G = Dic7⋊(C4⋊C4) order 448 = 26·7
2nd semidirect product of Dic7 and C4⋊C4 acting via C4⋊C4/C2×C4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊C4⋊6C4, C14.66(C4×D4), C14.19(C4×Q8), Dic7⋊2(C4⋊C4), (C2×C28).249D4, C14.24(C4⋊Q8), C22.18(Q8×D7), (C2×Dic7).14Q8, C22.104(D4×D7), (C22×C4).37D14, C14.87(C4⋊D4), C2.5(D14⋊Q8), (C2×Dic7).175D4, C2.5(Dic7.Q8), C14.43(C22⋊Q8), C2.9(Dic7⋊3Q8), C2.5(Dic7⋊D4), C2.1(Dic7⋊Q8), C22.55(C4○D28), C14.15(C42.C2), C23.286(C22×D7), C22.51(D4⋊2D7), C14.C42.14C2, (C22×C14).336C23, (C22×C28).345C22, C7⋊3(C23.65C23), (C22×Dic7).48C22, C2.19(D7×C4⋊C4), C14.18(C2×C4⋊C4), (C2×C4⋊C4).10D7, (C2×C4).40(C4×D7), C2.11(C4×C7⋊D4), (C14×C4⋊C4).26C2, (C2×C14).72(C2×Q8), (C2×C4×Dic7).34C2, C22.130(C2×C4×D7), (C2×C28).183(C2×C4), (C2×C14).445(C2×D4), (C2×C4).99(C7⋊D4), C22.60(C2×C7⋊D4), (C2×Dic7⋊C4).27C2, (C2×Dic7).61(C2×C4), (C2×C14).149(C4○D4), (C2×C14).112(C22×C4), SmallGroup(448,506)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊(C4⋊C4)
G = < a,b,c,d | a14=c4=d4=1, b2=a7, bab-1=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd-1=c-1 >
Subgroups: 644 in 170 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C22×C4, C22×C4, Dic7, Dic7, C28, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.65C23, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, C14.C42, C2×C4×Dic7, C2×Dic7⋊C4, C14×C4⋊C4, Dic7⋊(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D14, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C4×D7, C7⋊D4, C22×D7, C23.65C23, C2×C4×D7, C4○D28, D4×D7, D4⋊2D7, Q8×D7, C2×C7⋊D4, Dic7⋊3Q8, Dic7.Q8, D7×C4⋊C4, D14⋊Q8, C4×C7⋊D4, Dic7⋊D4, Dic7⋊Q8, Dic7⋊(C4⋊C4)
►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)
(1 274 8 267)(2 273 9 280)(3 272 10 279)(4 271 11 278)(5 270 12 277)(6 269 13 276)(7 268 14 275)(15 255 22 262)(16 254 23 261)(17 253 24 260)(18 266 25 259)(19 265 26 258)(20 264 27 257)(21 263 28 256)(29 239 36 246)(30 252 37 245)(31 251 38 244)(32 250 39 243)(33 249 40 242)(34 248 41 241)(35 247 42 240)(43 231 50 238)(44 230 51 237)(45 229 52 236)(46 228 53 235)(47 227 54 234)(48 226 55 233)(49 225 56 232)(57 329 64 336)(58 328 65 335)(59 327 66 334)(60 326 67 333)(61 325 68 332)(62 324 69 331)(63 323 70 330)(71 310 78 317)(72 309 79 316)(73 322 80 315)(74 321 81 314)(75 320 82 313)(76 319 83 312)(77 318 84 311)(85 307 92 300)(86 306 93 299)(87 305 94 298)(88 304 95 297)(89 303 96 296)(90 302 97 295)(91 301 98 308)(99 291 106 284)(100 290 107 283)(101 289 108 282)(102 288 109 281)(103 287 110 294)(104 286 111 293)(105 285 112 292)(113 386 120 379)(114 385 121 392)(115 384 122 391)(116 383 123 390)(117 382 124 389)(118 381 125 388)(119 380 126 387)(127 367 134 374)(128 366 135 373)(129 365 136 372)(130 378 137 371)(131 377 138 370)(132 376 139 369)(133 375 140 368)(141 355 148 362)(142 354 149 361)(143 353 150 360)(144 352 151 359)(145 351 152 358)(146 364 153 357)(147 363 154 356)(155 345 162 338)(156 344 163 337)(157 343 164 350)(158 342 165 349)(159 341 166 348)(160 340 167 347)(161 339 168 346)(169 441 176 448)(170 440 177 447)(171 439 178 446)(172 438 179 445)(173 437 180 444)(174 436 181 443)(175 435 182 442)(183 422 190 429)(184 421 191 428)(185 434 192 427)(186 433 193 426)(187 432 194 425)(188 431 195 424)(189 430 196 423)(197 417 204 410)(198 416 205 409)(199 415 206 408)(200 414 207 407)(201 413 208 420)(202 412 209 419)(203 411 210 418)(211 401 218 394)(212 400 219 393)(213 399 220 406)(214 398 221 405)(215 397 222 404)(216 396 223 403)(217 395 224 402)
