metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.8D14, C14.19C4≀C2, C4⋊C4.1Dic7, (C2×C28).232D4, C42.C2.1D7, (C4×C28).236C22, C42.D7.9C2, C2.7(D4⋊2Dic7), C14.8(C4.10D4), C2.3(C28.10D4), C7⋊2(C42.2C22), C22.40(C23.D7), (C7×C4⋊C4).1C4, (C2×C28).170(C2×C4), (C2×C4).10(C2×Dic7), (C7×C42.C2).7C2, (C2×C4).166(C7⋊D4), (C2×C14).101(C22⋊C4), SmallGroup(448,100)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.8D14
G = < a,b,c,d | a4=b4=1, c14=a2b2, d2=a-1b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=a2bc13 >
Subgroups: 188 in 60 conjugacy classes, 27 normal (13 characteristic)
C1, C2, C2, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, C14, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C28, C2×C14, C8⋊C4, C42.C2, C7⋊C8, C2×C28, C2×C28, C2×C28, C42.2C22, C2×C7⋊C8, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C42.D7, C7×C42.C2, C42.8D14
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, Dic7, D14, C4.10D4, C4≀C2, C2×Dic7, C7⋊D4, C42.2C22, C23.D7, C28.10D4, D4⋊2Dic7, C42.8D14
►Regular action on 448 points
Generators in S448
(1 197 312 388)(2 212 313 375)(3 199 314 390)(4 214 315 377)(5 201 316 392)(6 216 317 379)(7 203 318 366)(8 218 319 381)(9 205 320 368)(10 220 321 383)(11 207 322 370)(12 222 323 385)(13 209 324 372)(14 224 325 387)(15 211 326 374)(16 198 327 389)(17 213 328 376)(18 200 329 391)(19 215 330 378)(20 202 331 365)(21 217 332 380)(22 204 333 367)(23 219 334 382)(24 206 335 369)(25 221 336 384)(26 208 309 371)(27 223 310 386)(28 210 311 373)(29 275 135 405)(30 262 136 420)(31 277 137 407)(32 264 138 394)(33 279 139 409)(34 266 140 396)(35 253 113 411)(36 268 114 398)(37 255 115 413)(38 270 116 400)(39 257 117 415)(40 272 118 402)(41 259 119 417)(42 274 120 404)(43 261 121 419)(44 276 122 406)(45 263 123 393)(46 278 124 408)(47 265 125 395)(48 280 126 410)(49 267 127 397)(50 254 128 412)(51 269 129 399)(52 256 130 414)(53 271 131 401)(54 258 132 416)(55 273 133 403)(56 260 134 418)(57 446 92 283)(58 433 93 298)(59 448 94 285)(60 435 95 300)(61 422 96 287)(62 437 97 302)(63 424 98 289)(64 439 99 304)(65 426 100 291)(66 441 101 306)(67 428 102 293)(68 443 103 308)(69 430 104 295)(70 445 105 282)(71 432 106 297)(72 447 107 284)(73 434 108 299)(74 421 109 286)(75 436 110 301)(76 423 111 288)(77 438 112 303)(78 425 85 290)(79 440 86 305)(80 427 87 292)(81 442 88 307)(82 429 89 294)(83 444 90 281)(84 431 91 296)(141 246 349 176)(142 233 350 191)(143 248 351 178)(144 235 352 193)(145 250 353 180)(146 237 354 195)(147 252 355 182)(148 239 356 169)(149 226 357 184)(150 241 358 171)(151 228 359 186)(152 243 360 173)(153 230 361 188)(154 245 362 175)(155 232 363 190)(156 247 364 177)(157 234 337 192)(158 249 338 179)(159 236 339 194)(160 251 340 181)(161 238 341 196)(162 225 342 183)(163 240 343 170)(164 227 344 185)(165 242 345 172)(166 229 346 187)(167 244 347 174)(168 231 348 189)
(1 260 326 404)(2 405 327 261)(3 262 328 406)(4 407 329 263)(5 264 330 408)(6 409 331 265)(7 266 332 410)(8 411 333 267)(9 268 334 412)(10 413 335 269)(11 270 336 414)(12 415 309 271)(13 272 310 416)(14 417 311 273)(15 274 312 418)(16 419 313 275)(17 276 314 420)(18 393 315 277)(19 278 316 394)(20 395 317 279)(21 280 318 396)(22 397 319 253)(23 254 320 398)(24 399 321 255)(25 256 322 400)(26 401 323 257)(27 258 324 402)(28 403 325 259)(29 389 121 212)(30 213 122 390)(31 391 123 214)(32 215 124 392)(33 365 125 216)(34 217 126 366)(35 367 127 218)(36 219 128 368)(37 369 129 220)(38 221 130 370)(39 371 131 222)(40 223 132 372)(41 373 133 224)(42 197 134 374)(43 375 135 198)(44 199 136 376)(45 377 137 200)(46 201 138 378)(47 379 139 202)(48 203 140 380)(49 381 113 204)(50 205 114 382)(51 383 115 206)(52 207 116 384)(53 385 117 208)(54 209 118 386)(55 387 119 210)(56 211 120 388)(57 144 106 338)(58 339 107 145)(59 146 108 340)(60 341 109 147)(61 148 110 342)(62 343 111 149)(63 150 112 344)(64 345 85 151)(65 152 86 346)(66 347 87 153)(67 154 88 348)(68 349 89 155)(69 156 90 350)(70 351 91 157)(71 158 92 352)(72 353 93 159)(73 160 94 354)(74 355 95 161)(75 162 96 356)(76 357 97 163)(77 164 98 358)(78 359 99 165)(79 166 100 360)(80 361 101 167)(81 168 102 362)(82 363 103 141)(83 142 104 364)(84 337 105 143)(169 436 225 287)(170 288 226 437)(171 438 227 289)(172 290 228 439)(173 440 229 291)(174 292 230 441)(175 442 231 293)(176 294 232 443)(177 444 233 295)(178 296 234 445)(179 446 235 297)(180 298 236 447)(181 448 237 299)(182 300 238 421)(183 422 239 301)(184 302 240 423)(185 424 241 303)(186 304 242 425)(187 426 243 305)(188 306 244 427)(189 428 245 307)(190 308 246 429)(191 430 247 281)(192 282 248 431)(193 432 249 283)(194 284 250 433)(195 434 251 285)(196 286 252 435)
(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)
(1 103 56 176 15 89 42 190)(2 154 135 442 16 168 121 428)(3 101 30 174 17 87 44 188)(4 152 137 440 18 166 123 426)(5 99 32 172 19 85 46 186)(6 150 139 438 20 164 125 424)(7 97 34 170 21 111 48 184)(8 148 113 436 22 162 127 422)(9 95 36 196 23 109 50 182)(10 146 115 434 24 160 129 448)(11 93 38 194 25 107 52 180)(12 144 117 432 26 158 131 446)(13 91 40 192 27 105 54 178)(14 142 119 430 28 156 133 444)(29 307 327 348 43 293 313 362)(31 305 329 346 45 291 315 360)(33 303 331 344 47 289 317 358)(35 301 333 342 49 287 319 356)(37 299 335 340 51 285 321 354)(39 297 309 338 53 283 323 352)(41 295 311 364 55 281 325 350)(57 371 193 271 71 385 179 257)(58 116 236 336 72 130 250 322)(59 369 195 269 73 383 181 255)(60 114 238 334 74 128 252 320)(61 367 169 267 75 381 183 253)(62 140 240 332 76 126 226 318)(63 365 171 265 77 379 185 279)(64 138 242 330 78 124 228 316)(65 391 173 263 79 377 187 277)(66 136 244 328 80 122 230 314)(67 389 175 261 81 375 189 275)(68 134 246 326 82 120 232 312)(69 387 177 259 83 373 191 273)(70 132 248 324 84 118 234 310)(86 214 229 407 100 200 243 393)(88 212 231 405 102 198 245 419)(90 210 233 403 104 224 247 417)(92 208 235 401 106 222 249 415)(94 206 237 399 108 220 251 413)(96 204 239 397 110 218 225 411)(98 202 241 395 112 216 227 409)(141 197 294 260 155 211 308 274)(143 223 296 258 157 209 282 272)(145 221 298 256 159 207 284 270)(147 219 300 254 161 205 286 268)(149 217 302 280 163 203 288 266)(151 215 304 278 165 201 290 264)(153 213 306 276 167 199 292 262)(337 372 445 402 351 386 431 416)(339 370 447 400 353 384 433 414)(341 368 421 398 355 382 435 412)(343 366 423 396 357 380 437 410)(345 392 425 394 359 378 439 408)(347 390 427 420 361 376 441 406)(349 388 429 418 363 374 443 404)
G:=sub<Sym(448)| (1,197,312,388)(2,212,313,375)(3,199,314,390)(4,214,315,377)(5,201,316,392)(6,216,317,379)(7,203,318,366)(8,218,319,381)(9,205,320,368)(10,220,321,383)(11,207,322,370)(12,222,323,385)(13,209,324,372)(14,224,325,387)(15,211,326,374)(16,198,327,389)(17,213,328,376)(18,200,329,391)(19,215,330,378)(20,202,331,365)(21,217,332,380)(22,204,333,367)(23,219,334,382)(24,206,335,369)(25,221,336,384)(26,208,309,371)(27,223,310,386)(28,210,311,373)(29,275,135,405)(30,262,136,420)(31,277,137,407)(32,264,138,394)(33,279,139,409)(34,266,140,396)(35,253,113,411)(36,268,114,398)(37,255,115,413)(38,270,116,400)(39,257,117,415)(40,272,118,402)(41,259,119,417)(42,274,120,404)(43,261,121,419)(44,276,122,406)(45,263,123,393)(46,278,124,408)(47,265,125,395)(48,280,126,410)(49,267,127,397)(50,254,128,412)(51,269,129,399)(52,256,130,414)(53,271,131,401)(54,258,132,416)(55,273,133,403)(56,260,134,418)(57,446,92,283)(58,433,93,298)(59,448,94,285)(60,435,95,300)(61,422,96,287)(62,437,97,302)(63,424,98,289)(64,439,99,304)(65,426,100,291)(66,441,101,306)(67,428,102,293)(68,443,103,308)(69,430,104,295)(70,445,105,282)(71,432,106,297)(72,447,107,284)(73,434,108,299)(74,421,109,286)(75,436,110,301)(76,423,111,288)(77,438,112,303)(78,425,85,290)(79,440,86,305)(80,427,87,292)(81,442,88,307)(82,429,89,294)(83,444,90,281)(84,431,91,296)(141,246,349,176)(142,233,350,191)(143,248,351,178)(144,235,352,193)(145,250,353,180)(146,237,354,195)(147,252,355,182)(148,239,356,169)(149,226,357,184)(150,241,358,171)(151,228,359,186)(152,243,360,173)(153,230,361,188)(154,245,362,175)(155,232,363,190)(156,247,364,177)(157,234,337,192)(158,249,338,179)(159,236,339,194)(160,251,340,181)(161,238,341,196)(162,225,342,183)(163,240,343,170)(164,227,344,185)(165,242,345,172)(166,229,346,187)(167,244,347,174)(168,231,348,189), (1,260,326,404)(2,405,327,261)(3,262,328,406)(4,407,329,263)(5,264,330,408)(6,409,331,265)(7,266,332,410)(8,411,333,267)(9,268,334,412)(10,413,335,269)(11,270,336,414)(12,415,309,271)(13,272,310,416)(14,417,311,273)(15,274,312,418)(16,419,313,275)(17,276,314,420)(18,393,315,277)(19,278,316,394)(20,395,317,279)(21,280,318,396)(22,397,319,253)(23,254,320,398)(24,399,321,255)(25,256,322,400)(26,401,323,257)(27,258,324,402)(28,403,325,259)(29,389,121,212)(30,213,122,390)(31,391,123,214)(32,215,124,392)(33,365,125,216)(34,217,126,366)(35,367,127,218)(36,219,128,368)(37,369,129,220)(38,221,130,370)(39,371,131,222)(40,223,132,372)(41,373,133,224)(42,197,134,374)(43,375,135,198)(44,199,136,376)(45,377,137,200)(46,201,138,378)(47,379,139,202)(48,203,140,380)(49,381,113,204)(50,205,114,382)(51,383,115,206)(52,207,116,384)(53,385,117,208)(54,209,118,386)(55,387,119,210)(56,211,120,388)(57,144,106,338)(58,339,107,145)(59,146,108,340)(60,341,109,147)(61,148,110,342)(62,343,111,149)(63,150,112,344)(64,345,85,151)(65,152,86,346)(66,347,87,153)(67,154,88,348)(68,349,89,155)(69,156,90,350)(70,351,91,157)(71,158,92,352)(72,353,93,159)(73,160,94,354)(74,355,95,161)(75,162,96,356)(76,357,97,163)(77,164,98,358)(78,359,99,165)(79,166,100,360)(80,361,101,167)(81,168,102,362)(82,363,103,141)(83,142,104,364)(84,337,105,143)(169,436,225,287)(170,288,226,437)(171,438,227,289)(172,290,228,439)(173,440,229,291)(174,292,230,441)(175,442,231,293)(176,294,232,443)(177,444,233,295)(178,296,234,445)(179,446,235,297)(180,298,236,447)(181,448,237,299)(182,300,238,421)(183,422,239,301)(184,302,240,423)(185,424,241,303)(186,304,242,425)(187,426,243,305)(188,306,244,427)(189,428,245,307)(190,308,246,429)(191,430,247,281)(192,282,248,431)(193,432,249,283)(194,284,250,433)(195,434,251,285)(196,286,252,435), (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), (1,103,56,176,15,89,42,190)(2,154,135,442,16,168,121,428)(3,101,30,174,17,87,44,188)(4,152,137,440,18,166,123,426)(5,99,32,172,19,85,46,186)(6,150,139,438,20,164,125,424)(7,97,34,170,21,111,48,184)(8,148,113,436,22,162,127,422)(9,95,36,196,23,109,50,182)(10,146,115,434,24,160,129,448)(11,93,38,194,25,107,52,180)(12,144,117,432,26,158,131,446)(13,91,40,192,27,105,54,178)(14,142,119,430,28,156,133,444)(29,307,327,348,43,293,313,362)(31,305,329,346,45,291,315,360)(33,303,331,344,47,289,317,358)(35,301,333,342,49,287,319,356)(37,299,335,340,51,285,321,354)(39,297,309,338,53,283,323,352)(41,295,311,364,55,281,325,350)(57,371,193,271,71,385,179,257)(58,116,236,336,72,130,250,322)(59,369,195,269,73,383,181,255)(60,114,238,334,74,128,252,320)(61,367,169,267,75,381,183,253)(62,140,240,332,76,126,226,318)(63,365,171,265,77,379,185,279)(64,138,242,330,78,124,228,316)(65,391,173,263,79,377,187,277)(66,136,244,328,80,122,230,314)(67,389,175,261,81,375,189,275)(68,134,246,326,82,120,232,312)(69,387,177,259,83,373,191,273)(70,132,248,324,84,118,234,310)(86,214,229,407,100,200,243,393)(88,212,231,405,102,198,245,419)(90,210,233,403,104,224,247,417)(92,208,235,401,106,222,249,415)(94,206,237,399,108,220,251,413)(96,204,239,397,110,218,225,411)(98,202,241,395,112,216,227,409)(141,197,294,260,155,211,308,274)(143,223,296,258,157,209,282,272)(145,221,298,256,159,207,284,270)(147,219,300,254,161,205,286,268)(149,217,302,280,163,203,288,266)(151,215,304,278,165,201,290,264)(153,213,306,276,167,199,292,262)(337,372,445,402,351,386,431,416)(339,370,447,400,353,384,433,414)(341,368,421,398,355,382,435,412)(343,366,423,396,357,380,437,410)(345,392,425,394,359,378,439,408)(347,390,427,420,361,376,441,406)(349,388,429,418,363,374,443,404)>;
G:=Group( (1,197,312,388)(2,212,313,375)(3,199,314,390)(4,214,315,377)(5,201,316,392)(6,216,317,379)(7,203,318,366)(8,218,319,381)(9,205,320,368)(10,220,321,383)(11,207,322,370)(12,222,323,385)(13,209,324,372)(14,224,325,387)(15,211,326,374)(16,198,327,389)(17,213,328,376)(18,200,329,391)(19,215,330,378)(20,202,331,365)(21,217,332,380)(22,204,333,367)(23,219,334,382)(24,206,335,369)(25,221,336,384)(26,208,309,371)(27,223,310,386)(28,210,311,373)(29,275,135,405)(30,262,136,420)(31,277,137,407)(32,264,138,394)(33,279,139,409)(34,266,140,396)(35,253,113,411)(36,268,114,398)(37,255,115,413)(38,270,116,400)(39,257,117,415)(40,272,118,402)(41,259,119,417)(42,274,120,404)(43,261,121,419)(44,276,122,406)(45,263,123,393)(46,278,124,408)(47,265,125,395)(48,280,126,410)(49,267,127,397)(50,254,128,412)(51,269,129,399)(52,256,130,414)(53,271,131,401)(54,258,132,416)(55,273,133,403)(56,260,134,418)(57,446,92,283)(58,433,93,298)(59,448,94,285)(60,435,95,300)(61,422,96,287)(62,437,97,302)(63,424,98,289)(64,439,99,304)(65,426,100,291)(66,441,101,306)(67,428,102,293)(68,443,103,308)(69,430,104,295)(70,445,105,282)(71,432,106,297)(72,447,107,284)(73,434,108,299)(74,421,109,286)(75,436,110,301)(76,423,111,288)(77,438,112,303)(78,425,85,290)(79,440,86,305)(80,427,87,292)(81,442,88,307)(82,429,89,294)(83,444,90,281)(84,431,91,296)(141,246,349,176)(142,233,350,191)(143,248,351,178)(144,235,352,193)(145,250,353,180)(146,237,354,195)(147,252,355,182)(148,239,356,169)(149,226,357,184)(150,241,358,171)(151,228,359,186)(152,243,360,173)(153,230,361,188)(154,245,362,175)(155,232,363,190)(156,247,364,177)(157,234,337,192)(158,249,338,179)(159,236,339,194)(160,251,340,181)(161,238,341,196)(162,225,342,183)(163,240,343,170)(164,227,344,185)(165,242,345,172)(166,229,346,187)(167,244,347,174)(168,231,348,189), (1,260,326,404)(2,405,327,261)(3,262,328,406)(4,407,329,263)(5,264,330,408)(6,409,331,265)(7,266,332,410)(8,411,333,267)(9,268,334,412)(10,413,335,269)(11,270,336,414)(12,415,309,271)(13,272,310,416)(14,417,311,273)(15,274,312,418)(16,419,313,275)(17,276,314,420)(18,393,315,277)(19,278,316,394)(20,395,317,279)(21,280,318,396)(22,397,319,253)(23,254,320,398)(24,399,321,255)(25,256,322,400)(26,401,323,257)(27,258,324,402)(28,403,325,259)(29,389,121,212)(30,213,122,390)(31,391,123,214)(32,215,124,392)(33,365,125,216)(34,217,126,366)(35,367,127,218)(36,219,128,368)(37,369,129,220)(38,221,130,370)(39,371,131,222)(40,223,132,372)(41,373,133,224)(42,197,134,374)(43,375,135,198)(44,199,136,376)(45,377,137,200)(46,201,138,378)(47,379,139,202)(48,203,140,380)(49,381,113,204)(50,205,114,382)(51,383,115,206)(52,207,116,384)(53,385,117,208)(54,209,118,386)(55,387,119,210)(56,211,120,388)(57,144,106,338)(58,339,107,145)(59,146,108,340)(60,341,109,147)(61,148,110,342)(62,343,111,149)(63,150,112,344)(64,345,85,151)(65,152,86,346)(66,347,87,153)(67,154,88,348)(68,349,89,155)(69,156,90,350)(70,351,91,157)(71,158,92,352)(72,353,93,159)(73,160,94,354)(74,355,95,161)(75,162,96,356)(76,357,97,163)(77,164,98,358)(78,359,99,165)(79,166,100,360)(80,361,101,167)(81,168,102,362)(82,363,103,141)(83,142,104,364)(84,337,105,143)(169,436,225,287)(170,288,226,437)(171,438,227,289)(172,290,228,439)(173,440,229,291)(174,292,230,441)(175,442,231,293)(176,294,232,443)(177,444,233,295)(178,296,234,445)(179,446,235,297)(180,298,236,447)(181,448,237,299)(182,300,238,421)(183,422,239,301)(184,302,240,423)(185,424,241,303)(186,304,242,425)(187,426,243,305)(188,306,244,427)(189,428,245,307)(190,308,246,429)(191,430,247,281)(192,282,248,431)(193,432,249,283)(194,284,250,433)(195,434,251,285)(196,286,252,435), (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), (1,103,56,176,15,89,42,190)(2,154,135,442,16,168,121,428)(3,101,30,174,17,87,44,188)(4,152,137,440,18,166,123,426)(5,99,32,172,19,85,46,186)(6,150,139,438,20,164,125,424)(7,97,34,170,21,111,48,184)(8,148,113,436,22,162,127,422)(9,95,36,196,23,109,50,182)(10,146,115,434,24,160,129,448)(11,93,38,194,25,107,52,180)(12,144,117,432,26,158,131,446)(13,91,40,192,27,105,54,178)(14,142,119,430,28,156,133,444)(29,307,327,348,43,293,313,362)(31,305,329,346,45,291,315,360)(33,303,331,344,47,289,317,358)(35,301,333,342,49,287,319,356)(37,299,335,340,51,285,321,354)(39,297,309,338,53,283,323,352)(41,295,311,364,55,281,325,350)(57,371,193,271,71,385,179,257)(58,116,236,336,72,130,250,322)(59,369,195,269,73,383,181,255)(60,114,238,334,74,128,252,320)(61,367,169,267,75,381,183,253)(62,140,240,332,76,126,226,318)(63,365,171,265,77,379,185,279)(64,138,242,330,78,124,228,316)(65,391,173,263,79,377,187,277)(66,136,244,328,80,122,230,314)(67,389,175,261,81,375,189,275)(68,134,246,326,82,120,232,312)(69,387,177,259,83,373,191,273)(70,132,248,324,84,118,234,310)(86,214,229,407,100,200,243,393)(88,212,231,405,102,198,245,419)(90,210,233,403,104,224,247,417)(92,208,235,401,106,222,249,415)(94,206,237,399,108,220,251,413)(96,204,239,397,110,218,225,411)(98,202,241,395,112,216,227,409)(141,197,294,260,155,211,308,274)(143,223,296,258,157,209,282,272)(145,221,298,256,159,207,284,270)(147,219,300,254,161,205,286,268)(149,217,302,280,163,203,288,266)(151,215,304,278,165,201,290,264)(153,213,306,276,167,199,292,262)(337,372,445,402,351,386,431,416)(339,370,447,400,353,384,433,414)(341,368,421,398,355,382,435,412)(343,366,423,396,357,380,437,410)(345,392,425,394,359,378,439,408)(347,390,427,420,361,376,441,406)(349,388,429,418,363,374,443,404) );
G=PermutationGroup([[(1,197,312,388),(2,212,313,375),(3,199,314,390),(4,214,315,377),(5,201,316,392),(6,216,317,379),(7,203,318,366),(8,218,319,381),(9,205,320,368),(10,220,321,383),(11,207,322,370),(12,222,323,385),(13,209,324,372),(14,224,325,387),(15,211,326,374),(16,198,327,389),(17,213,328,376),(18,200,329,391),(19,215,330,378),(20,202,331,365),(21,217,332,380),(22,204,333,367),(23,219,334,382),(24,206,335,369),(25,221,336,384),(26,208,309,371),(27,223,310,386),(28,210,311,373),(29,275,135,405),(30,262,136,420),(31,277,137,407),(32,264,138,394),(33,279,139,409),(34,266,140,396),(35,253,113,411),(36,268,114,398),(37,255,115,413),(38,270,116,400),(39,257,117,415),(40,272,118,402),(41,259,119,417),(42,274,120,404),(43,261,121,419),(44,276,122,406),(45,263,123,393),(46,278,124,408),(47,265,125,395),(48,280,126,410),(49,267,127,397),(50,254,128,412),(51,269,129,399),(52,256,130,414),(53,271,131,401),(54,258,132,416),(55,273,133,403),(56,260,134,418),(57,446,92,283),(58,433,93,298),(59,448,94,285),(60,435,95,300),(61,422,96,287),(62,437,97,302),(63,424,98,289),(64,439,99,304),(65,426,100,291),(66,441,101,306),(67,428,102,293),(68,443,103,308),(69,430,104,295),(70,445,105,282),(71,432,106,297),(72,447,107,284),(73,434,108,299),(74,421,109,286),(75,436,110,301),(76,423,111,288),(77,438,112,303),(78,425,85,290),(79,440,86,305),(80,427,87,292),(81,442,88,307),(82,429,89,294),(83,444,90,281),(84,431,91,296),(141,246,349,176),(142,233,350,191),(143,248,351,178),(144,235,352,193),(145,250,353,180),(146,237,354,195),(147,252,355,182),(148,239,356,169),(149,226,357,184),(150,241,358,171),(151,228,359,186),(152,243,360,173),(153,230,361,188),(154,245,362,175),(155,232,363,190),(156,247,364,177),(157,234,337,192),(158,249,338,179),(159,236,339,194),(160,251,340,181),(161,238,341,196),(162,225,342,183),(163,240,343,170),(164,227,344,185),(165,242,345,172),(166,229,346,187),(167,244,347,174),(168,231,348,189)], [(1,260,326,404),(2,405,327,261),(3,262,328,406),(4,407,329,263),(5,264,330,408),(6,409,331,265),(7,266,332,410),(8,411,333,267),(9,268,334,412),(10,413,335,269),(11,270,336,414),(12,415,309,271),(13,272,310,416),(14,417,311,273),(15,274,312,418),(16,419,313,275),(17,276,314,420),(18,393,315,277),(19,278,316,394),(20,395,317,279),(21,280,318,396),(22,397,319,253),(23,254,320,398),(24,399,321,255),(25,256,322,400),(26,401,323,257),(27,258,324,402),(28,403,325,259),(29,389,121,212),(30,213,122,390),(31,391,123,214),(32,215,124,392),(33,365,125,216),(34,217,126,366),(35,367,127,218),(36,219,128,368),(37,369,129,220),(38,221,130,370),(39,371,131,222),(40,223,132,372),(41,373,133,224),(42,197,134,374),(43,375,135,198),(44,199,136,376),(45,377,137,200),(46,201,138,378),(47,379,139,202),(48,203,140,380),(49,381,113,204),(50,205,114,382),(51,383,115,206),(52,207,116,384),(53,385,117,208),(54,209,118,386),(55,387,119,210),(56,211,120,388),(57,144,106,338),(58,339,107,145),(59,146,108,340),(60,341,109,147),(61,148,110,342),(62,343,111,149),(63,150,112,344),(64,345,85,151),(65,152,86,346),(66,347,87,153),(67,154,88,348),(68,349,89,155),(69,156,90,350),(70,351,91,157),(71,158,92,352),(72,353,93,159),(73,160,94,354),(74,355,95,161),(75,162,96,356),(76,357,97,163),(77,164,98,358),(78,359,99,165),(79,166,100,360),(80,361,101,167),(81,168,102,362),(82,363,103,141),(83,142,104,364),(84,337,105,143),(169,436,225,287),(170,288,226,437),(171,438,227,289),(172,290,228,439),(173,440,229,291),(174,292,230,441),(175,442,231,293),(176,294,232,443),(177,444,233,295),(178,296,234,445),(179,446,235,297),(180,298,236,447),(181,448,237,299),(182,300,238,421),(183,422,239,301),(184,302,240,423),(185,424,241,303),(186,304,242,425),(187,426,243,305),(188,306,244,427),(189,428,245,307),(190,308,246,429),(191,430,247,281),(192,282,248,431),(193,432,249,283),(194,284,250,433),(195,434,251,285),(196,286,252,435)], [(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)], [(1,103,56,176,15,89,42,190),(2,154,135,442,16,168,121,428),(3,101,30,174,17,87,44,188),(4,152,137,440,18,166,123,426),(5,99,32,172,19,85,46,186),(6,150,139,438,20,164,125,424),(7,97,34,170,21,111,48,184),(8,148,113,436,22,162,127,422),(9,95,36,196,23,109,50,182),(10,146,115,434,24,160,129,448),(11,93,38,194,25,107,52,180),(12,144,117,432,26,158,131,446),(13,91,40,192,27,105,54,178),(14,142,119,430,28,156,133,444),(29,307,327,348,43,293,313,362),(31,305,329,346,45,291,315,360),(33,303,331,344,47,289,317,358),(35,301,333,342,49,287,319,356),(37,299,335,340,51,285,321,354),(39,297,309,338,53,283,323,352),(41,295,311,364,55,281,325,350),(57,371,193,271,71,385,179,257),(58,116,236,336,72,130,250,322),(59,369,195,269,73,383,181,255),(60,114,238,334,74,128,252,320),(61,367,169,267,75,381,183,253),(62,140,240,332,76,126,226,318),(63,365,171,265,77,379,185,279),(64,138,242,330,78,124,228,316),(65,391,173,263,79,377,187,277),(66,136,244,328,80,122,230,314),(67,389,175,261,81,375,189,275),(68,134,246,326,82,120,232,312),(69,387,177,259,83,373,191,273),(70,132,248,324,84,118,234,310),(86,214,229,407,100,200,243,393),(88,212,231,405,102,198,245,419),(90,210,233,403,104,224,247,417),(92,208,235,401,106,222,249,415),(94,206,237,399,108,220,251,413),(96,204,239,397,110,218,225,411),(98,202,241,395,112,216,227,409),(141,197,294,260,155,211,308,274),(143,223,296,258,157,209,282,272),(145,221,298,256,159,207,284,270),(147,219,300,254,161,205,286,268),(149,217,302,280,163,203,288,266),(151,215,304,278,165,201,290,264),(153,213,306,276,167,199,292,262),(337,372,445,402,351,386,431,416),(339,370,447,400,353,384,433,414),(341,368,421,398,355,382,435,412),(343,366,423,396,357,380,437,410),(345,392,425,394,359,378,439,408),(347,390,427,420,361,376,441,406),(349,388,429,418,363,374,443,404)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C4 | D4 | D7 | D14 | Dic7 | C4≀C2 | C7⋊D4 | C4.10D4 | C28.10D4 | D4⋊2Dic7 |
| kernel | C42.8D14 | C42.D7 | C7×C42.C2 | C7×C4⋊C4 | C2×C28 | C42.C2 | C42 | C4⋊C4 | C14 | C2×C4 | C14 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 3 | 3 | 6 | 8 | 12 | 1 | 6 | 12 |
Matrix representation of C42.8D14 ►in GL6(𝔽113)
| 15 | 83 | 0 | 0 | 0 | 0 |
| 15 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 0 | 98 |
| 1 | 111 | 0 | 0 | 0 | 0 |
| 1 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 93 | 104 | 0 | 0 | 0 | 0 |
| 32 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 88 | 0 | 0 |
| 0 | 0 | 25 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 53 | 19 | 0 | 0 | 0 | 0 |
| 6 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 109 | 103 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 106 |
| 0 | 0 | 0 | 0 | 7 | 7 |
G:=sub<GL(6,GF(113))| [15,15,0,0,0,0,83,98,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,98,0,0,0,0,0,0,98],[1,1,0,0,0,0,111,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[93,32,0,0,0,0,104,20,0,0,0,0,0,0,88,25,0,0,0,0,88,34,0,0,0,0,0,0,0,112,0,0,0,0,112,0],[53,6,0,0,0,0,19,60,0,0,0,0,0,0,109,13,0,0,0,0,103,4,0,0,0,0,0,0,7,7,0,0,0,0,106,7] >;
C42.8D14 in GAP, Magma, Sage, TeX
C_4^2._8D_{14} % in TeX
G:=Group("C4^2.8D14"); // GroupNames label
G:=SmallGroup(448,100);
// by ID
G=gap.SmallGroup(448,100);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,120,219,268,1571,570,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^2*b^2,d^2=a^-1*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*b*c^13>;
// generators/relations