metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7⋊C4⋊5C4, C14.21(C4×D4), C14.14(C4×Q8), C22.54(D4×D7), C22.13(Q8×D7), (C2×Dic7).11Q8, C2.2(D14⋊Q8), (C2×Dic7).127D4, (C22×C4).295D14, C2.7(C42⋊D7), C2.3(Dic7.Q8), C14.18(C22⋊Q8), (C22×C28).5C22, C14.7(C42.C2), C2.C42.3D7, C14.7(C42⋊2C2), C14.3(C42⋊C2), C2.6(Dic7⋊4D4), C2.5(Dic7⋊3Q8), C2.1(D14.D4), C22.29(C4○D28), C14.C42.3C2, C23.249(C22×D7), C22.31(D4⋊2D7), (C22×C14).277C23, C7⋊1(C23.63C23), C2.2(C23.D14), C14.1(C22.D4), (C22×Dic7).173C22, (C2×C4).59(C4×D7), C22.84(C2×C4×D7), (C2×C14).60(C2×Q8), (C2×C4×Dic7).25C2, (C2×C28).141(C2×C4), (C2×C14).191(C2×D4), (C2×Dic7⋊C4).4C2, (C2×Dic7).7(C2×C4), (C2×C14).55(C4○D4), (C2×C14).43(C22×C4), (C7×C2.C42).23C2, SmallGroup(448,186)
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, dbd-1=a7bc2, dcd-1=a7c >
Subgroups: 604 in 154 conjugacy classes, 67 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, Dic7⋊C4, C22×Dic7, C22×C28, C14.C42, C7×C2.C42, C2×C4×Dic7, C2×Dic7⋊C4, Dic7⋊C4⋊C4
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, C22×D7, C23.63C23, C2×C4×D7, C4○D28, D4×D7, D4⋊2D7, Q8×D7, C42⋊D7, C23.D14, Dic7⋊4D4, D14.D4, Dic7⋊3Q8, Dic7.Q8, D14⋊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)
(1 316 8 309)(2 315 9 322)(3 314 10 321)(4 313 11 320)(5 312 12 319)(6 311 13 318)(7 310 14 317)(15 74 22 81)(16 73 23 80)(17 72 24 79)(18 71 25 78)(19 84 26 77)(20 83 27 76)(21 82 28 75)(29 399 36 406)(30 398 37 405)(31 397 38 404)(32 396 39 403)(33 395 40 402)(34 394 41 401)(35 393 42 400)(43 287 50 294)(44 286 51 293)(45 285 52 292)(46 284 53 291)(47 283 54 290)(48 282 55 289)(49 281 56 288)(57 89 64 96)(58 88 65 95)(59 87 66 94)(60 86 67 93)(61 85 68 92)(62 98 69 91)(63 97 70 90)(99 350 106 343)(100 349 107 342)(101 348 108 341)(102 347 109 340)(103 346 110 339)(104 345 111 338)(105 344 112 337)(113 156 120 163)(114 155 121 162)(115 168 122 161)(116 167 123 160)(117 166 124 159)(118 165 125 158)(119 164 126 157)(127 386 134 379)(128 385 135 392)(129 384 136 391)(130 383 137 390)(131 382 138 389)(132 381 139 388)(133 380 140 387)(141 368 148 375)(142 367 149 374)(143 366 150 373)(144 365 151 372)(145 378 152 371)(146 377 153 370)(147 376 154 369)(169 185 176 192)(170 184 177 191)(171 183 178 190)(172 196 179 189)(173 195 180 188)(174 194 181 187)(175 193 182 186)(197 329 204 336)(198 328 205 335)(199 327 206 334)(200 326 207 333)(201 325 208 332)(202 324 209 331)(203 323 210 330)(211 409 218 416)(212 408 219 415)(213 407 220 414)(214 420 221 413)(215 419 222 412)(216 418 223 411)(217 417 224 410)(225 266 232 259)(226 265 233 258)(227 264 234 257)(228 263 235 256)(229 262 236 255)(230 261 237 254)(231 260 238 253)(239 433 246 426)(240 432 247 425)(241 431 248 424)(242 430 249 423)(243 429 250 422)(244 428 251 421)(245 427 252 434)(267 301 274 308)(268 300 275 307)(269 299 276 306)(270 298 277 305)(271 297 278 304)(272 296 279 303)(273 295 280 302)(351 442 358 435)(352 441 359 448)(353 440 360 447)(354 439 361 446)(355 438 362 445)(356 437 363 444)(357 436 364 443)
(1 21 42 366)(2 22 29 367)(3 23 30 368)(4 24 31 369)(5 25 32 370)(6 26 33 371)(7 27 34 372)(8 28 35 373)(9 15 36 374)(10 16 37 375)(11 17 38 376)(12 18 39 377)(13 19 40 378)(14 20 41 365)(43 233 129 110)(44 234 130 111)(45 235 131 112)(46 236 132 99)(47 237 133 100)(48 238 134 101)(49 225 135 102)(50 226 136 103)(51 227 137 104)(52 228 138 105)(53 229 139 106)(54 230 140 107)(55 231 127 108)(56 232 128 109)(57 197 242 272)(58 198 243 273)(59 199 244 274)(60 200 245 275)(61 201 246 276)(62 202 247 277)(63 203 248 278)(64 204 249 279)(65 205 250 280)(66 206 251 267)(67 207 252 268)(68 208 239 269)(69 209 240 270)(70 210 241 271)(71 396 153 312)(72 397 154 313)(73 398 141 314)(74 399 142 315)(75 400 143 316)(76 401 144 317)(77 402 145 318)(78 403 146 319)(79 404 147 320)(80 405 148 321)(81 406 149 322)(82 393 150 309)(83 394 151 310)(84 395 152 311)(85 332 426 299)(86 333 427 300)(87 334 428 301)(88 335 429 302)(89 336 430 303)(90 323 431 304)(91 324 432 305)(92 325 433 306)(93 326 434 307)(94 327 421 308)(95 328 422 295)(96 329 423 296)(97 330 424 297)(98 331 425 298)(113 414 441 175)(114 415 442 176)(115 416 443 177)(116 417 444 178)(117 418 445 179)(118 419 446 180)(119 420 447 181)(120 407 448 182)(121 408 435 169)(122 409 436 170)(123 410 437 171)(124 411 438 172)(125 412 439 173)(126 413 440 174)(155 219 358 185)(156 220 359 186)(157 221 360 187)(158 222 361 188)(159 223 362 189)(160 224 363 190)(161 211 364 191)(162 212 351 192)(163 213 352 193)(164 214 353 194)(165 215 354 195)(166 216 355 196)(167 217 356 183)(168 218 357 184)(253 386 348 289)(254 387 349 290)(255 388 350 291)(256 389 337 292)(257 390 338 293)(258 391 339 294)(259 392 340 281)(260 379 341 282)(261 380 342 283)(262 381 343 284)(263 382 344 285)(264 383 345 286)(265 384 346 287)(266 385 347 288)
(1 202 169 259)(2 203 170 260)(3 204 171 261)(4 205 172 262)(5 206 173 263)(6 207 174 264)(7 208 175 265)(8 209 176 266)(9 210 177 253)(10 197 178 254)(11 198 179 255)(12 199 180 256)(13 200 181 257)(14 201 182 258)(15 248 115 379)(16 249 116 380)(17 250 117 381)(18 251 118 382)(19 252 119 383)(20 239 120 384)(21 240 121 385)(22 241 122 386)(23 242 123 387)(24 243 124 388)(25 244 125 389)(26 245 126 390)(27 246 113 391)(28 247 114 392)(29 278 409 341)(30 279 410 342)(31 280 411 343)(32 267 412 344)(33 268 413 345)(34 269 414 346)(35 270 415 347)(36 271 416 348)(37 272 417 349)(38 273 418 350)(39 274 419 337)(40 275 420 338)(41 276 407 339)(42 277 408 340)(43 83 85 163)(44 84 86 164)(45 71 87 165)(46 72 88 166)(47 73 89 167)(48 74 90 168)(49 75 91 155)(50 76 92 156)(51 77 93 157)(52 78 94 158)(53 79 95 159)(54 80 96 160)(55 81 97 161)(56 82 98 162)(57 437 290 368)(58 438 291 369)(59 439 292 370)(60 440 293 371)(61 441 294 372)(62 442 281 373)(63 443 282 374)(64 444 283 375)(65 445 284 376)(66 446 285 377)(67 447 286 378)(68 448 287 365)(69 435 288 366)(70 436 289 367)(99 320 302 189)(100 321 303 190)(101 322 304 191)(102 309 305 192)(103 310 306 193)(104 311 307 194)(105 312 308 195)(106 313 295 196)(107 314 296 183)(108 315 297 184)(109 316 298 185)(110 317 299 186)(111 318 300 187)(112 319 301 188)(127 149 424 364)(128 150 425 351)(129 151 426 352)(130 152 427 353)(131 153 428 354)(132 154 429 355)(133 141 430 356)(134 142 431 357)(135 143 432 358)(136 144 433 359)(137 145 434 360)(138 146 421 361)(139 147 422 362)(140 148 423 363)(211 238 406 323)(212 225 393 324)(213 226 394 325)(214 227 395 326)(215 228 396 327)(216 229 397 328)(217 230 398 329)(218 231 399 330)(219 232 400 331)(220 233 401 332)(221 234 402 333)(222 235 403 334)(223 236 404 335)(224 237 405 336)
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), (1,316,8,309)(2,315,9,322)(3,314,10,321)(4,313,11,320)(5,312,12,319)(6,311,13,318)(7,310,14,317)(15,74,22,81)(16,73,23,80)(17,72,24,79)(18,71,25,78)(19,84,26,77)(20,83,27,76)(21,82,28,75)(29,399,36,406)(30,398,37,405)(31,397,38,404)(32,396,39,403)(33,395,40,402)(34,394,41,401)(35,393,42,400)(43,287,50,294)(44,286,51,293)(45,285,52,292)(46,284,53,291)(47,283,54,290)(48,282,55,289)(49,281,56,288)(57,89,64,96)(58,88,65,95)(59,87,66,94)(60,86,67,93)(61,85,68,92)(62,98,69,91)(63,97,70,90)(99,350,106,343)(100,349,107,342)(101,348,108,341)(102,347,109,340)(103,346,110,339)(104,345,111,338)(105,344,112,337)(113,156,120,163)(114,155,121,162)(115,168,122,161)(116,167,123,160)(117,166,124,159)(118,165,125,158)(119,164,126,157)(127,386,134,379)(128,385,135,392)(129,384,136,391)(130,383,137,390)(131,382,138,389)(132,381,139,388)(133,380,140,387)(141,368,148,375)(142,367,149,374)(143,366,150,373)(144,365,151,372)(145,378,152,371)(146,377,153,370)(147,376,154,369)(169,185,176,192)(170,184,177,191)(171,183,178,190)(172,196,179,189)(173,195,180,188)(174,194,181,187)(175,193,182,186)(197,329,204,336)(198,328,205,335)(199,327,206,334)(200,326,207,333)(201,325,208,332)(202,324,209,331)(203,323,210,330)(211,409,218,416)(212,408,219,415)(213,407,220,414)(214,420,221,413)(215,419,222,412)(216,418,223,411)(217,417,224,410)(225,266,232,259)(226,265,233,258)(227,264,234,257)(228,263,235,256)(229,262,236,255)(230,261,237,254)(231,260,238,253)(239,433,246,426)(240,432,247,425)(241,431,248,424)(242,430,249,423)(243,429,250,422)(244,428,251,421)(245,427,252,434)(267,301,274,308)(268,300,275,307)(269,299,276,306)(270,298,277,305)(271,297,278,304)(272,296,279,303)(273,295,280,302)(351,442,358,435)(352,441,359,448)(353,440,360,447)(354,439,361,446)(355,438,362,445)(356,437,363,444)(357,436,364,443), (1,21,42,366)(2,22,29,367)(3,23,30,368)(4,24,31,369)(5,25,32,370)(6,26,33,371)(7,27,34,372)(8,28,35,373)(9,15,36,374)(10,16,37,375)(11,17,38,376)(12,18,39,377)(13,19,40,378)(14,20,41,365)(43,233,129,110)(44,234,130,111)(45,235,131,112)(46,236,132,99)(47,237,133,100)(48,238,134,101)(49,225,135,102)(50,226,136,103)(51,227,137,104)(52,228,138,105)(53,229,139,106)(54,230,140,107)(55,231,127,108)(56,232,128,109)(57,197,242,272)(58,198,243,273)(59,199,244,274)(60,200,245,275)(61,201,246,276)(62,202,247,277)(63,203,248,278)(64,204,249,279)(65,205,250,280)(66,206,251,267)(67,207,252,268)(68,208,239,269)(69,209,240,270)(70,210,241,271)(71,396,153,312)(72,397,154,313)(73,398,141,314)(74,399,142,315)(75,400,143,316)(76,401,144,317)(77,402,145,318)(78,403,146,319)(79,404,147,320)(80,405,148,321)(81,406,149,322)(82,393,150,309)(83,394,151,310)(84,395,152,311)(85,332,426,299)(86,333,427,300)(87,334,428,301)(88,335,429,302)(89,336,430,303)(90,323,431,304)(91,324,432,305)(92,325,433,306)(93,326,434,307)(94,327,421,308)(95,328,422,295)(96,329,423,296)(97,330,424,297)(98,331,425,298)(113,414,441,175)(114,415,442,176)(115,416,443,177)(116,417,444,178)(117,418,445,179)(118,419,446,180)(119,420,447,181)(120,407,448,182)(121,408,435,169)(122,409,436,170)(123,410,437,171)(124,411,438,172)(125,412,439,173)(126,413,440,174)(155,219,358,185)(156,220,359,186)(157,221,360,187)(158,222,361,188)(159,223,362,189)(160,224,363,190)(161,211,364,191)(162,212,351,192)(163,213,352,193)(164,214,353,194)(165,215,354,195)(166,216,355,196)(167,217,356,183)(168,218,357,184)(253,386,348,289)(254,387,349,290)(255,388,350,291)(256,389,337,292)(257,390,338,293)(258,391,339,294)(259,392,340,281)(260,379,341,282)(261,380,342,283)(262,381,343,284)(263,382,344,285)(264,383,345,286)(265,384,346,287)(266,385,347,288), (1,202,169,259)(2,203,170,260)(3,204,171,261)(4,205,172,262)(5,206,173,263)(6,207,174,264)(7,208,175,265)(8,209,176,266)(9,210,177,253)(10,197,178,254)(11,198,179,255)(12,199,180,256)(13,200,181,257)(14,201,182,258)(15,248,115,379)(16,249,116,380)(17,250,117,381)(18,251,118,382)(19,252,119,383)(20,239,120,384)(21,240,121,385)(22,241,122,386)(23,242,123,387)(24,243,124,388)(25,244,125,389)(26,245,126,390)(27,246,113,391)(28,247,114,392)(29,278,409,341)(30,279,410,342)(31,280,411,343)(32,267,412,344)(33,268,413,345)(34,269,414,346)(35,270,415,347)(36,271,416,348)(37,272,417,349)(38,273,418,350)(39,274,419,337)(40,275,420,338)(41,276,407,339)(42,277,408,340)(43,83,85,163)(44,84,86,164)(45,71,87,165)(46,72,88,166)(47,73,89,167)(48,74,90,168)(49,75,91,155)(50,76,92,156)(51,77,93,157)(52,78,94,158)(53,79,95,159)(54,80,96,160)(55,81,97,161)(56,82,98,162)(57,437,290,368)(58,438,291,369)(59,439,292,370)(60,440,293,371)(61,441,294,372)(62,442,281,373)(63,443,282,374)(64,444,283,375)(65,445,284,376)(66,446,285,377)(67,447,286,378)(68,448,287,365)(69,435,288,366)(70,436,289,367)(99,320,302,189)(100,321,303,190)(101,322,304,191)(102,309,305,192)(103,310,306,193)(104,311,307,194)(105,312,308,195)(106,313,295,196)(107,314,296,183)(108,315,297,184)(109,316,298,185)(110,317,299,186)(111,318,300,187)(112,319,301,188)(127,149,424,364)(128,150,425,351)(129,151,426,352)(130,152,427,353)(131,153,428,354)(132,154,429,355)(133,141,430,356)(134,142,431,357)(135,143,432,358)(136,144,433,359)(137,145,434,360)(138,146,421,361)(139,147,422,362)(140,148,423,363)(211,238,406,323)(212,225,393,324)(213,226,394,325)(214,227,395,326)(215,228,396,327)(216,229,397,328)(217,230,398,329)(218,231,399,330)(219,232,400,331)(220,233,401,332)(221,234,402,333)(222,235,403,334)(223,236,404,335)(224,237,405,336)>;
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), (1,316,8,309)(2,315,9,322)(3,314,10,321)(4,313,11,320)(5,312,12,319)(6,311,13,318)(7,310,14,317)(15,74,22,81)(16,73,23,80)(17,72,24,79)(18,71,25,78)(19,84,26,77)(20,83,27,76)(21,82,28,75)(29,399,36,406)(30,398,37,405)(31,397,38,404)(32,396,39,403)(33,395,40,402)(34,394,41,401)(35,393,42,400)(43,287,50,294)(44,286,51,293)(45,285,52,292)(46,284,53,291)(47,283,54,290)(48,282,55,289)(49,281,56,288)(57,89,64,96)(58,88,65,95)(59,87,66,94)(60,86,67,93)(61,85,68,92)(62,98,69,91)(63,97,70,90)(99,350,106,343)(100,349,107,342)(101,348,108,341)(102,347,109,340)(103,346,110,339)(104,345,111,338)(105,344,112,337)(113,156,120,163)(114,155,121,162)(115,168,122,161)(116,167,123,160)(117,166,124,159)(118,165,125,158)(119,164,126,157)(127,386,134,379)(128,385,135,392)(129,384,136,391)(130,383,137,390)(131,382,138,389)(132,381,139,388)(133,380,140,387)(141,368,148,375)(142,367,149,374)(143,366,150,373)(144,365,151,372)(145,378,152,371)(146,377,153,370)(147,376,154,369)(169,185,176,192)(170,184,177,191)(171,183,178,190)(172,196,179,189)(173,195,180,188)(174,194,181,187)(175,193,182,186)(197,329,204,336)(198,328,205,335)(199,327,206,334)(200,326,207,333)(201,325,208,332)(202,324,209,331)(203,323,210,330)(211,409,218,416)(212,408,219,415)(213,407,220,414)(214,420,221,413)(215,419,222,412)(216,418,223,411)(217,417,224,410)(225,266,232,259)(226,265,233,258)(227,264,234,257)(228,263,235,256)(229,262,236,255)(230,261,237,254)(231,260,238,253)(239,433,246,426)(240,432,247,425)(241,431,248,424)(242,430,249,423)(243,429,250,422)(244,428,251,421)(245,427,252,434)(267,301,274,308)(268,300,275,307)(269,299,276,306)(270,298,277,305)(271,297,278,304)(272,296,279,303)(273,295,280,302)(351,442,358,435)(352,441,359,448)(353,440,360,447)(354,439,361,446)(355,438,362,445)(356,437,363,444)(357,436,364,443), (1,21,42,366)(2,22,29,367)(3,23,30,368)(4,24,31,369)(5,25,32,370)(6,26,33,371)(7,27,34,372)(8,28,35,373)(9,15,36,374)(10,16,37,375)(11,17,38,376)(12,18,39,377)(13,19,40,378)(14,20,41,365)(43,233,129,110)(44,234,130,111)(45,235,131,112)(46,236,132,99)(47,237,133,100)(48,238,134,101)(49,225,135,102)(50,226,136,103)(51,227,137,104)(52,228,138,105)(53,229,139,106)(54,230,140,107)(55,231,127,108)(56,232,128,109)(57,197,242,272)(58,198,243,273)(59,199,244,274)(60,200,245,275)(61,201,246,276)(62,202,247,277)(63,203,248,278)(64,204,249,279)(65,205,250,280)(66,206,251,267)(67,207,252,268)(68,208,239,269)(69,209,240,270)(70,210,241,271)(71,396,153,312)(72,397,154,313)(73,398,141,314)(74,399,142,315)(75,400,143,316)(76,401,144,317)(77,402,145,318)(78,403,146,319)(79,404,147,320)(80,405,148,321)(81,406,149,322)(82,393,150,309)(83,394,151,310)(84,395,152,311)(85,332,426,299)(86,333,427,300)(87,334,428,301)(88,335,429,302)(89,336,430,303)(90,323,431,304)(91,324,432,305)(92,325,433,306)(93,326,434,307)(94,327,421,308)(95,328,422,295)(96,329,423,296)(97,330,424,297)(98,331,425,298)(113,414,441,175)(114,415,442,176)(115,416,443,177)(116,417,444,178)(117,418,445,179)(118,419,446,180)(119,420,447,181)(120,407,448,182)(121,408,435,169)(122,409,436,170)(123,410,437,171)(124,411,438,172)(125,412,439,173)(126,413,440,174)(155,219,358,185)(156,220,359,186)(157,221,360,187)(158,222,361,188)(159,223,362,189)(160,224,363,190)(161,211,364,191)(162,212,351,192)(163,213,352,193)(164,214,353,194)(165,215,354,195)(166,216,355,196)(167,217,356,183)(168,218,357,184)(253,386,348,289)(254,387,349,290)(255,388,350,291)(256,389,337,292)(257,390,338,293)(258,391,339,294)(259,392,340,281)(260,379,341,282)(261,380,342,283)(262,381,343,284)(263,382,344,285)(264,383,345,286)(265,384,346,287)(266,385,347,288), (1,202,169,259)(2,203,170,260)(3,204,171,261)(4,205,172,262)(5,206,173,263)(6,207,174,264)(7,208,175,265)(8,209,176,266)(9,210,177,253)(10,197,178,254)(11,198,179,255)(12,199,180,256)(13,200,181,257)(14,201,182,258)(15,248,115,379)(16,249,116,380)(17,250,117,381)(18,251,118,382)(19,252,119,383)(20,239,120,384)(21,240,121,385)(22,241,122,386)(23,242,123,387)(24,243,124,388)(25,244,125,389)(26,245,126,390)(27,246,113,391)(28,247,114,392)(29,278,409,341)(30,279,410,342)(31,280,411,343)(32,267,412,344)(33,268,413,345)(34,269,414,346)(35,270,415,347)(36,271,416,348)(37,272,417,349)(38,273,418,350)(39,274,419,337)(40,275,420,338)(41,276,407,339)(42,277,408,340)(43,83,85,163)(44,84,86,164)(45,71,87,165)(46,72,88,166)(47,73,89,167)(48,74,90,168)(49,75,91,155)(50,76,92,156)(51,77,93,157)(52,78,94,158)(53,79,95,159)(54,80,96,160)(55,81,97,161)(56,82,98,162)(57,437,290,368)(58,438,291,369)(59,439,292,370)(60,440,293,371)(61,441,294,372)(62,442,281,373)(63,443,282,374)(64,444,283,375)(65,445,284,376)(66,446,285,377)(67,447,286,378)(68,448,287,365)(69,435,288,366)(70,436,289,367)(99,320,302,189)(100,321,303,190)(101,322,304,191)(102,309,305,192)(103,310,306,193)(104,311,307,194)(105,312,308,195)(106,313,295,196)(107,314,296,183)(108,315,297,184)(109,316,298,185)(110,317,299,186)(111,318,300,187)(112,319,301,188)(127,149,424,364)(128,150,425,351)(129,151,426,352)(130,152,427,353)(131,153,428,354)(132,154,429,355)(133,141,430,356)(134,142,431,357)(135,143,432,358)(136,144,433,359)(137,145,434,360)(138,146,421,361)(139,147,422,362)(140,148,423,363)(211,238,406,323)(212,225,393,324)(213,226,394,325)(214,227,395,326)(215,228,396,327)(216,229,397,328)(217,230,398,329)(218,231,399,330)(219,232,400,331)(220,233,401,332)(221,234,402,333)(222,235,403,334)(223,236,404,335)(224,237,405,336) );
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)], [(1,316,8,309),(2,315,9,322),(3,314,10,321),(4,313,11,320),(5,312,12,319),(6,311,13,318),(7,310,14,317),(15,74,22,81),(16,73,23,80),(17,72,24,79),(18,71,25,78),(19,84,26,77),(20,83,27,76),(21,82,28,75),(29,399,36,406),(30,398,37,405),(31,397,38,404),(32,396,39,403),(33,395,40,402),(34,394,41,401),(35,393,42,400),(43,287,50,294),(44,286,51,293),(45,285,52,292),(46,284,53,291),(47,283,54,290),(48,282,55,289),(49,281,56,288),(57,89,64,96),(58,88,65,95),(59,87,66,94),(60,86,67,93),(61,85,68,92),(62,98,69,91),(63,97,70,90),(99,350,106,343),(100,349,107,342),(101,348,108,341),(102,347,109,340),(103,346,110,339),(104,345,111,338),(105,344,112,337),(113,156,120,163),(114,155,121,162),(115,168,122,161),(116,167,123,160),(117,166,124,159),(118,165,125,158),(119,164,126,157),(127,386,134,379),(128,385,135,392),(129,384,136,391),(130,383,137,390),(131,382,138,389),(132,381,139,388),(133,380,140,387),(141,368,148,375),(142,367,149,374),(143,366,150,373),(144,365,151,372),(145,378,152,371),(146,377,153,370),(147,376,154,369),(169,185,176,192),(170,184,177,191),(171,183,178,190),(172,196,179,189),(173,195,180,188),(174,194,181,187),(175,193,182,186),(197,329,204,336),(198,328,205,335),(199,327,206,334),(200,326,207,333),(201,325,208,332),(202,324,209,331),(203,323,210,330),(211,409,218,416),(212,408,219,415),(213,407,220,414),(214,420,221,413),(215,419,222,412),(216,418,223,411),(217,417,224,410),(225,266,232,259),(226,265,233,258),(227,264,234,257),(228,263,235,256),(229,262,236,255),(230,261,237,254),(231,260,238,253),(239,433,246,426),(240,432,247,425),(241,431,248,424),(242,430,249,423),(243,429,250,422),(244,428,251,421),(245,427,252,434),(267,301,274,308),(268,300,275,307),(269,299,276,306),(270,298,277,305),(271,297,278,304),(272,296,279,303),(273,295,280,302),(351,442,358,435),(352,441,359,448),(353,440,360,447),(354,439,361,446),(355,438,362,445),(356,437,363,444),(357,436,364,443)], [(1,21,42,366),(2,22,29,367),(3,23,30,368),(4,24,31,369),(5,25,32,370),(6,26,33,371),(7,27,34,372),(8,28,35,373),(9,15,36,374),(10,16,37,375),(11,17,38,376),(12,18,39,377),(13,19,40,378),(14,20,41,365),(43,233,129,110),(44,234,130,111),(45,235,131,112),(46,236,132,99),(47,237,133,100),(48,238,134,101),(49,225,135,102),(50,226,136,103),(51,227,137,104),(52,228,138,105),(53,229,139,106),(54,230,140,107),(55,231,127,108),(56,232,128,109),(57,197,242,272),(58,198,243,273),(59,199,244,274),(60,200,245,275),(61,201,246,276),(62,202,247,277),(63,203,248,278),(64,204,249,279),(65,205,250,280),(66,206,251,267),(67,207,252,268),(68,208,239,269),(69,209,240,270),(70,210,241,271),(71,396,153,312),(72,397,154,313),(73,398,141,314),(74,399,142,315),(75,400,143,316),(76,401,144,317),(77,402,145,318),(78,403,146,319),(79,404,147,320),(80,405,148,321),(81,406,149,322),(82,393,150,309),(83,394,151,310),(84,395,152,311),(85,332,426,299),(86,333,427,300),(87,334,428,301),(88,335,429,302),(89,336,430,303),(90,323,431,304),(91,324,432,305),(92,325,433,306),(93,326,434,307),(94,327,421,308),(95,328,422,295),(96,329,423,296),(97,330,424,297),(98,331,425,298),(113,414,441,175),(114,415,442,176),(115,416,443,177),(116,417,444,178),(117,418,445,179),(118,419,446,180),(119,420,447,181),(120,407,448,182),(121,408,435,169),(122,409,436,170),(123,410,437,171),(124,411,438,172),(125,412,439,173),(126,413,440,174),(155,219,358,185),(156,220,359,186),(157,221,360,187),(158,222,361,188),(159,223,362,189),(160,224,363,190),(161,211,364,191),(162,212,351,192),(163,213,352,193),(164,214,353,194),(165,215,354,195),(166,216,355,196),(167,217,356,183),(168,218,357,184),(253,386,348,289),(254,387,349,290),(255,388,350,291),(256,389,337,292),(257,390,338,293),(258,391,339,294),(259,392,340,281),(260,379,341,282),(261,380,342,283),(262,381,343,284),(263,382,344,285),(264,383,345,286),(265,384,346,287),(266,385,347,288)], [(1,202,169,259),(2,203,170,260),(3,204,171,261),(4,205,172,262),(5,206,173,263),(6,207,174,264),(7,208,175,265),(8,209,176,266),(9,210,177,253),(10,197,178,254),(11,198,179,255),(12,199,180,256),(13,200,181,257),(14,201,182,258),(15,248,115,379),(16,249,116,380),(17,250,117,381),(18,251,118,382),(19,252,119,383),(20,239,120,384),(21,240,121,385),(22,241,122,386),(23,242,123,387),(24,243,124,388),(25,244,125,389),(26,245,126,390),(27,246,113,391),(28,247,114,392),(29,278,409,341),(30,279,410,342),(31,280,411,343),(32,267,412,344),(33,268,413,345),(34,269,414,346),(35,270,415,347),(36,271,416,348),(37,272,417,349),(38,273,418,350),(39,274,419,337),(40,275,420,338),(41,276,407,339),(42,277,408,340),(43,83,85,163),(44,84,86,164),(45,71,87,165),(46,72,88,166),(47,73,89,167),(48,74,90,168),(49,75,91,155),(50,76,92,156),(51,77,93,157),(52,78,94,158),(53,79,95,159),(54,80,96,160),(55,81,97,161),(56,82,98,162),(57,437,290,368),(58,438,291,369),(59,439,292,370),(60,440,293,371),(61,441,294,372),(62,442,281,373),(63,443,282,374),(64,444,283,375),(65,445,284,376),(66,446,285,377),(67,447,286,378),(68,448,287,365),(69,435,288,366),(70,436,289,367),(99,320,302,189),(100,321,303,190),(101,322,304,191),(102,309,305,192),(103,310,306,193),(104,311,307,194),(105,312,308,195),(106,313,295,196),(107,314,296,183),(108,315,297,184),(109,316,298,185),(110,317,299,186),(111,318,300,187),(112,319,301,188),(127,149,424,364),(128,150,425,351),(129,151,426,352),(130,152,427,353),(131,153,428,354),(132,154,429,355),(133,141,430,356),(134,142,431,357),(135,143,432,358),(136,144,433,359),(137,145,434,360),(138,146,421,361),(139,147,422,362),(140,148,423,363),(211,238,406,323),(212,225,393,324),(213,226,394,325),(214,227,395,326),(215,228,396,327),(216,229,397,328),(217,230,398,329),(218,231,399,330),(219,232,400,331),(220,233,401,332),(221,234,402,333),(222,235,403,334),(223,236,404,335),(224,237,405,336)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D7 | C4○D4 | D14 | C4×D7 | C4○D28 | D4×D7 | D4⋊2D7 | Q8×D7 |
| kernel | Dic7⋊C4⋊C4 | C14.C42 | C7×C2.C42 | C2×C4×Dic7 | C2×Dic7⋊C4 | Dic7⋊C4 | C2×Dic7 | C2×Dic7 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 1 | 1 | 2 | 8 | 2 | 2 | 3 | 8 | 9 | 12 | 24 | 3 | 6 | 3 |
Matrix representation of Dic7⋊C4⋊C4 ►in GL6(𝔽29)
| 18 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 2 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 11 | 2 | 0 | 0 | 0 | 0 |
| 27 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 27 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [18,1,0,0,0,0,28,0,0,0,0,0,0,0,18,1,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,11,28,0,0,0,0,0,0,1,0,0,0,0,0,11,28,0,0,0,0,0,0,17,1,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,1,0,0,0,0,2,12],[11,27,0,0,0,0,2,18,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,27,17] >;
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(448,186);
// by ID
G=gap.SmallGroup(448,186);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,64,254,219,184,18822]);
// 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,d*b*d^-1=a^7*b*c^2,d*c*d^-1=a^7*c>;
// generators/relations