(1 163 24 145)(2 164 25 146)(3 165 26 147)(4 166 27 148)(5 167 28 149)(6 168 15 150)(7 155 16 151)(8 156 17 152)(9 157 18 153)(10 158 19 154)(11 159 20 141)(12 160 21 142)(13 161 22 143)(14 162 23 144)(29 136 56 113)(30 137 43 114)(31 138 44 115)(32 139 45 116)(33 140 46 117)(34 127 47 118)(35 128 48 119)(36 129 49 120)(37 130 50 121)(38 131 51 122)(39 132 52 123)(40 133 53 124)(41 134 54 125)(42 135 55 126)(57 220 80 201)(58 221 81 202)(59 222 82 203)(60 223 83 204)(61 224 84 205)(62 211 71 206)(63 212 72 207)(64 213 73 208)(65 214 74 209)(66 215 75 210)(67 216 76 197)(68 217 77 198)(69 218 78 199)(70 219 79 200)(85 193 104 170)(86 194 105 171)(87 195 106 172)(88 196 107 173)(89 183 108 174)(90 184 109 175)(91 185 110 176)(92 186 111 177)(93 187 112 178)(94 188 99 179)(95 189 100 180)(96 190 101 181)(97 191 102 182)(98 192 103 169)(225 386 246 372)(226 387 247 373)(227 388 248 374)(228 389 249 375)(229 390 250 376)(230 391 251 377)(231 392 252 378)(232 379 239 365)(233 380 240 366)(234 381 241 367)(235 382 242 368)(236 383 243 369)(237 384 244 370)(238 385 245 371)(253 351 267 337)(254 352 268 338)(255 353 269 339)(256 354 270 340)(257 355 271 341)(258 356 272 342)(259 357 273 343)(260 358 274 344)(261 359 275 345)(262 360 276 346)(263 361 277 347)(264 362 278 348)(265 363 279 349)(266 364 280 350)(281 442 302 428)(282 443 303 429)(283 444 304 430)(284 445 305 431)(285 446 306 432)(286 447 307 433)(287 448 308 434)(288 435 295 421)(289 436 296 422)(290 437 297 423)(291 438 298 424)(292 439 299 425)(293 440 300 426)(294 441 301 427)(309 414 323 393)(310 415 324 394)(311 416 325 395)(312 417 326 396)(313 418 327 397)(314 419 328 398)(315 420 329 399)(316 407 330 400)(317 408 331 401)(318 409 332 402)(319 410 333 403)(320 411 334 404)(321 412 335 405)(322 413 336 406)
(1 97 29 70)(2 98 30 57)(3 85 31 58)(4 86 32 59)(5 87 33 60)(6 88 34 61)(7 89 35 62)(8 90 36 63)(9 91 37 64)(10 92 38 65)(11 93 39 66)(12 94 40 67)(13 95 41 68)(14 96 42 69)(15 107 47 84)(16 108 48 71)(17 109 49 72)(18 110 50 73)(19 111 51 74)(20 112 52 75)(21 99 53 76)(22 100 54 77)(23 101 55 78)(24 102 56 79)(25 103 43 80)(26 104 44 81)(27 105 45 82)(28 106 46 83)(113 219 145 191)(114 220 146 192)(115 221 147 193)(116 222 148 194)(117 223 149 195)(118 224 150 196)(119 211 151 183)(120 212 152 184)(121 213 153 185)(122 214 154 186)(123 215 141 187)(124 216 142 188)(125 217 143 189)(126 218 144 190)(127 205 168 173)(128 206 155 174)(129 207 156 175)(130 208 157 176)(131 209 158 177)(132 210 159 178)(133 197 160 179)(134 198 161 180)(135 199 162 181)(136 200 163 182)(137 201 164 169)(138 202 165 170)(139 203 166 171)(140 204 167 172)(225 309 253 281)(226 310 254 282)(227 311 255 283)(228 312 256 284)(229 313 257 285)(230 314 258 286)(231 315 259 287)(232 316 260 288)(233 317 261 289)(234 318 262 290)(235 319 263 291)(236 320 264 292)(237 321 265 293)(238 322 266 294)(239 330 274 295)(240 331 275 296)(241 332 276 297)(242 333 277 298)(243 334 278 299)(244 335 279 300)(245 336 280 301)(246 323 267 302)(247 324 268 303)(248 325 269 304)(249 326 270 305)(250 327 271 306)(251 328 272 307)(252 329 273 308)(337 442 372 414)(338 443 373 415)(339 444 374 416)(340 445 375 417)(341 446 376 418)(342 447 377 419)(343 448 378 420)(344 435 365 407)(345 436 366 408)(346 437 367 409)(347 438 368 410)(348 439 369 411)(349 440 370 412)(350 441 371 413)(351 428 386 393)(352 429 387 394)(353 430 388 395)(354 431 389 396)(355 432 390 397)(356 433 391 398)(357 434 392 399)(358 421 379 400)(359 422 380 401)(360 423 381 402)(361 424 382 403)(362 425 383 404)(363 426 384 405)(364 427 385 406)
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), (1,274,8,267)(2,273,9,280)(3,272,10,279)(4,271,11,278)(5,270,12,277)(6,269,13,276)(7,268,14,275)(15,255,22,262)(16,254,23,261)(17,253,24,260)(18,266,25,259)(19,265,26,258)(20,264,27,257)(21,263,28,256)(29,239,36,246)(30,252,37,245)(31,251,38,244)(32,250,39,243)(33,249,40,242)(34,248,41,241)(35,247,42,240)(43,231,50,238)(44,230,51,237)(45,229,52,236)(46,228,53,235)(47,227,54,234)(48,226,55,233)(49,225,56,232)(57,329,64,336)(58,328,65,335)(59,327,66,334)(60,326,67,333)(61,325,68,332)(62,324,69,331)(63,323,70,330)(71,310,78,317)(72,309,79,316)(73,322,80,315)(74,321,81,314)(75,320,82,313)(76,319,83,312)(77,318,84,311)(85,307,92,300)(86,306,93,299)(87,305,94,298)(88,304,95,297)(89,303,96,296)(90,302,97,295)(91,301,98,308)(99,291,106,284)(100,290,107,283)(101,289,108,282)(102,288,109,281)(103,287,110,294)(104,286,111,293)(105,285,112,292)(113,386,120,379)(114,385,121,392)(115,384,122,391)(116,383,123,390)(117,382,124,389)(118,381,125,388)(119,380,126,387)(127,367,134,374)(128,366,135,373)(129,365,136,372)(130,378,137,371)(131,377,138,370)(132,376,139,369)(133,375,140,368)(141,355,148,362)(142,354,149,361)(143,353,150,360)(144,352,151,359)(145,351,152,358)(146,364,153,357)(147,363,154,356)(155,345,162,338)(156,344,163,337)(157,343,164,350)(158,342,165,349)(159,341,166,348)(160,340,167,347)(161,339,168,346)(169,441,176,448)(170,440,177,447)(171,439,178,446)(172,438,179,445)(173,437,180,444)(174,436,181,443)(175,435,182,442)(183,422,190,429)(184,421,191,428)(185,434,192,427)(186,433,193,426)(187,432,194,425)(188,431,195,424)(189,430,196,423)(197,417,204,410)(198,416,205,409)(199,415,206,408)(200,414,207,407)(201,413,208,420)(202,412,209,419)(203,411,210,418)(211,401,218,394)(212,400,219,393)(213,399,220,406)(214,398,221,405)(215,397,222,404)(216,396,223,403)(217,395,224,402), (1,163,24,145)(2,164,25,146)(3,165,26,147)(4,166,27,148)(5,167,28,149)(6,168,15,150)(7,155,16,151)(8,156,17,152)(9,157,18,153)(10,158,19,154)(11,159,20,141)(12,160,21,142)(13,161,22,143)(14,162,23,144)(29,136,56,113)(30,137,43,114)(31,138,44,115)(32,139,45,116)(33,140,46,117)(34,127,47,118)(35,128,48,119)(36,129,49,120)(37,130,50,121)(38,131,51,122)(39,132,52,123)(40,133,53,124)(41,134,54,125)(42,135,55,126)(57,220,80,201)(58,221,81,202)(59,222,82,203)(60,223,83,204)(61,224,84,205)(62,211,71,206)(63,212,72,207)(64,213,73,208)(65,214,74,209)(66,215,75,210)(67,216,76,197)(68,217,77,198)(69,218,78,199)(70,219,79,200)(85,193,104,170)(86,194,105,171)(87,195,106,172)(88,196,107,173)(89,183,108,174)(90,184,109,175)(91,185,110,176)(92,186,111,177)(93,187,112,178)(94,188,99,179)(95,189,100,180)(96,190,101,181)(97,191,102,182)(98,192,103,169)(225,386,246,372)(226,387,247,373)(227,388,248,374)(228,389,249,375)(229,390,250,376)(230,391,251,377)(231,392,252,378)(232,379,239,365)(233,380,240,366)(234,381,241,367)(235,382,242,368)(236,383,243,369)(237,384,244,370)(238,385,245,371)(253,351,267,337)(254,352,268,338)(255,353,269,339)(256,354,270,340)(257,355,271,341)(258,356,272,342)(259,357,273,343)(260,358,274,344)(261,359,275,345)(262,360,276,346)(263,361,277,347)(264,362,278,348)(265,363,279,349)(266,364,280,350)(281,442,302,428)(282,443,303,429)(283,444,304,430)(284,445,305,431)(285,446,306,432)(286,447,307,433)(287,448,308,434)(288,435,295,421)(289,436,296,422)(290,437,297,423)(291,438,298,424)(292,439,299,425)(293,440,300,426)(294,441,301,427)(309,414,323,393)(310,415,324,394)(311,416,325,395)(312,417,326,396)(313,418,327,397)(314,419,328,398)(315,420,329,399)(316,407,330,400)(317,408,331,401)(318,409,332,402)(319,410,333,403)(320,411,334,404)(321,412,335,405)(322,413,336,406), (1,97,29,70)(2,98,30,57)(3,85,31,58)(4,86,32,59)(5,87,33,60)(6,88,34,61)(7,89,35,62)(8,90,36,63)(9,91,37,64)(10,92,38,65)(11,93,39,66)(12,94,40,67)(13,95,41,68)(14,96,42,69)(15,107,47,84)(16,108,48,71)(17,109,49,72)(18,110,50,73)(19,111,51,74)(20,112,52,75)(21,99,53,76)(22,100,54,77)(23,101,55,78)(24,102,56,79)(25,103,43,80)(26,104,44,81)(27,105,45,82)(28,106,46,83)(113,219,145,191)(114,220,146,192)(115,221,147,193)(116,222,148,194)(117,223,149,195)(118,224,150,196)(119,211,151,183)(120,212,152,184)(121,213,153,185)(122,214,154,186)(123,215,141,187)(124,216,142,188)(125,217,143,189)(126,218,144,190)(127,205,168,173)(128,206,155,174)(129,207,156,175)(130,208,157,176)(131,209,158,177)(132,210,159,178)(133,197,160,179)(134,198,161,180)(135,199,162,181)(136,200,163,182)(137,201,164,169)(138,202,165,170)(139,203,166,171)(140,204,167,172)(225,309,253,281)(226,310,254,282)(227,311,255,283)(228,312,256,284)(229,313,257,285)(230,314,258,286)(231,315,259,287)(232,316,260,288)(233,317,261,289)(234,318,262,290)(235,319,263,291)(236,320,264,292)(237,321,265,293)(238,322,266,294)(239,330,274,295)(240,331,275,296)(241,332,276,297)(242,333,277,298)(243,334,278,299)(244,335,279,300)(245,336,280,301)(246,323,267,302)(247,324,268,303)(248,325,269,304)(249,326,270,305)(250,327,271,306)(251,328,272,307)(252,329,273,308)(337,442,372,414)(338,443,373,415)(339,444,374,416)(340,445,375,417)(341,446,376,418)(342,447,377,419)(343,448,378,420)(344,435,365,407)(345,436,366,408)(346,437,367,409)(347,438,368,410)(348,439,369,411)(349,440,370,412)(350,441,371,413)(351,428,386,393)(352,429,387,394)(353,430,388,395)(354,431,389,396)(355,432,390,397)(356,433,391,398)(357,434,392,399)(358,421,379,400)(359,422,380,401)(360,423,381,402)(361,424,382,403)(362,425,383,404)(363,426,384,405)(364,427,385,406)>;
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), (1,274,8,267)(2,273,9,280)(3,272,10,279)(4,271,11,278)(5,270,12,277)(6,269,13,276)(7,268,14,275)(15,255,22,262)(16,254,23,261)(17,253,24,260)(18,266,25,259)(19,265,26,258)(20,264,27,257)(21,263,28,256)(29,239,36,246)(30,252,37,245)(31,251,38,244)(32,250,39,243)(33,249,40,242)(34,248,41,241)(35,247,42,240)(43,231,50,238)(44,230,51,237)(45,229,52,236)(46,228,53,235)(47,227,54,234)(48,226,55,233)(49,225,56,232)(57,329,64,336)(58,328,65,335)(59,327,66,334)(60,326,67,333)(61,325,68,332)(62,324,69,331)(63,323,70,330)(71,310,78,317)(72,309,79,316)(73,322,80,315)(74,321,81,314)(75,320,82,313)(76,319,83,312)(77,318,84,311)(85,307,92,300)(86,306,93,299)(87,305,94,298)(88,304,95,297)(89,303,96,296)(90,302,97,295)(91,301,98,308)(99,291,106,284)(100,290,107,283)(101,289,108,282)(102,288,109,281)(103,287,110,294)(104,286,111,293)(105,285,112,292)(113,386,120,379)(114,385,121,392)(115,384,122,391)(116,383,123,390)(117,382,124,389)(118,381,125,388)(119,380,126,387)(127,367,134,374)(128,366,135,373)(129,365,136,372)(130,378,137,371)(131,377,138,370)(132,376,139,369)(133,375,140,368)(141,355,148,362)(142,354,149,361)(143,353,150,360)(144,352,151,359)(145,351,152,358)(146,364,153,357)(147,363,154,356)(155,345,162,338)(156,344,163,337)(157,343,164,350)(158,342,165,349)(159,341,166,348)(160,340,167,347)(161,339,168,346)(169,441,176,448)(170,440,177,447)(171,439,178,446)(172,438,179,445)(173,437,180,444)(174,436,181,443)(175,435,182,442)(183,422,190,429)(184,421,191,428)(185,434,192,427)(186,433,193,426)(187,432,194,425)(188,431,195,424)(189,430,196,423)(197,417,204,410)(198,416,205,409)(199,415,206,408)(200,414,207,407)(201,413,208,420)(202,412,209,419)(203,411,210,418)(211,401,218,394)(212,400,219,393)(213,399,220,406)(214,398,221,405)(215,397,222,404)(216,396,223,403)(217,395,224,402), (1,163,24,145)(2,164,25,146)(3,165,26,147)(4,166,27,148)(5,167,28,149)(6,168,15,150)(7,155,16,151)(8,156,17,152)(9,157,18,153)(10,158,19,154)(11,159,20,141)(12,160,21,142)(13,161,22,143)(14,162,23,144)(29,136,56,113)(30,137,43,114)(31,138,44,115)(32,139,45,116)(33,140,46,117)(34,127,47,118)(35,128,48,119)(36,129,49,120)(37,130,50,121)(38,131,51,122)(39,132,52,123)(40,133,53,124)(41,134,54,125)(42,135,55,126)(57,220,80,201)(58,221,81,202)(59,222,82,203)(60,223,83,204)(61,224,84,205)(62,211,71,206)(63,212,72,207)(64,213,73,208)(65,214,74,209)(66,215,75,210)(67,216,76,197)(68,217,77,198)(69,218,78,199)(70,219,79,200)(85,193,104,170)(86,194,105,171)(87,195,106,172)(88,196,107,173)(89,183,108,174)(90,184,109,175)(91,185,110,176)(92,186,111,177)(93,187,112,178)(94,188,99,179)(95,189,100,180)(96,190,101,181)(97,191,102,182)(98,192,103,169)(225,386,246,372)(226,387,247,373)(227,388,248,374)(228,389,249,375)(229,390,250,376)(230,391,251,377)(231,392,252,378)(232,379,239,365)(233,380,240,366)(234,381,241,367)(235,382,242,368)(236,383,243,369)(237,384,244,370)(238,385,245,371)(253,351,267,337)(254,352,268,338)(255,353,269,339)(256,354,270,340)(257,355,271,341)(258,356,272,342)(259,357,273,343)(260,358,274,344)(261,359,275,345)(262,360,276,346)(263,361,277,347)(264,362,278,348)(265,363,279,349)(266,364,280,350)(281,442,302,428)(282,443,303,429)(283,444,304,430)(284,445,305,431)(285,446,306,432)(286,447,307,433)(287,448,308,434)(288,435,295,421)(289,436,296,422)(290,437,297,423)(291,438,298,424)(292,439,299,425)(293,440,300,426)(294,441,301,427)(309,414,323,393)(310,415,324,394)(311,416,325,395)(312,417,326,396)(313,418,327,397)(314,419,328,398)(315,420,329,399)(316,407,330,400)(317,408,331,401)(318,409,332,402)(319,410,333,403)(320,411,334,404)(321,412,335,405)(322,413,336,406), (1,97,29,70)(2,98,30,57)(3,85,31,58)(4,86,32,59)(5,87,33,60)(6,88,34,61)(7,89,35,62)(8,90,36,63)(9,91,37,64)(10,92,38,65)(11,93,39,66)(12,94,40,67)(13,95,41,68)(14,96,42,69)(15,107,47,84)(16,108,48,71)(17,109,49,72)(18,110,50,73)(19,111,51,74)(20,112,52,75)(21,99,53,76)(22,100,54,77)(23,101,55,78)(24,102,56,79)(25,103,43,80)(26,104,44,81)(27,105,45,82)(28,106,46,83)(113,219,145,191)(114,220,146,192)(115,221,147,193)(116,222,148,194)(117,223,149,195)(118,224,150,196)(119,211,151,183)(120,212,152,184)(121,213,153,185)(122,214,154,186)(123,215,141,187)(124,216,142,188)(125,217,143,189)(126,218,144,190)(127,205,168,173)(128,206,155,174)(129,207,156,175)(130,208,157,176)(131,209,158,177)(132,210,159,178)(133,197,160,179)(134,198,161,180)(135,199,162,181)(136,200,163,182)(137,201,164,169)(138,202,165,170)(139,203,166,171)(140,204,167,172)(225,309,253,281)(226,310,254,282)(227,311,255,283)(228,312,256,284)(229,313,257,285)(230,314,258,286)(231,315,259,287)(232,316,260,288)(233,317,261,289)(234,318,262,290)(235,319,263,291)(236,320,264,292)(237,321,265,293)(238,322,266,294)(239,330,274,295)(240,331,275,296)(241,332,276,297)(242,333,277,298)(243,334,278,299)(244,335,279,300)(245,336,280,301)(246,323,267,302)(247,324,268,303)(248,325,269,304)(249,326,270,305)(250,327,271,306)(251,328,272,307)(252,329,273,308)(337,442,372,414)(338,443,373,415)(339,444,374,416)(340,445,375,417)(341,446,376,418)(342,447,377,419)(343,448,378,420)(344,435,365,407)(345,436,366,408)(346,437,367,409)(347,438,368,410)(348,439,369,411)(349,440,370,412)(350,441,371,413)(351,428,386,393)(352,429,387,394)(353,430,388,395)(354,431,389,396)(355,432,390,397)(356,433,391,398)(357,434,392,399)(358,421,379,400)(359,422,380,401)(360,423,381,402)(361,424,382,403)(362,425,383,404)(363,426,384,405)(364,427,385,406) );
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)], [(1,274,8,267),(2,273,9,280),(3,272,10,279),(4,271,11,278),(5,270,12,277),(6,269,13,276),(7,268,14,275),(15,255,22,262),(16,254,23,261),(17,253,24,260),(18,266,25,259),(19,265,26,258),(20,264,27,257),(21,263,28,256),(29,239,36,246),(30,252,37,245),(31,251,38,244),(32,250,39,243),(33,249,40,242),(34,248,41,241),(35,247,42,240),(43,231,50,238),(44,230,51,237),(45,229,52,236),(46,228,53,235),(47,227,54,234),(48,226,55,233),(49,225,56,232),(57,329,64,336),(58,328,65,335),(59,327,66,334),(60,326,67,333),(61,325,68,332),(62,324,69,331),(63,323,70,330),(71,310,78,317),(72,309,79,316),(73,322,80,315),(74,321,81,314),(75,320,82,313),(76,319,83,312),(77,318,84,311),(85,307,92,300),(86,306,93,299),(87,305,94,298),(88,304,95,297),(89,303,96,296),(90,302,97,295),(91,301,98,308),(99,291,106,284),(100,290,107,283),(101,289,108,282),(102,288,109,281),(103,287,110,294),(104,286,111,293),(105,285,112,292),(113,386,120,379),(114,385,121,392),(115,384,122,391),(116,383,123,390),(117,382,124,389),(118,381,125,388),(119,380,126,387),(127,367,134,374),(128,366,135,373),(129,365,136,372),(130,378,137,371),(131,377,138,370),(132,376,139,369),(133,375,140,368),(141,355,148,362),(142,354,149,361),(143,353,150,360),(144,352,151,359),(145,351,152,358),(146,364,153,357),(147,363,154,356),(155,345,162,338),(156,344,163,337),(157,343,164,350),(158,342,165,349),(159,341,166,348),(160,340,167,347),(161,339,168,346),(169,441,176,448),(170,440,177,447),(171,439,178,446),(172,438,179,445),(173,437,180,444),(174,436,181,443),(175,435,182,442),(183,422,190,429),(184,421,191,428),(185,434,192,427),(186,433,193,426),(187,432,194,425),(188,431,195,424),(189,430,196,423),(197,417,204,410),(198,416,205,409),(199,415,206,408),(200,414,207,407),(201,413,208,420),(202,412,209,419),(203,411,210,418),(211,401,218,394),(212,400,219,393),(213,399,220,406),(214,398,221,405),(215,397,222,404),(216,396,223,403),(217,395,224,402)], [(1,163,24,145),(2,164,25,146),(3,165,26,147),(4,166,27,148),(5,167,28,149),(6,168,15,150),(7,155,16,151),(8,156,17,152),(9,157,18,153),(10,158,19,154),(11,159,20,141),(12,160,21,142),(13,161,22,143),(14,162,23,144),(29,136,56,113),(30,137,43,114),(31,138,44,115),(32,139,45,116),(33,140,46,117),(34,127,47,118),(35,128,48,119),(36,129,49,120),(37,130,50,121),(38,131,51,122),(39,132,52,123),(40,133,53,124),(41,134,54,125),(42,135,55,126),(57,220,80,201),(58,221,81,202),(59,222,82,203),(60,223,83,204),(61,224,84,205),(62,211,71,206),(63,212,72,207),(64,213,73,208),(65,214,74,209),(66,215,75,210),(67,216,76,197),(68,217,77,198),(69,218,78,199),(70,219,79,200),(85,193,104,170),(86,194,105,171),(87,195,106,172),(88,196,107,173),(89,183,108,174),(90,184,109,175),(91,185,110,176),(92,186,111,177),(93,187,112,178),(94,188,99,179),(95,189,100,180),(96,190,101,181),(97,191,102,182),(98,192,103,169),(225,386,246,372),(226,387,247,373),(227,388,248,374),(228,389,249,375),(229,390,250,376),(230,391,251,377),(231,392,252,378),(232,379,239,365),(233,380,240,366),(234,381,241,367),(235,382,242,368),(236,383,243,369),(237,384,244,370),(238,385,245,371),(253,351,267,337),(254,352,268,338),(255,353,269,339),(256,354,270,340),(257,355,271,341),(258,356,272,342),(259,357,273,343),(260,358,274,344),(261,359,275,345),(262,360,276,346),(263,361,277,347),(264,362,278,348),(265,363,279,349),(266,364,280,350),(281,442,302,428),(282,443,303,429),(283,444,304,430),(284,445,305,431),(285,446,306,432),(286,447,307,433),(287,448,308,434),(288,435,295,421),(289,436,296,422),(290,437,297,423),(291,438,298,424),(292,439,299,425),(293,440,300,426),(294,441,301,427),(309,414,323,393),(310,415,324,394),(311,416,325,395),(312,417,326,396),(313,418,327,397),(314,419,328,398),(315,420,329,399),(316,407,330,400),(317,408,331,401),(318,409,332,402),(319,410,333,403),(320,411,334,404),(321,412,335,405),(322,413,336,406)], [(1,97,29,70),(2,98,30,57),(3,85,31,58),(4,86,32,59),(5,87,33,60),(6,88,34,61),(7,89,35,62),(8,90,36,63),(9,91,37,64),(10,92,38,65),(11,93,39,66),(12,94,40,67),(13,95,41,68),(14,96,42,69),(15,107,47,84),(16,108,48,71),(17,109,49,72),(18,110,50,73),(19,111,51,74),(20,112,52,75),(21,99,53,76),(22,100,54,77),(23,101,55,78),(24,102,56,79),(25,103,43,80),(26,104,44,81),(27,105,45,82),(28,106,46,83),(113,219,145,191),(114,220,146,192),(115,221,147,193),(116,222,148,194),(117,223,149,195),(118,224,150,196),(119,211,151,183),(120,212,152,184),(121,213,153,185),(122,214,154,186),(123,215,141,187),(124,216,142,188),(125,217,143,189),(126,218,144,190),(127,205,168,173),(128,206,155,174),(129,207,156,175),(130,208,157,176),(131,209,158,177),(132,210,159,178),(133,197,160,179),(134,198,161,180),(135,199,162,181),(136,200,163,182),(137,201,164,169),(138,202,165,170),(139,203,166,171),(140,204,167,172),(225,309,253,281),(226,310,254,282),(227,311,255,283),(228,312,256,284),(229,313,257,285),(230,314,258,286),(231,315,259,287),(232,316,260,288),(233,317,261,289),(234,318,262,290),(235,319,263,291),(236,320,264,292),(237,321,265,293),(238,322,266,294),(239,330,274,295),(240,331,275,296),(241,332,276,297),(242,333,277,298),(243,334,278,299),(244,335,279,300),(245,336,280,301),(246,323,267,302),(247,324,268,303),(248,325,269,304),(249,326,270,305),(250,327,271,306),(251,328,272,307),(252,329,273,308),(337,442,372,414),(338,443,373,415),(339,444,374,416),(340,445,375,417),(341,446,376,418),(342,447,377,419),(343,448,378,420),(344,435,365,407),(345,436,366,408),(346,437,367,409),(347,438,368,410),(348,439,369,411),(349,440,370,412),(350,441,371,413),(351,428,386,393),(352,429,387,394),(353,430,388,395),(354,431,389,396),(355,432,390,397),(356,433,391,398),(357,434,392,399),(358,421,379,400),(359,422,380,401),(360,423,381,402),(361,424,382,403),(362,425,383,404),(363,426,384,405),(364,427,385,406)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D7 | C4○D4 | D14 | C4×D7 | C7⋊D4 | C4○D28 | D4×D7 | D4⋊2D7 | Q8×D7 |
| kernel | Dic7⋊(C4⋊C4) | C14.C42 | C2×C4×Dic7 | C2×Dic7⋊C4 | C14×C4⋊C4 | Dic7⋊C4 | C2×Dic7 | C2×Dic7 | C2×C28 | C2×C4⋊C4 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 8 | 2 | 4 | 2 | 3 | 4 | 9 | 12 | 12 | 12 | 3 | 3 | 6 |
Matrix representation of Dic7⋊(C4⋊C4) ►in GL6(𝔽29)
| 3 | 28 | 0 | 0 | 0 | 0 |
| 19 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 28 | 0 | 0 |
| 0 | 0 | 11 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 9 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 12 | 0 | 0 |
| 0 | 0 | 22 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 11 |
| 0 | 0 | 0 | 0 | 4 | 10 |
| 15 | 16 | 0 | 0 | 0 | 0 |
| 15 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 16 | 0 | 0 |
| 0 | 0 | 27 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 2 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 13 |
| 0 | 0 | 0 | 0 | 10 | 25 |
G:=sub<GL(6,GF(29))| [3,19,0,0,0,0,28,23,0,0,0,0,0,0,24,11,0,0,0,0,28,2,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[11,9,0,0,0,0,9,18,0,0,0,0,0,0,24,22,0,0,0,0,12,5,0,0,0,0,0,0,19,4,0,0,0,0,11,10],[15,15,0,0,0,0,16,14,0,0,0,0,0,0,27,27,0,0,0,0,16,2,0,0,0,0,0,0,12,0,0,0,0,0,2,17],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,4,10,0,0,0,0,13,25] >;
Dic7⋊(C4⋊C4) in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes (C_4\rtimes C_4) % in TeX
G:=Group("Dic7:(C4:C4)"); // GroupNames label
G:=SmallGroup(448,506);
// by ID
G=gap.SmallGroup(448,506);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,477,120,219,58,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^4=d^4=1,b^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations