direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8×Dic14, C56⋊10Q8, C42.250D14, C7⋊1(C8×Q8), C4.9(C8×D7), (C4×C8).4D7, C14.5(C4×Q8), C28.19(C2×C8), (C4×C56).21C2, C28.79(C2×Q8), C14.1(C8○D4), (C2×C8).337D14, C4⋊Dic7.22C4, C14.1(C22×C8), Dic7.1(C2×C8), C28⋊C8.24C2, C2.1(C4×Dic14), Dic7⋊C4.18C4, Dic7⋊C8.16C2, (C8×Dic7).10C2, C4.44(C2×Dic14), C28.238(C4○D4), C4.122(C4○D28), (C2×C28).800C23, (C2×C56).338C22, (C4×C28).321C22, (C2×Dic14).18C4, (C4×Dic14).27C2, C2.1(D28.2C4), (C4×Dic7).262C22, C2.4(D7×C2×C8), C22.33(C2×C4×D7), (C2×C4).101(C4×D7), (C2×C28).218(C2×C4), (C2×C7⋊C8).288C22, (C2×C14).55(C22×C4), (C2×Dic7).45(C2×C4), (C2×C4).742(C22×D7), SmallGroup(448,212)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8×Dic14
G = < a,b,c | a8=b28=1, c2=b14, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 324 in 102 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C4, C4, C4, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic7, Dic7, C28, C28, C28, C2×C14, C4×C8, C4×C8, C4⋊C8, C4×Q8, C7⋊C8, C56, C56, Dic14, C2×Dic7, C2×C28, C8×Q8, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×C56, C2×Dic14, C28⋊C8, C8×Dic7, Dic7⋊C8, C4×C56, C4×Dic14, C8×Dic14
Quotients: C1, C2, C4, C22, C8, C2×C4, Q8, C23, D7, C2×C8, C22×C4, C2×Q8, C4○D4, D14, C4×Q8, C22×C8, C8○D4, Dic14, C4×D7, C22×D7, C8×Q8, C8×D7, C2×Dic14, C2×C4×D7, C4○D28, C4×Dic14, D7×C2×C8, D28.2C4, C8×Dic14
►Regular action on 448 points
Generators in S448
(1 131 172 443 111 301 160 386)(2 132 173 444 112 302 161 387)(3 133 174 445 85 303 162 388)(4 134 175 446 86 304 163 389)(5 135 176 447 87 305 164 390)(6 136 177 448 88 306 165 391)(7 137 178 421 89 307 166 392)(8 138 179 422 90 308 167 365)(9 139 180 423 91 281 168 366)(10 140 181 424 92 282 141 367)(11 113 182 425 93 283 142 368)(12 114 183 426 94 284 143 369)(13 115 184 427 95 285 144 370)(14 116 185 428 96 286 145 371)(15 117 186 429 97 287 146 372)(16 118 187 430 98 288 147 373)(17 119 188 431 99 289 148 374)(18 120 189 432 100 290 149 375)(19 121 190 433 101 291 150 376)(20 122 191 434 102 292 151 377)(21 123 192 435 103 293 152 378)(22 124 193 436 104 294 153 379)(23 125 194 437 105 295 154 380)(24 126 195 438 106 296 155 381)(25 127 196 439 107 297 156 382)(26 128 169 440 108 298 157 383)(27 129 170 441 109 299 158 384)(28 130 171 442 110 300 159 385)(29 327 204 364 279 64 399 236)(30 328 205 337 280 65 400 237)(31 329 206 338 253 66 401 238)(32 330 207 339 254 67 402 239)(33 331 208 340 255 68 403 240)(34 332 209 341 256 69 404 241)(35 333 210 342 257 70 405 242)(36 334 211 343 258 71 406 243)(37 335 212 344 259 72 407 244)(38 336 213 345 260 73 408 245)(39 309 214 346 261 74 409 246)(40 310 215 347 262 75 410 247)(41 311 216 348 263 76 411 248)(42 312 217 349 264 77 412 249)(43 313 218 350 265 78 413 250)(44 314 219 351 266 79 414 251)(45 315 220 352 267 80 415 252)(46 316 221 353 268 81 416 225)(47 317 222 354 269 82 417 226)(48 318 223 355 270 83 418 227)(49 319 224 356 271 84 419 228)(50 320 197 357 272 57 420 229)(51 321 198 358 273 58 393 230)(52 322 199 359 274 59 394 231)(53 323 200 360 275 60 395 232)(54 324 201 361 276 61 396 233)(55 325 202 362 277 62 397 234)(56 326 203 363 278 63 398 235)
