276,295),(241,330,277,296),(242,331,278,297),(243,332,279,298),(244,333,280,299),(245,334,267,300),(246,335,268,301),(247,336,269,302),(248,323,270,303),(249,324,271,304),(250,325,272,305),(251,326,273,306),(252,327,274,307),(337,428,367,400),(338,429,368,401),(339,430,369,402),(340,431,370,403),(341,432,371,404),(342,433,372,405),(343,434,373,406),(344,421,374,393),(345,422,375,394),(346,423,376,395),(347,424,377,396),(348,425,378,397),(349,426,365,398),(350,427,366,399),(351,443,380,409),(352,444,381,410),(353,445,382,411),(354,446,383,412),(355,447,384,413),(356,448,385,414),(357,435,386,415),(358,436,387,416),(359,437,388,417),(360,438,389,418),(361,439,390,419),(362,440,391,420),(363,441,392,407),(364,442,379,408)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | D7 | C4○D4 | D14 | C4×D7 | D28 | C4○D28 | D4⋊2D7 | Q8×D7 |
| kernel | C4⋊Dic7⋊7C4 | C14.C42 | C7×C2.C42 | C2×C4×Dic7 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C4⋊Dic7 | C2×Dic7 | C2×C28 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 3 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 3 | 8 | 9 | 12 | 12 | 12 | 9 | 3 |
Matrix representation of C4⋊Dic7⋊7C4 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 13 | 20 | 0 | 0 |
| 0 | 9 | 16 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 16 | 17 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 9 | 24 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 16 | 17 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,13,9,0,0,0,20,16,0,0,0,0,0,12,16,0,0,0,0,17],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,23,9,0,0,0,0,24],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,16,0,0,0,11,17],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12] >;
C4⋊Dic7⋊7C4 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_7\rtimes_7C_4 % in TeX
G:=Group("C4:Dic7:7C4"); // GroupNames label
G:=SmallGroup(448,187);
// by ID
G=gap.SmallGroup(448,187);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^14=d^4=1,c^2=b^7,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^7,c*b*c^-1=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations