metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C4×Dic7)⋊9C4, C28.17(C4⋊C4), (C2×C28).19Q8, C7⋊3(C42⋊8C4), (C2×C28).139D4, (C2×C4).30Dic14, (C22×C4).39D14, C4.17(Dic7⋊C4), C2.4(C28.3Q8), C2.1(C28.23D4), C2.3(C28.17D4), C14.49(C4.4D4), C14.17(C42.C2), C22.28(C2×Dic14), C23.291(C22×D7), C14.35(C42⋊C2), C22.55(D4⋊2D7), C14.C42.17C2, (C22×C14).341C23, (C22×C28).142C22, C22.23(Q8⋊2D7), (C22×Dic7).189C22, C14.37(C2×C4⋊C4), (C2×C4⋊C4).14D7, (C14×C4⋊C4).13C2, (C2×C4×Dic7).7C2, (C2×C28).84(C2×C4), (C2×C4).153(C4×D7), (C2×C14).36(C2×Q8), C22.134(C2×C4×D7), (C2×C14).446(C2×D4), (C2×C4⋊Dic7).34C2, C2.12(C2×Dic7⋊C4), C22.64(C2×C7⋊D4), C2.11(C4⋊C4⋊7D7), (C2×C4).128(C7⋊D4), (C2×Dic7).95(C2×C4), (C2×C14).152(C4○D4), (C2×C14).117(C22×C4), SmallGroup(448,511)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C4×Dic7)⋊9C4
G = < a,b,c,d | a4=b14=d4=1, c2=b7, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2b7c >
Subgroups: 580 in 154 conjugacy classes, 79 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C42⋊8C4, C4×Dic7, C4⋊Dic7, C7×C4⋊C4, C22×Dic7, C22×C28, C22×C28, C14.C42, C2×C4×Dic7, C2×C4⋊Dic7, C14×C4⋊C4, (C4×Dic7)⋊9C4
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, C42⋊C2, C4.4D4, C42.C2, Dic14, C4×D7, C7⋊D4, C22×D7, C42⋊8C4, Dic7⋊C4, C2×Dic14, C2×C4×D7, D4⋊2D7, Q8⋊2D7, C2×C7⋊D4, C28.3Q8, C4⋊C4⋊7D7, C2×Dic7⋊C4, C28.17D4, C28.23D4, (C4×Dic7)⋊9C4
►Regular action on 448 points
Generators in S448
(1 419 126 374)(2 420 113 375)(3 407 114 376)(4 408 115 377)(5 409 116 378)(6 410 117 365)(7 411 118 366)(8 412 119 367)(9 413 120 368)(10 414 121 369)(11 415 122 370)(12 416 123 371)(13 417 124 372)(14 418 125 373)(15 273 46 249)(16 274 47 250)(17 275 48 251)(18 276 49 252)(19 277 50 239)(20 278 51 240)(21 279 52 241)(22 280 53 242)(23 267 54 243)(24 268 55 244)(25 269 56 245)(26 270 43 246)(27 271 44 247)(28 272 45 248)(29 334 60 295)(30 335 61 296)(31 336 62 297)(32 323 63 298)(33 324 64 299)(34 325 65 300)(35 326 66 301)(36 327 67 302)(37 328 68 303)(38 329 69 304)(39 330 70 305)(40 331 57 306)(41 332 58 307)(42 333 59 308)(71 183 321 439)(72 184 322 440)(73 185 309 441)(74 186 310 442)(75 187 311 443)(76 188 312 444)(77 189 313 445)(78 190 314 446)(79 191 315 447)(80 192 316 448)(81 193 317 435)(82 194 318 436)(83 195 319 437)(84 196 320 438)(85 265 361 397)(86 266 362 398)(87 253 363 399)(88 254 364 400)(89 255 351 401)(90 256 352 402)(91 257 353 403)(92 258 354 404)(93 259 355 405)(94 260 356 406)(95 261 357 393)(96 262 358 394)(97 263 359 395)(98 264 360 396)(99 286 229 434)(100 287 230 421)(101 288 231 422)(102 289 232 423)(103 290 233 424)(104 291 234 425)(105 292 235 426)(106 293 236 427)(107 294 237 428)(108 281 238 429)(109 282 225 430)(110 283 226 431)(111 284 227 432)(112 285 228 433)(127 159 203 383)(128 160 204 384)(129 161 205 385)(130 162 206 386)(131 163 207 387)(132 164 208 388)(133 165 209 389)(134 166 210 390)(135 167 197 391)(136 168 198 392)(137 155 199 379)(138 156 200 380)(139 157 201 381)(140 158 202 382)(141 339 214 180)(142 340 215 181)(143 341 216 182)(144 342 217 169)(145 343 218 170)(146 344 219 171)(147 345 220 172)(148 346 221 173)(149 347 222 174)(150 348 223 175)(151 349 224 176)(152 350 211 177)(153 337 212 178)(154 338 213 179)
(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)
(1 441 8 448)(2 440 9 447)(3 439 10 446)(4 438 11 445)(5 437 12 444)(6 436 13 443)(7 435 14 442)(15 210 22 203)(16 209 23 202)(17 208 24 201)(18 207 25 200)(19 206 26 199)(20 205 27 198)(21 204 28 197)(29 294 36 287)(30 293 37 286)(31 292 38 285)(32 291 39 284)(33 290 40 283)(34 289 41 282)(35 288 42 281)(43 137 50 130)(44 136 51 129)(45 135 52 128)(46 134 53 127)(47 133 54 140)(48 132 55 139)(49 131 56 138)(57 431 64 424)(58 430 65 423)(59 429 66 422)(60 428 67 421)(61 427 68 434)(62 426 69 433)(63 425 70 432)(71 414 78 407)(72 413 79 420)(73 412 80 419)(74 411 81 418)(75 410 82 417)(76 409 83 416)(77 408 84 415)(85 341 92 348)(86 340 93 347)(87 339 94 346)(88 338 95 345)(89 337 96 344)(90 350 97 343)(91 349 98 342)(99 296 106 303)(100 295 107 302)(101 308 108 301)(102 307 109 300)(103 306 110 299)(104 305 111 298)(105 304 112 297)(113 184 120 191)(114 183 121 190)(115 196 122 189)(116 195 123 188)(117 194 124 187)(118 193 125 186)(119 192 126 185)(141 406 148 399)(142 405 149 398)(143 404 150 397)(144 403 151 396)(145 402 152 395)(146 401 153 394)(147 400 154 393)(155 239 162 246)(156 252 163 245)(157 251 164 244)(158 250 165 243)(159 249 166 242)(160 248 167 241)(161 247 168 240)(169 353 176 360)(170 352 177 359)(171 351 178 358)(172 364 179 357)(173 363 180 356)(174 362 181 355)(175 361 182 354)(211 263 218 256)(212 262 219 255)(213 261 220 254)(214 260 221 253)(215 259 222 266)(216 258 223 265)(217 257 224 264)(225 325 232 332)(226 324 233 331)(227 323 234 330)(228 336 235 329)(229 335 236 328)(230 334 237 327)(231 333 238 326)(267 382 274 389)(268 381 275 388)(269 380 276 387)(270 379 277 386)(271 392 278 385)(272 391 279 384)(273 390 280 383)(309 367 316 374)(310 366 317 373)(311 365 318 372)(312 378 319 371)(313 377 320 370)(314 376 321 369)(315 375 322 368)
(1 328 278 345)(2 329 279 346)(3 330 280 347)(4 331 267 348)(5 332 268 349)(6 333 269 350)(7 334 270 337)(8 335 271 338)(9 336 272 339)(10 323 273 340)(11 324 274 341)(12 325 275 342)(13 326 276 343)(14 327 277 344)(15 215 369 63)(16 216 370 64)(17 217 371 65)(18 218 372 66)(19 219 373 67)(20 220 374 68)(21 221 375 69)(22 222 376 70)(23 223 377 57)(24 224 378 58)(25 211 365 59)(26 212 366 60)(27 213 367 61)(28 214 368 62)(29 43 153 411)(30 44 154 412)(31 45 141 413)(32 46 142 414)(33 47 143 415)(34 48 144 416)(35 49 145 417)(36 50 146 418)(37 51 147 419)(38 52 148 420)(39 53 149 407)(40 54 150 408)(41 55 151 409)(42 56 152 410)(71 425 127 259)(72 426 128 260)(73 427 129 261)(74 428 130 262)(75 429 131 263)(76 430 132 264)(77 431 133 265)(78 432 134 266)(79 433 135 253)(80 434 136 254)(81 421 137 255)(82 422 138 256)(83 423 139 257)(84 424 140 258)(85 189 226 165)(86 190 227 166)(87 191 228 167)(88 192 229 168)(89 193 230 155)(90 194 231 156)(91 195 232 157)(92 196 233 158)(93 183 234 159)(94 184 235 160)(95 185 236 161)(96 186 237 162)(97 187 238 163)(98 188 225 164)(99 392 364 448)(100 379 351 435)(101 380 352 436)(102 381 353 437)(103 382 354 438)(104 383 355 439)(105 384 356 440)(106 385 357 441)(107 386 358 442)(108 387 359 443)(109 388 360 444)(110 389 361 445)(111 390 362 446)(112 391 363 447)(113 304 241 173)(114 305 242 174)(115 306 243 175)(116 307 244 176)(117 308 245 177)(118 295 246 178)(119 296 247 179)(120 297 248 180)(121 298 249 181)(122 299 250 182)(123 300 251 169)(124 301 252 170)(125 302 239 171)(126 303 240 172)(197 399 315 285)(198 400 316 286)(199 401 317 287)(200 402 318 288)(201 403 319 289)(202 404 320 290)(203 405 321 291)(204 406 322 292)(205 393 309 293)(206 394 310 294)(207 395 311 281)(208 396 312 282)(209 397 313 283)(210 398 314 284)
G:=sub<Sym(448)| (1,419,126,374)(2,420,113,375)(3,407,114,376)(4,408,115,377)(5,409,116,378)(6,410,117,365)(7,411,118,366)(8,412,119,367)(9,413,120,368)(10,414,121,369)(11,415,122,370)(12,416,123,371)(13,417,124,372)(14,418,125,373)(15,273,46,249)(16,274,47,250)(17,275,48,251)(18,276,49,252)(19,277,50,239)(20,278,51,240)(21,279,52,241)(22,280,53,242)(23,267,54,243)(24,268,55,244)(25,269,56,245)(26,270,43,246)(27,271,44,247)(28,272,45,248)(29,334,60,295)(30,335,61,296)(31,336,62,297)(32,323,63,298)(33,324,64,299)(34,325,65,300)(35,326,66,301)(36,327,67,302)(37,328,68,303)(38,329,69,304)(39,330,70,305)(40,331,57,306)(41,332,58,307)(42,333,59,308)(71,183,321,439)(72,184,322,440)(73,185,309,441)(74,186,310,442)(75,187,311,443)(76,188,312,444)(77,189,313,445)(78,190,314,446)(79,191,315,447)(80,192,316,448)(81,193,317,435)(82,194,318,436)(83,195,319,437)(84,196,320,438)(85,265,361,397)(86,266,362,398)(87,253,363,399)(88,254,364,400)(89,255,351,401)(90,256,352,402)(91,257,353,403)(92,258,354,404)(93,259,355,405)(94,260,356,406)(95,261,357,393)(96,262,358,394)(97,263,359,395)(98,264,360,396)(99,286,229,434)(100,287,230,421)(101,288,231,422)(102,289,232,423)(103,290,233,424)(104,291,234,425)(105,292,235,426)(106,293,236,427)(107,294,237,428)(108,281,238,429)(109,282,225,430)(110,283,226,431)(111,284,227,432)(112,285,228,433)(127,159,203,383)(128,160,204,384)(129,161,205,385)(130,162,206,386)(131,163,207,387)(132,164,208,388)(133,165,209,389)(134,166,210,390)(135,167,197,391)(136,168,198,392)(137,155,199,379)(138,156,200,380)(139,157,201,381)(140,158,202,382)(141,339,214,180)(142,340,215,181)(143,341,216,182)(144,342,217,169)(145,343,218,170)(146,344,219,171)(147,345,220,172)(148,346,221,173)(149,347,222,174)(150,348,223,175)(151,349,224,176)(152,350,211,177)(153,337,212,178)(154,338,213,179), (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), (1,441,8,448)(2,440,9,447)(3,439,10,446)(4,438,11,445)(5,437,12,444)(6,436,13,443)(7,435,14,442)(15,210,22,203)(16,209,23,202)(17,208,24,201)(18,207,25,200)(19,206,26,199)(20,205,27,198)(21,204,28,197)(29,294,36,287)(30,293,37,286)(31,292,38,285)(32,291,39,284)(33,290,40,283)(34,289,41,282)(35,288,42,281)(43,137,50,130)(44,136,51,129)(45,135,52,128)(46,134,53,127)(47,133,54,140)(48,132,55,139)(49,131,56,138)(57,431,64,424)(58,430,65,423)(59,429,66,422)(60,428,67,421)(61,427,68,434)(62,426,69,433)(63,425,70,432)(71,414,78,407)(72,413,79,420)(73,412,80,419)(74,411,81,418)(75,410,82,417)(76,409,83,416)(77,408,84,415)(85,341,92,348)(86,340,93,347)(87,339,94,346)(88,338,95,345)(89,337,96,344)(90,350,97,343)(91,349,98,342)(99,296,106,303)(100,295,107,302)(101,308,108,301)(102,307,109,300)(103,306,110,299)(104,305,111,298)(105,304,112,297)(113,184,120,191)(114,183,121,190)(115,196,122,189)(116,195,123,188)(117,194,124,187)(118,193,125,186)(119,192,126,185)(141,406,148,399)(142,405,149,398)(143,404,150,397)(144,403,151,396)(145,402,152,395)(146,401,153,394)(147,400,154,393)(155,239,162,246)(156,252,163,245)(157,251,164,244)(158,250,165,243)(159,249,166,242)(160,248,167,241)(161,247,168,240)(169,353,176,360)(170,352,177,359)(171,351,178,358)(172,364,179,357)(173,363,180,356)(174,362,181,355)(175,361,182,354)(211,263,218,256)(212,262,219,255)(213,261,220,254)(214,260,221,253)(215,259,222,266)(216,258,223,265)(217,257,224,264)(225,325,232,332)(226,324,233,331)(227,323,234,330)(228,336,235,329)(229,335,236,328)(230,334,237,327)(231,333,238,326)(267,382,274,389)(268,381,275,388)(269,380,276,387)(270,379,277,386)(271,392,278,385)(272,391,279,384)(273,390,280,383)(309,367,316,374)(310,366,317,373)(311,365,318,372)(312,378,319,371)(313,377,320,370)(314,376,321,369)(315,375,322,368), (1,328,278,345)(2,329,279,346)(3,330,280,347)(4,331,267,348)(5,332,268,349)(6,333,269,350)(7,334,270,337)(8,335,271,338)(9,336,272,339)(10,323,273,340)(11,324,274,341)(12,325,275,342)(13,326,276,343)(14,327,277,344)(15,215,369,63)(16,216,370,64)(17,217,371,65)(18,218,372,66)(19,219,373,67)(20,220,374,68)(21,221,375,69)(22,222,376,70)(23,223,377,57)(24,224,378,58)(25,211,365,59)(26,212,366,60)(27,213,367,61)(28,214,368,62)(29,43,153,411)(30,44,154,412)(31,45,141,413)(32,46,142,414)(33,47,143,415)(34,48,144,416)(35,49,145,417)(36,50,146,418)(37,51,147,419)(38,52,148,420)(39,53,149,407)(40,54,150,408)(41,55,151,409)(42,56,152,410)(71,425,127,259)(72,426,128,260)(73,427,129,261)(74,428,130,262)(75,429,131,263)(76,430,132,264)(77,431,133,265)(78,432,134,266)(79,433,135,253)(80,434,136,254)(81,421,137,255)(82,422,138,256)(83,423,139,257)(84,424,140,258)(85,189,226,165)(86,190,227,166)(87,191,228,167)(88,192,229,168)(89,193,230,155)(90,194,231,156)(91,195,232,157)(92,196,233,158)(93,183,234,159)(94,184,235,160)(95,185,236,161)(96,186,237,162)(97,187,238,163)(98,188,225,164)(99,392,364,448)(100,379,351,435)(101,380,352,436)(102,381,353,437)(103,382,354,438)(104,383,355,439)(105,384,356,440)(106,385,357,441)(107,386,358,442)(108,387,359,443)(109,388,360,444)(110,389,361,445)(111,390,362,446)(112,391,363,447)(113,304,241,173)(114,305,242,174)(115,306,243,175)(116,307,244,176)(117,308,245,177)(118,295,246,178)(119,296,247,179)(120,297,248,180)(121,298,249,181)(122,299,250,182)(123,300,251,169)(124,301,252,170)(125,302,239,171)(126,303,240,172)(197,399,315,285)(198,400,316,286)(199,401,317,287)(200,402,318,288)(201,403,319,289)(202,404,320,290)(203,405,321,291)(204,406,322,292)(205,393,309,293)(206,394,310,294)(207,395,311,281)(208,396,312,282)(209,397,313,283)(210,398,314,284)>;
G:=Group( (1,419,126,374)(2,420,113,375)(3,407,114,376)(4,408,115,377)(5,409,116,378)(6,410,117,365)(7,411,118,366)(8,412,119,367)(9,413,120,368)(10,414,121,369)(11,415,122,370)(12,416,123,371)(13,417,124,372)(14,418,125,373)(15,273,46,249)(16,274,47,250)(17,275,48,251)(18,276,49,252)(19,277,50,239)(20,278,51,240)(21,279,52,241)(22,280,53,242)(23,267,54,243)(24,268,55,244)(25,269,56,245)(26,270,43,246)(27,271,44,247)(28,272,45,248)(29,334,60,295)(30,335,61,296)(31,336,62,297)(32,323,63,298)(33,324,64,299)(34,325,65,300)(35,326,66,301)(36,327,67,302)(37,328,68,303)(38,329,69,304)(39,330,70,305)(40,331,57,306)(41,332,58,307)(42,333,59,308)(71,183,321,439)(72,184,322,440)(73,185,309,441)(74,186,310,442)(75,187,311,443)(76,188,312,444)(77,189,313,445)(78,190,314,446)(79,191,315,447)(80,192,316,448)(81,193,317,435)(82,194,318,436)(83,195,319,437)(84,196,320,438)(85,265,361,397)(86,266,362,398)(87,253,363,399)(88,254,364,400)(89,255,351,401)(90,256,352,402)(91,257,353,403)(92,258,354,404)(93,259,355,405)(94,260,356,406)(95,261,357,393)(96,262,358,394)(97,263,359,395)(98,264,360,396)(99,286,229,434)(100,287,230,421)(101,288,231,422)(102,289,232,423)(103,290,233,424)(104,291,234,425)(105,292,235,426)(106,293,236,427)(107,294,237,428)(108,281,238,429)(109,282,225,430)(110,283,226,431)(111,284,227,432)(112,285,228,433)(127,159,203,383)(128,160,204,384)(129,161,205,385)(130,162,206,386)(131,163,207,387)(132,164,208,388)(133,165,209,389)(134,166,210,390)(135,167,197,391)(136,168,198,392)(137,155,199,379)(138,156,200,380)(139,157,201,381)(140,158,202,382)(141,339,214,180)(142,340,215,181)(143,341,216,182)(144,342,217,169)(145,343,218,170)(146,344,219,171)(147,345,220,172)(148,346,221,173)(149,347,222,174)(150,348,223,175)(151,349,224,176)(152,350,211,177)(153,337,212,178)(154,338,213,179), (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), (1,441,8,448)(2,440,9,447)(3,439,10,446)(4,438,11,445)(5,437,12,444)(6,436,13,443)(7,435,14,442)(15,210,22,203)(16,209,23,202)(17,208,24,201)(18,207,25,200)(19,206,26,199)(20,205,27,198)(21,204,28,197)(29,294,36,287)(30,293,37,286)(31,292,38,285)(32,291,39,284)(33,290,40,283)(34,289,41,282)(35,288,42,281)(43,137,50,130)(44,136,51,129)(45,135,52,128)(46,134,53,127)(47,133,54,140)(48,132,55,139)(49,131,56,138)(57,431,64,424)(58,430,65,423)(59,429,66,422)(60,428,67,421)(61,427,68,434)(62,426,69,433)(63,425,70,432)(71,414,78,407)(72,413,79,420)(73,412,80,419)(74,411,81,418)(75,410,82,417)(76,409,83,416)(77,408,84,415)(85,341,92,348)(86,340,93,347)(87,339,94,346)(88,338,95,345)(89,337,96,344)(90,350,97,343)(91,349,98,342)(99,296,106,303)(100,295,107,302)(101,308,108,301)(102,307,109,300)(103,306,110,299)(104,305,111,298)(105,304,112,297)(113,184,120,191)(114,183,121,190)(115,196,122,189)(116,195,123,188)(117,194,124,187)(118,193,125,186)(119,192,126,185)(141,406,148,399)(142,405,149,398)(143,404,150,397)(144,403,151,396)(145,402,152,395)(146,401,153,394)(147,400,154,393)(155,239,162,246)(156,252,163,245)(157,251,164,244)(158,250,165,243)(159,249,166,242)(160,248,167,241)(161,247,168,240)(169,353,176,360)(170,352,177,359)(171,351,178,358)(172,364,179,357)(173,363,180,356)(174,362,181,355)(175,361,182,354)(211,263,218,256)(212,262,219,255)(213,261,220,254)(214,260,221,253)(215,259,222,266)(216,258,223,265)(217,257,224,264)(225,325,232,332)(226,324,233,331)(227,323,234,330)(228,336,235,329)(229,335,236,328)(230,334,237,327)(231,333,238,326)(267,382,274,389)(268,381,275,388)(269,380,276,387)(270,379,277,386)(271,392,278,385)(272,391,279,384)(273,390,280,383)(309,367,316,374)(310,366,317,373)(311,365,318,372)(312,378,319,371)(313,377,320,370)(314,376,321,369)(315,375,322,368), (1,328,278,345)(2,329,279,346)(3,330,280,347)(4,331,267,348)(5,332,268,349)(6,333,269,350)(7,334,270,337)(8,335,271,338)(9,336,272,339)(10,323,273,340)(11,324,274,341)(12,325,275,342)(13,326,276,343)(14,327,277,344)(15,215,369,63)(16,216,370,64)(17,217,371,65)(18,218,372,66)(19,219,373,67)(20,220,374,68)(21,221,375,69)(22,222,376,70)(23,223,377,57)(24,224,378,58)(25,211,365,59)(26,212,366,60)(27,213,367,61)(28,214,368,62)(29,43,153,411)(30,44,154,412)(31,45,141,413)(32,46,142,414)(33,47,143,415)(34,48,144,416)(35,49,145,417)(36,50,146,418)(37,51,147,419)(38,52,148,420)(39,53,149,407)(40,54,150,408)(41,55,151,409)(42,56,152,410)(71,425,127,259)(72,426,128,260)(73,427,129,261)(74,428,130,262)(75,429,131,263)(76,430,132,264)(77,431,133,265)(78,432,134,266)(79,433,135,253)(80,434,136,254)(81,421,137,255)(82,422,138,256)(83,423,139,257)(84,424,140,258)(85,189,226,165)(86,190,227,166)(87,191,228,167)(88,192,229,168)(89,193,230,155)(90,194,231,156)(91,195,232,157)(92,196,233,158)(93,183,234,159)(94,184,235,160)(95,185,236,161)(96,186,237,162)(97,187,238,163)(98,188,225,164)(99,392,364,448)(100,379,351,435)(101,380,352,436)(102,381,353,437)(103,382,354,438)(104,383,355,439)(105,384,356,440)(106,385,357,441)(107,386,358,442)(108,387,359,443)(109,388,360,444)(110,389,361,445)(111,390,362,446)(112,391,363,447)(113,304,241,173)(114,305,242,174)(115,306,243,175)(116,307,244,176)(117,308,245,177)(118,295,246,178)(119,296,247,179)(120,297,248,180)(121,298,249,181)(122,299,250,182)(123,300,251,169)(124,301,252,170)(125,302,239,171)(126,303,240,172)(197,399,315,285)(198,400,316,286)(199,401,317,287)(200,402,318,288)(201,403,319,289)(202,404,320,290)(203,405,321,291)(204,406,322,292)(205,393,309,293)(206,394,310,294)(207,395,311,281)(208,396,312,282)(209,397,313,283)(210,398,314,284) );
G=PermutationGroup([[(1,419,126,374),(2,420,113,375),(3,407,114,376),(4,408,115,377),(5,409,116,378),(6,410,117,365),(7,411,118,366),(8,412,119,367),(9,413,120,368),(10,414,121,369),(11,415,122,370),(12,416,123,371),(13,417,124,372),(14,418,125,373),(15,273,46,249),(16,274,47,250),(17,275,48,251),(18,276,49,252),(19,277,50,239),(20,278,51,240),(21,279,52,241),(22,280,53,242),(23,267,54,243),(24,268,55,244),(25,269,56,245),(26,270,43,246),(27,271,44,247),(28,272,45,248),(29,334,60,295),(30,335,61,296),(31,336,62,297),(32,323,63,298),(33,324,64,299),(34,325,65,300),(35,326,66,301),(36,327,67,302),(37,328,68,303),(38,329,69,304),(39,330,70,305),(40,331,57,306),(41,332,58,307),(42,333,59,308),(71,183,321,439),(72,184,322,440),(73,185,309,441),(74,186,310,442),(75,187,311,443),(76,188,312,444),(77,189,313,445),(78,190,314,446),(79,191,315,447),(80,192,316,448),(81,193,317,435),(82,194,318,436),(83,195,319,437),(84,196,320,438),(85,265,361,397),(86,266,362,398),(87,253,363,399),(88,254,364,400),(89,255,351,401),(90,256,352,402),(91,257,353,403),(92,258,354,404),(93,259,355,405),(94,260,356,406),(95,261,357,393),(96,262,358,394),(97,263,359,395),(98,264,360,396),(99,286,229,434),(100,287,230,421),(101,288,231,422),(102,289,232,423),(103,290,233,424),(104,291,234,425),(105,292,235,426),(106,293,236,427),(107,294,237,428),(108,281,238,429),(109,282,225,430),(110,283,226,431),(111,284,227,432),(112,285,228,433),(127,159,203,383),(128,160,204,384),(129,161,205,385),(130,162,206,386),(131,163,207,387),(132,164,208,388),(133,165,209,389),(134,166,210,390),(135,167,197,391),(136,168,198,392),(137,155,199,379),(138,156,200,380),(139,157,201,381),(140,158,202,382),(141,339,214,180),(142,340,215,181),(143,341,216,182),(144,342,217,169),(145,343,218,170),(146,344,219,171),(147,345,220,172),(148,346,221,173),(149,347,222,174),(150,348,223,175),(151,349,224,176),(152,350,211,177),(153,337,212,178),(154,338,213,179)], [(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)], [(1,441,8,448),(2,440,9,447),(3,439,10,446),(4,438,11,445),(5,437,12,444),(6,436,13,443),(7,435,14,442),(15,210,22,203),(16,209,23,202),(17,208,24,201),(18,207,25,200),(19,206,26,199),(20,205,27,198),(21,204,28,197),(29,294,36,287),(30,293,37,286),(31,292,38,285),(32,291,39,284),(33,290,40,283),(34,289,41,282),(35,288,42,281),(43,137,50,130),(44,136,51,129),(45,135,52,128),(46,134,53,127),(47,133,54,140),(48,132,55,139),(49,131,56,138),(57,431,64,424),(58,430,65,423),(59,429,66,422),(60,428,67,421),(61,427,68,434),(62,426,69,433),(63,425,70,432),(71,414,78,407),(72,413,79,420),(73,412,80,419),(74,411,81,418),(75,410,82,417),(76,409,83,416),(77,408,84,415),(85,341,92,348),(86,340,93,347),(87,339,94,346),(88,338,95,345),(89,337,96,344),(90,350,97,343),(91,349,98,342),(99,296,106,303),(100,295,107,302),(101,308,108,301),(102,307,109,300),(103,306,110,299),(104,305,111,298),(105,304,112,297),(113,184,120,191),(114,183,121,190),(115,196,122,189),(116,195,123,188),(117,194,124,187),(118,193,125,186),(119,192,126,185),(141,406,148,399),(142,405,149,398),(143,404,150,397),(144,403,151,396),(145,402,152,395),(146,401,153,394),(147,400,154,393),(155,239,162,246),(156,252,163,245),(157,251,164,244),(158,250,165,243),(159,249,166,242),(160,248,167,241),(161,247,168,240),(169,353,176,360),(170,352,177,359),(171,351,178,358),(172,364,179,357),(173,363,180,356),(174,362,181,355),(175,361,182,354),(211,263,218,256),(212,262,219,255),(213,261,220,254),(214,260,221,253),(215,259,222,266),(216,258,223,265),(217,257,224,264),(225,325,232,332),(226,324,233,331),(227,323,234,330),(228,336,235,329),(229,335,236,328),(230,334,237,327),(231,333,238,326),(267,382,274,389),(268,381,275,388),(269,380,276,387),(270,379,277,386),(271,392,278,385),(272,391,279,384),(273,390,280,383),(309,367,316,374),(310,366,317,373),(311,365,318,372),(312,378,319,371),(313,377,320,370),(314,376,321,369),(315,375,322,368)], [(1,328,278,345),(2,329,279,346),(3,330,280,347),(4,331,267,348),(5,332,268,349),(6,333,269,350),(7,334,270,337),(8,335,271,338),(9,336,272,339),(10,323,273,340),(11,324,274,341),(12,325,275,342),(13,326,276,343),(14,327,277,344),(15,215,369,63),(16,216,370,64),(17,217,371,65),(18,218,372,66),(19,219,373,67),(20,220,374,68),(21,221,375,69),(22,222,376,70),(23,223,377,57),(24,224,378,58),(25,211,365,59),(26,212,366,60),(27,213,367,61),(28,214,368,62),(29,43,153,411),(30,44,154,412),(31,45,141,413),(32,46,142,414),(33,47,143,415),(34,48,144,416),(35,49,145,417),(36,50,146,418),(37,51,147,419),(38,52,148,420),(39,53,149,407),(40,54,150,408),(41,55,151,409),(42,56,152,410),(71,425,127,259),(72,426,128,260),(73,427,129,261),(74,428,130,262),(75,429,131,263),(76,430,132,264),(77,431,133,265),(78,432,134,266),(79,433,135,253),(80,434,136,254),(81,421,137,255),(82,422,138,256),(83,423,139,257),(84,424,140,258),(85,189,226,165),(86,190,227,166),(87,191,228,167),(88,192,229,168),(89,193,230,155),(90,194,231,156),(91,195,232,157),(92,196,233,158),(93,183,234,159),(94,184,235,160),(95,185,236,161),(96,186,237,162),(97,187,238,163),(98,188,225,164),(99,392,364,448),(100,379,351,435),(101,380,352,436),(102,381,353,437),(103,382,354,438),(104,383,355,439),(105,384,356,440),(106,385,357,441),(107,386,358,442),(108,387,359,443),(109,388,360,444),(110,389,361,445),(111,390,362,446),(112,391,363,447),(113,304,241,173),(114,305,242,174),(115,306,243,175),(116,307,244,176),(117,308,245,177),(118,295,246,178),(119,296,247,179),(120,297,248,180),(121,298,249,181),(122,299,250,182),(123,300,251,169),(124,301,252,170),(125,302,239,171),(126,303,240,172),(197,399,315,285),(198,400,316,286),(199,401,317,287),(200,402,318,288),(201,403,319,289),(202,404,320,290),(203,405,321,291),(204,406,322,292),(205,393,309,293),(206,394,310,294),(207,395,311,281),(208,396,312,282),(209,397,313,283),(210,398,314,284)]])
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 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D7 | C4○D4 | D14 | Dic14 | C4×D7 | C7⋊D4 | D4⋊2D7 | Q8⋊2D7 |
| kernel | (C4×Dic7)⋊9C4 | C14.C42 | C2×C4×Dic7 | C2×C4⋊Dic7 | C14×C4⋊C4 | C4×Dic7 | C2×C28 | C2×C28 | C2×C4⋊C4 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 2 | 2 | 3 | 8 | 9 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of (C4×Dic7)⋊9C4 ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 11 | 1 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 22 | 7 | 0 | 0 |
| 0 | 26 | 7 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 16 | 24 | 0 | 0 |
| 0 | 5 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,17,0,0,0,0,0,12],[1,0,0,0,0,0,11,28,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,22,26,0,0,0,7,7,0,0,0,0,0,28,0,0,0,0,0,1],[17,0,0,0,0,0,16,5,0,0,0,24,13,0,0,0,0,0,0,1,0,0,0,1,0] >;
(C4×Dic7)⋊9C4 in GAP, Magma, Sage, TeX
(C_4\times {\rm Dic}_7)\rtimes_9C_4 % in TeX
G:=Group("(C4xDic7):9C4"); // GroupNames label
G:=SmallGroup(448,511);
// by ID
G=gap.SmallGroup(448,511);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,1094,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^14=d^4=1,c^2=b^7,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^7*c>;
// generators/relations