(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 67 15 81)(2 66 16 80)(3 65 17 79)(4 64 18 78)(5 63 19 77)(6 62 20 76)(7 61 21 75)(8 60 22 74)(9 59 23 73)(10 58 24 72)(11 57 25 71)(12 84 26 70)(13 83 27 69)(14 82 28 68)(29 432 43 446)(30 431 44 445)(31 430 45 444)(32 429 46 443)(33 428 47 442)(34 427 48 441)(35 426 49 440)(36 425 50 439)(37 424 51 438)(38 423 52 437)(39 422 53 436)(40 421 54 435)(41 448 55 434)(42 447 56 433)(85 328 99 314)(86 327 100 313)(87 326 101 312)(88 325 102 311)(89 324 103 310)(90 323 104 309)(91 322 105 336)(92 321 106 335)(93 320 107 334)(94 319 108 333)(95 318 109 332)(96 317 110 331)(97 316 111 330)(98 315 112 329)(113 420 127 406)(114 419 128 405)(115 418 129 404)(116 417 130 403)(117 416 131 402)(118 415 132 401)(119 414 133 400)(120 413 134 399)(121 412 135 398)(122 411 136 397)(123 410 137 396)(124 409 138 395)(125 408 139 394)(126 407 140 393)(141 358 155 344)(142 357 156 343)(143 356 157 342)(144 355 158 341)(145 354 159 340)(146 353 160 339)(147 352 161 338)(148 351 162 337)(149 350 163 364)(150 349 164 363)(151 348 165 362)(152 347 166 361)(153 346 167 360)(154 345 168 359)(169 242 183 228)(170 241 184 227)(171 240 185 226)(172 239 186 225)(173 238 187 252)(174 237 188 251)(175 236 189 250)(176 235 190 249)(177 234 191 248)(178 233 192 247)(179 232 193 246)(180 231 194 245)(181 230 195 244)(182 229 196 243)(197 297 211 283)(198 296 212 282)(199 295 213 281)(200 294 214 308)(201 293 215 307)(202 292 216 306)(203 291 217 305)(204 290 218 304)(205 289 219 303)(206 288 220 302)(207 287 221 301)(208 286 222 300)(209 285 223 299)(210 284 224 298)(253 373 267 387)(254 372 268 386)(255 371 269 385)(256 370 270 384)(257 369 271 383)(258 368 272 382)(259 367 273 381)(260 366 274 380)(261 365 275 379)(262 392 276 378)(263 391 277 377)(264 390 278 376)(265 389 279 375)(266 388 280 374)
G:=sub<Sym(448)| (1,131,172,443,111,301,160,386)(2,132,173,444,112,302,161,387)(3,133,174,445,85,303,162,388)(4,134,175,446,86,304,163,389)(5,135,176,447,87,305,164,390)(6,136,177,448,88,306,165,391)(7,137,178,421,89,307,166,392)(8,138,179,422,90,308,167,365)(9,139,180,423,91,281,168,366)(10,140,181,424,92,282,141,367)(11,113,182,425,93,283,142,368)(12,114,183,426,94,284,143,369)(13,115,184,427,95,285,144,370)(14,116,185,428,96,286,145,371)(15,117,186,429,97,287,146,372)(16,118,187,430,98,288,147,373)(17,119,188,431,99,289,148,374)(18,120,189,432,100,290,149,375)(19,121,190,433,101,291,150,376)(20,122,191,434,102,292,151,377)(21,123,192,435,103,293,152,378)(22,124,193,436,104,294,153,379)(23,125,194,437,105,295,154,380)(24,126,195,438,106,296,155,381)(25,127,196,439,107,297,156,382)(26,128,169,440,108,298,157,383)(27,129,170,441,109,299,158,384)(28,130,171,442,110,300,159,385)(29,327,204,364,279,64,399,236)(30,328,205,337,280,65,400,237)(31,329,206,338,253,66,401,238)(32,330,207,339,254,67,402,239)(33,331,208,340,255,68,403,240)(34,332,209,341,256,69,404,241)(35,333,210,342,257,70,405,242)(36,334,211,343,258,71,406,243)(37,335,212,344,259,72,407,244)(38,336,213,345,260,73,408,245)(39,309,214,346,261,74,409,246)(40,310,215,347,262,75,410,247)(41,311,216,348,263,76,411,248)(42,312,217,349,264,77,412,249)(43,313,218,350,265,78,413,250)(44,314,219,351,266,79,414,251)(45,315,220,352,267,80,415,252)(46,316,221,353,268,81,416,225)(47,317,222,354,269,82,417,226)(48,318,223,355,270,83,418,227)(49,319,224,356,271,84,419,228)(50,320,197,357,272,57,420,229)(51,321,198,358,273,58,393,230)(52,322,199,359,274,59,394,231)(53,323,200,360,275,60,395,232)(54,324,201,361,276,61,396,233)(55,325,202,362,277,62,397,234)(56,326,203,363,278,63,398,235), (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), (1,67,15,81)(2,66,16,80)(3,65,17,79)(4,64,18,78)(5,63,19,77)(6,62,20,76)(7,61,21,75)(8,60,22,74)(9,59,23,73)(10,58,24,72)(11,57,25,71)(12,84,26,70)(13,83,27,69)(14,82,28,68)(29,432,43,446)(30,431,44,445)(31,430,45,444)(32,429,46,443)(33,428,47,442)(34,427,48,441)(35,426,49,440)(36,425,50,439)(37,424,51,438)(38,423,52,437)(39,422,53,436)(40,421,54,435)(41,448,55,434)(42,447,56,433)(85,328,99,314)(86,327,100,313)(87,326,101,312)(88,325,102,311)(89,324,103,310)(90,323,104,309)(91,322,105,336)(92,321,106,335)(93,320,107,334)(94,319,108,333)(95,318,109,332)(96,317,110,331)(97,316,111,330)(98,315,112,329)(113,420,127,406)(114,419,128,405)(115,418,129,404)(116,417,130,403)(117,416,131,402)(118,415,132,401)(119,414,133,400)(120,413,134,399)(121,412,135,398)(122,411,136,397)(123,410,137,396)(124,409,138,395)(125,408,139,394)(126,407,140,393)(141,358,155,344)(142,357,156,343)(143,356,157,342)(144,355,158,341)(145,354,159,340)(146,353,160,339)(147,352,161,338)(148,351,162,337)(149,350,163,364)(150,349,164,363)(151,348,165,362)(152,347,166,361)(153,346,167,360)(154,345,168,359)(169,242,183,228)(170,241,184,227)(171,240,185,226)(172,239,186,225)(173,238,187,252)(174,237,188,251)(175,236,189,250)(176,235,190,249)(177,234,191,248)(178,233,192,247)(179,232,193,246)(180,231,194,245)(181,230,195,244)(182,229,196,243)(197,297,211,283)(198,296,212,282)(199,295,213,281)(200,294,214,308)(201,293,215,307)(202,292,216,306)(203,291,217,305)(204,290,218,304)(205,289,219,303)(206,288,220,302)(207,287,221,301)(208,286,222,300)(209,285,223,299)(210,284,224,298)(253,373,267,387)(254,372,268,386)(255,371,269,385)(256,370,270,384)(257,369,271,383)(258,368,272,382)(259,367,273,381)(260,366,274,380)(261,365,275,379)(262,392,276,378)(263,391,277,377)(264,390,278,376)(265,389,279,375)(266,388,280,374)>;
G:=Group( (1,131,172,443,111,301,160,386)(2,132,173,444,112,302,161,387)(3,133,174,445,85,303,162,388)(4,134,175,446,86,304,163,389)(5,135,176,447,87,305,164,390)(6,136,177,448,88,306,165,391)(7,137,178,421,89,307,166,392)(8,138,179,422,90,308,167,365)(9,139,180,423,91,281,168,366)(10,140,181,424,92,282,141,367)(11,113,182,425,93,283,142,368)(12,114,183,426,94,284,143,369)(13,115,184,427,95,285,144,370)(14,116,185,428,96,286,145,371)(15,117,186,429,97,287,146,372)(16,118,187,430,98,288,147,373)(17,119,188,431,99,289,148,374)(18,120,189,432,100,290,149,375)(19,121,190,433,101,291,150,376)(20,122,191,434,102,292,151,377)(21,123,192,435,103,293,152,378)(22,124,193,436,104,294,153,379)(23,125,194,437,105,295,154,380)(24,126,195,438,106,296,155,381)(25,127,196,439,107,297,156,382)(26,128,169,440,108,298,157,383)(27,129,170,441,109,299,158,384)(28,130,171,442,110,300,159,385)(29,327,204,364,279,64,399,236)(30,328,205,337,280,65,400,237)(31,329,206,338,253,66,401,238)(32,330,207,339,254,67,402,239)(33,331,208,340,255,68,403,240)(34,332,209,341,256,69,404,241)(35,333,210,342,257,70,405,242)(36,334,211,343,258,71,406,243)(37,335,212,344,259,72,407,244)(38,336,213,345,260,73,408,245)(39,309,214,346,261,74,409,246)(40,310,215,347,262,75,410,247)(41,311,216,348,263,76,411,248)(42,312,217,349,264,77,412,249)(43,313,218,350,265,78,413,250)(44,314,219,351,266,79,414,251)(45,315,220,352,267,80,415,252)(46,316,221,353,268,81,416,225)(47,317,222,354,269,82,417,226)(48,318,223,355,270,83,418,227)(49,319,224,356,271,84,419,228)(50,320,197,357,272,57,420,229)(51,321,198,358,273,58,393,230)(52,322,199,359,274,59,394,231)(53,323,200,360,275,60,395,232)(54,324,201,361,276,61,396,233)(55,325,202,362,277,62,397,234)(56,326,203,363,278,63,398,235), (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), (1,67,15,81)(2,66,16,80)(3,65,17,79)(4,64,18,78)(5,63,19,77)(6,62,20,76)(7,61,21,75)(8,60,22,74)(9,59,23,73)(10,58,24,72)(11,57,25,71)(12,84,26,70)(13,83,27,69)(14,82,28,68)(29,432,43,446)(30,431,44,445)(31,430,45,444)(32,429,46,443)(33,428,47,442)(34,427,48,441)(35,426,49,440)(36,425,50,439)(37,424,51,438)(38,423,52,437)(39,422,53,436)(40,421,54,435)(41,448,55,434)(42,447,56,433)(85,328,99,314)(86,327,100,313)(87,326,101,312)(88,325,102,311)(89,324,103,310)(90,323,104,309)(91,322,105,336)(92,321,106,335)(93,320,107,334)(94,319,108,333)(95,318,109,332)(96,317,110,331)(97,316,111,330)(98,315,112,329)(113,420,127,406)(114,419,128,405)(115,418,129,404)(116,417,130,403)(117,416,131,402)(118,415,132,401)(119,414,133,400)(120,413,134,399)(121,412,135,398)(122,411,136,397)(123,410,137,396)(124,409,138,395)(125,408,139,394)(126,407,140,393)(141,358,155,344)(142,357,156,343)(143,356,157,342)(144,355,158,341)(145,354,159,340)(146,353,160,339)(147,352,161,338)(148,351,162,337)(149,350,163,364)(150,349,164,363)(151,348,165,362)(152,347,166,361)(153,346,167,360)(154,345,168,359)(169,242,183,228)(170,241,184,227)(171,240,185,226)(172,239,186,225)(173,238,187,252)(174,237,188,251)(175,236,189,250)(176,235,190,249)(177,234,191,248)(178,233,192,247)(179,232,193,246)(180,231,194,245)(181,230,195,244)(182,229,196,243)(197,297,211,283)(198,296,212,282)(199,295,213,281)(200,294,214,308)(201,293,215,307)(202,292,216,306)(203,291,217,305)(204,290,218,304)(205,289,219,303)(206,288,220,302)(207,287,221,301)(208,286,222,300)(209,285,223,299)(210,284,224,298)(253,373,267,387)(254,372,268,386)(255,371,269,385)(256,370,270,384)(257,369,271,383)(258,368,272,382)(259,367,273,381)(260,366,274,380)(261,365,275,379)(262,392,276,378)(263,391,277,377)(264,390,278,376)(265,389,279,375)(266,388,280,374) );
G=PermutationGroup([[(1,131,172,443,111,301,160,386),(2,132,173,444,112,302,161,387),(3,133,174,445,85,303,162,388),(4,134,175,446,86,304,163,389),(5,135,176,447,87,305,164,390),(6,136,177,448,88,306,165,391),(7,137,178,421,89,307,166,392),(8,138,179,422,90,308,167,365),(9,139,180,423,91,281,168,366),(10,140,181,424,92,282,141,367),(11,113,182,425,93,283,142,368),(12,114,183,426,94,284,143,369),(13,115,184,427,95,285,144,370),(14,116,185,428,96,286,145,371),(15,117,186,429,97,287,146,372),(16,118,187,430,98,288,147,373),(17,119,188,431,99,289,148,374),(18,120,189,432,100,290,149,375),(19,121,190,433,101,291,150,376),(20,122,191,434,102,292,151,377),(21,123,192,435,103,293,152,378),(22,124,193,436,104,294,153,379),(23,125,194,437,105,295,154,380),(24,126,195,438,106,296,155,381),(25,127,196,439,107,297,156,382),(26,128,169,440,108,298,157,383),(27,129,170,441,109,299,158,384),(28,130,171,442,110,300,159,385),(29,327,204,364,279,64,399,236),(30,328,205,337,280,65,400,237),(31,329,206,338,253,66,401,238),(32,330,207,339,254,67,402,239),(33,331,208,340,255,68,403,240),(34,332,209,341,256,69,404,241),(35,333,210,342,257,70,405,242),(36,334,211,343,258,71,406,243),(37,335,212,344,259,72,407,244),(38,336,213,345,260,73,408,245),(39,309,214,346,261,74,409,246),(40,310,215,347,262,75,410,247),(41,311,216,348,263,76,411,248),(42,312,217,349,264,77,412,249),(43,313,218,350,265,78,413,250),(44,314,219,351,266,79,414,251),(45,315,220,352,267,80,415,252),(46,316,221,353,268,81,416,225),(47,317,222,354,269,82,417,226),(48,318,223,355,270,83,418,227),(49,319,224,356,271,84,419,228),(50,320,197,357,272,57,420,229),(51,321,198,358,273,58,393,230),(52,322,199,359,274,59,394,231),(53,323,200,360,275,60,395,232),(54,324,201,361,276,61,396,233),(55,325,202,362,277,62,397,234),(56,326,203,363,278,63,398,235)], [(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)], [(1,67,15,81),(2,66,16,80),(3,65,17,79),(4,64,18,78),(5,63,19,77),(6,62,20,76),(7,61,21,75),(8,60,22,74),(9,59,23,73),(10,58,24,72),(11,57,25,71),(12,84,26,70),(13,83,27,69),(14,82,28,68),(29,432,43,446),(30,431,44,445),(31,430,45,444),(32,429,46,443),(33,428,47,442),(34,427,48,441),(35,426,49,440),(36,425,50,439),(37,424,51,438),(38,423,52,437),(39,422,53,436),(40,421,54,435),(41,448,55,434),(42,447,56,433),(85,328,99,314),(86,327,100,313),(87,326,101,312),(88,325,102,311),(89,324,103,310),(90,323,104,309),(91,322,105,336),(92,321,106,335),(93,320,107,334),(94,319,108,333),(95,318,109,332),(96,317,110,331),(97,316,111,330),(98,315,112,329),(113,420,127,406),(114,419,128,405),(115,418,129,404),(116,417,130,403),(117,416,131,402),(118,415,132,401),(119,414,133,400),(120,413,134,399),(121,412,135,398),(122,411,136,397),(123,410,137,396),(124,409,138,395),(125,408,139,394),(126,407,140,393),(141,358,155,344),(142,357,156,343),(143,356,157,342),(144,355,158,341),(145,354,159,340),(146,353,160,339),(147,352,161,338),(148,351,162,337),(149,350,163,364),(150,349,164,363),(151,348,165,362),(152,347,166,361),(153,346,167,360),(154,345,168,359),(169,242,183,228),(170,241,184,227),(171,240,185,226),(172,239,186,225),(173,238,187,252),(174,237,188,251),(175,236,189,250),(176,235,190,249),(177,234,191,248),(178,233,192,247),(179,232,193,246),(180,231,194,245),(181,230,195,244),(182,229,196,243),(197,297,211,283),(198,296,212,282),(199,295,213,281),(200,294,214,308),(201,293,215,307),(202,292,216,306),(203,291,217,305),(204,290,218,304),(205,289,219,303),(206,288,220,302),(207,287,221,301),(208,286,222,300),(209,285,223,299),(210,284,224,298),(253,373,267,387),(254,372,268,386),(255,371,269,385),(256,370,270,384),(257,369,271,383),(258,368,272,382),(259,367,273,381),(260,366,274,380),(261,365,275,379),(262,392,276,378),(263,391,277,377),(264,390,278,376),(265,389,279,375),(266,388,280,374)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | ··· | 8T | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | Q8 | D7 | C4○D4 | D14 | D14 | C8○D4 | Dic14 | C4×D7 | C8×D7 | C4○D28 | D28.2C4 |
| kernel | C8×Dic14 | C28⋊C8 | C8×Dic7 | Dic7⋊C8 | C4×C56 | C4×Dic14 | Dic7⋊C4 | C4⋊Dic7 | C2×Dic14 | Dic14 | C56 | C4×C8 | C28 | C42 | C2×C8 | C14 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 16 | 2 | 3 | 2 | 3 | 6 | 4 | 12 | 12 | 24 | 12 | 24 |
Matrix representation of C8×Dic14 ►in GL3(𝔽113) generated by
| 69 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 1 | 0 | 0 |
| 0 | 46 | 104 |
| 0 | 9 | 13 |
| 1 | 0 | 0 |
| 0 | 69 | 48 |
| 0 | 75 | 44 |
G:=sub<GL(3,GF(113))| [69,0,0,0,112,0,0,0,112],[1,0,0,0,46,9,0,104,13],[1,0,0,0,69,75,0,48,44] >;
C8×Dic14 in GAP, Magma, Sage, TeX
C_8\times {\rm Dic}_{14} % in TeX
G:=Group("C8xDic14"); // GroupNames label
G:=SmallGroup(448,212);
// by ID
G=gap.SmallGroup(448,212);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=1,c^2=b^14,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations