metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.2Q8, C14.6D16, C14.3Q32, C28.3Q16, C8.4Dic14, C7⋊C16⋊5C4, C56.7(C2×C4), C8.24(C4×D7), C7⋊1(C16⋊3C4), C28.2(C4⋊C4), (C2×C28).89D4, (C2×C14).31D8, C2.D8.1D7, (C2×C8).219D14, C2.1(C7⋊D16), C56⋊1C4.11C2, C14.2(C2.D8), C2.1(C7⋊Q32), C4.1(C7⋊Q16), C4.2(Dic7⋊C4), (C2×C56).71C22, C22.12(D4⋊D7), C2.3(C28.Q8), (C2×C7⋊C16).2C2, (C7×C2.D8).1C2, (C2×C4).113(C7⋊D4), SmallGroup(448,46)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.4Dic14
G = < a,b,c | a8=b28=1, c2=ab14, bab-1=a-1, ac=ca, cbc-1=ab-1 >
Smallest permutation representation of C8.4Dic14
►Regular action on 448 points
Generators in S448
(1 67 289 39 313 190 367 261)(2 262 368 191 314 40 290 68)(3 69 291 41 315 192 369 263)(4 264 370 193 316 42 292 70)(5 71 293 43 317 194 371 265)(6 266 372 195 318 44 294 72)(7 73 295 45 319 196 373 267)(8 268 374 169 320 46 296 74)(9 75 297 47 321 170 375 269)(10 270 376 171 322 48 298 76)(11 77 299 49 323 172 377 271)(12 272 378 173 324 50 300 78)(13 79 301 51 325 174 379 273)(14 274 380 175 326 52 302 80)(15 81 303 53 327 176 381 275)(16 276 382 177 328 54 304 82)(17 83 305 55 329 178 383 277)(18 278 384 179 330 56 306 84)(19 57 307 29 331 180 385 279)(20 280 386 181 332 30 308 58)(21 59 281 31 333 182 387 253)(22 254 388 183 334 32 282 60)(23 61 283 33 335 184 389 255)(24 256 390 185 336 34 284 62)(25 63 285 35 309 186 391 257)(26 258 392 187 310 36 286 64)(27 65 287 37 311 188 365 259)(28 260 366 189 312 38 288 66)(85 446 416 148 119 210 246 360)(86 361 247 211 120 149 417 447)(87 448 418 150 121 212 248 362)(88 363 249 213 122 151 419 421)(89 422 420 152 123 214 250 364)(90 337 251 215 124 153 393 423)(91 424 394 154 125 216 252 338)(92 339 225 217 126 155 395 425)(93 426 396 156 127 218 226 340)(94 341 227 219 128 157 397 427)(95 428 398 158 129 220 228 342)(96 343 229 221 130 159 399 429)(97 430 400 160 131 222 230 344)(98 345 231 223 132 161 401 431)(99 432 402 162 133 224 232 346)(100 347 233 197 134 163 403 433)(101 434 404 164 135 198 234 348)(102 349 235 199 136 165 405 435)(103 436 406 166 137 200 236 350)(104 351 237 201 138 167 407 437)(105 438 408 168 139 202 238 352)(106 353 239 203 140 141 409 439)(107 440 410 142 113 204 240 354)(108 355 241 205 114 143 411 441)(109 442 412 144 115 206 242 356)(110 357 243 207 116 145 413 443)(111 444 414 146 117 208 244 358)(112 359 245 209 118 147 415 445)
(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)
(1 118 81 161 289 415 53 431 313 112 176 345 367 245 275 223)(2 208 276 230 368 358 177 97 314 444 54 400 290 146 82 131)(3 116 83 159 291 413 55 429 315 110 178 343 369 243 277 221)(4 206 278 228 370 356 179 95 316 442 56 398 292 144 84 129)(5 114 57 157 293 411 29 427 317 108 180 341 371 241 279 219)(6 204 280 226 372 354 181 93 318 440 30 396 294 142 58 127)(7 140 59 155 295 409 31 425 319 106 182 339 373 239 253 217)(8 202 254 252 374 352 183 91 320 438 32 394 296 168 60 125)(9 138 61 153 297 407 33 423 321 104 184 337 375 237 255 215)(10 200 256 250 376 350 185 89 322 436 34 420 298 166 62 123)(11 136 63 151 299 405 35 421 323 102 186 363 377 235 257 213)(12 198 258 248 378 348 187 87 324 434 36 418 300 164 64 121)(13 134 65 149 301 403 37 447 325 100 188 361 379 233 259 211)(14 224 260 246 380 346 189 85 326 432 38 416 302 162 66 119)(15 132 67 147 303 401 39 445 327 98 190 359 381 231 261 209)(16 222 262 244 382 344 191 111 328 430 40 414 304 160 68 117)(17 130 69 145 305 399 41 443 329 96 192 357 383 229 263 207)(18 220 264 242 384 342 193 109 330 428 42 412 306 158 70 115)(19 128 71 143 307 397 43 441 331 94 194 355 385 227 265 205)(20 218 266 240 386 340 195 107 332 426 44 410 308 156 72 113)(21 126 73 141 281 395 45 439 333 92 196 353 387 225 267 203)(22 216 268 238 388 338 169 105 334 424 46 408 282 154 74 139)(23 124 75 167 283 393 47 437 335 90 170 351 389 251 269 201)(24 214 270 236 390 364 171 103 336 422 48 406 284 152 76 137)(25 122 77 165 285 419 49 435 309 88 172 349 391 249 271 199)(26 212 272 234 392 362 173 101 310 448 50 404 286 150 78 135)(27 120 79 163 287 417 51 433 311 86 174 347 365 247 273 197)(28 210 274 232 366 360 175 99 312 446 52 402 288 148 80 133)
G:=sub<Sym(448)| (1,67,289,39,313,190,367,261)(2,262,368,191,314,40,290,68)(3,69,291,41,315,192,369,263)(4,264,370,193,316,42,292,70)(5,71,293,43,317,194,371,265)(6,266,372,195,318,44,294,72)(7,73,295,45,319,196,373,267)(8,268,374,169,320,46,296,74)(9,75,297,47,321,170,375,269)(10,270,376,171,322,48,298,76)(11,77,299,49,323,172,377,271)(12,272,378,173,324,50,300,78)(13,79,301,51,325,174,379,273)(14,274,380,175,326,52,302,80)(15,81,303,53,327,176,381,275)(16,276,382,177,328,54,304,82)(17,83,305,55,329,178,383,277)(18,278,384,179,330,56,306,84)(19,57,307,29,331,180,385,279)(20,280,386,181,332,30,308,58)(21,59,281,31,333,182,387,253)(22,254,388,183,334,32,282,60)(23,61,283,33,335,184,389,255)(24,256,390,185,336,34,284,62)(25,63,285,35,309,186,391,257)(26,258,392,187,310,36,286,64)(27,65,287,37,311,188,365,259)(28,260,366,189,312,38,288,66)(85,446,416,148,119,210,246,360)(86,361,247,211,120,149,417,447)(87,448,418,150,121,212,248,362)(88,363,249,213,122,151,419,421)(89,422,420,152,123,214,250,364)(90,337,251,215,124,153,393,423)(91,424,394,154,125,216,252,338)(92,339,225,217,126,155,395,425)(93,426,396,156,127,218,226,340)(94,341,227,219,128,157,397,427)(95,428,398,158,129,220,228,342)(96,343,229,221,130,159,399,429)(97,430,400,160,131,222,230,344)(98,345,231,223,132,161,401,431)(99,432,402,162,133,224,232,346)(100,347,233,197,134,163,403,433)(101,434,404,164,135,198,234,348)(102,349,235,199,136,165,405,435)(103,436,406,166,137,200,236,350)(104,351,237,201,138,167,407,437)(105,438,408,168,139,202,238,352)(106,353,239,203,140,141,409,439)(107,440,410,142,113,204,240,354)(108,355,241,205,114,143,411,441)(109,442,412,144,115,206,242,356)(110,357,243,207,116,145,413,443)(111,444,414,146,117,208,244,358)(112,359,245,209,118,147,415,445), (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), (1,118,81,161,289,415,53,431,313,112,176,345,367,245,275,223)(2,208,276,230,368,358,177,97,314,444,54,400,290,146,82,131)(3,116,83,159,291,413,55,429,315,110,178,343,369,243,277,221)(4,206,278,228,370,356,179,95,316,442,56,398,292,144,84,129)(5,114,57,157,293,411,29,427,317,108,180,341,371,241,279,219)(6,204,280,226,372,354,181,93,318,440,30,396,294,142,58,127)(7,140,59,155,295,409,31,425,319,106,182,339,373,239,253,217)(8,202,254,252,374,352,183,91,320,438,32,394,296,168,60,125)(9,138,61,153,297,407,33,423,321,104,184,337,375,237,255,215)(10,200,256,250,376,350,185,89,322,436,34,420,298,166,62,123)(11,136,63,151,299,405,35,421,323,102,186,363,377,235,257,213)(12,198,258,248,378,348,187,87,324,434,36,418,300,164,64,121)(13,134,65,149,301,403,37,447,325,100,188,361,379,233,259,211)(14,224,260,246,380,346,189,85,326,432,38,416,302,162,66,119)(15,132,67,147,303,401,39,445,327,98,190,359,381,231,261,209)(16,222,262,244,382,344,191,111,328,430,40,414,304,160,68,117)(17,130,69,145,305,399,41,443,329,96,192,357,383,229,263,207)(18,220,264,242,384,342,193,109,330,428,42,412,306,158,70,115)(19,128,71,143,307,397,43,441,331,94,194,355,385,227,265,205)(20,218,266,240,386,340,195,107,332,426,44,410,308,156,72,113)(21,126,73,141,281,395,45,439,333,92,196,353,387,225,267,203)(22,216,268,238,388,338,169,105,334,424,46,408,282,154,74,139)(23,124,75,167,283,393,47,437,335,90,170,351,389,251,269,201)(24,214,270,236,390,364,171,103,336,422,48,406,284,152,76,137)(25,122,77,165,285,419,49,435,309,88,172,349,391,249,271,199)(26,212,272,234,392,362,173,101,310,448,50,404,286,150,78,135)(27,120,79,163,287,417,51,433,311,86,174,347,365,247,273,197)(28,210,274,232,366,360,175,99,312,446,52,402,288,148,80,133)>;
G:=Group( (1,67,289,39,313,190,367,261)(2,262,368,191,314,40,290,68)(3,69,291,41,315,192,369,263)(4,264,370,193,316,42,292,70)(5,71,293,43,317,194,371,265)(6,266,372,195,318,44,294,72)(7,73,295,45,319,196,373,267)(8,268,374,169,320,46,296,74)(9,75,297,47,321,170,375,269)(10,270,376,171,322,48,298,76)(11,77,299,49,323,172,377,271)(12,272,378,173,324,50,300,78)(13,79,301,51,325,174,379,273)(14,274,380,175,326,52,302,80)(15,81,303,53,327,176,381,275)(16,276,382,177,328,54,304,82)(17,83,305,55,329,178,383,277)(18,278,384,179,330,56,306,84)(19,57,307,29,331,180,385,279)(20,280,386,181,332,30,308,58)(21,59,281,31,333,182,387,253)(22,254,388,183,334,32,282,60)(23,61,283,33,335,184,389,255)(24,256,390,185,336,34,284,62)(25,63,285,35,309,186,391,257)(26,258,392,187,310,36,286,64)(27,65,287,37,311,188,365,259)(28,260,366,189,312,38,288,66)(85,446,416,148,119,210,246,360)(86,361,247,211,120,149,417,447)(87,448,418,150,121,212,248,362)(88,363,249,213,122,151,419,421)(89,422,420,152,123,214,250,364)(90,337,251,215,124,153,393,423)(91,424,394,154,125,216,252,338)(92,339,225,217,126,155,395,425)(93,426,396,156,127,218,226,340)(94,341,227,219,128,157,397,427)(95,428,398,158,129,220,228,342)(96,343,229,221,130,159,399,429)(97,430,400,160,131,222,230,344)(98,345,231,223,132,161,401,431)(99,432,402,162,133,224,232,346)(100,347,233,197,134,163,403,433)(101,434,404,164,135,198,234,348)(102,349,235,199,136,165,405,435)(103,436,406,166,137,200,236,350)(104,351,237,201,138,167,407,437)(105,438,408,168,139,202,238,352)(106,353,239,203,140,141,409,439)(107,440,410,142,113,204,240,354)(108,355,241,205,114,143,411,441)(109,442,412,144,115,206,242,356)(110,357,243,207,116,145,413,443)(111,444,414,146,117,208,244,358)(112,359,245,209,118,147,415,445), (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), (1,118,81,161,289,415,53,431,313,112,176,345,367,245,275,223)(2,208,276,230,368,358,177,97,314,444,54,400,290,146,82,131)(3,116,83,159,291,413,55,429,315,110,178,343,369,243,277,221)(4,206,278,228,370,356,179,95,316,442,56,398,292,144,84,129)(5,114,57,157,293,411,29,427,317,108,180,341,371,241,279,219)(6,204,280,226,372,354,181,93,318,440,30,396,294,142,58,127)(7,140,59,155,295,409,31,425,319,106,182,339,373,239,253,217)(8,202,254,252,374,352,183,91,320,438,32,394,296,168,60,125)(9,138,61,153,297,407,33,423,321,104,184,337,375,237,255,215)(10,200,256,250,376,350,185,89,322,436,34,420,298,166,62,123)(11,136,63,151,299,405,35,421,323,102,186,363,377,235,257,213)(12,198,258,248,378,348,187,87,324,434,36,418,300,164,64,121)(13,134,65,149,301,403,37,447,325,100,188,361,379,233,259,211)(14,224,260,246,380,346,189,85,326,432,38,416,302,162,66,119)(15,132,67,147,303,401,39,445,327,98,190,359,381,231,261,209)(16,222,262,244,382,344,191,111,328,430,40,414,304,160,68,117)(17,130,69,145,305,399,41,443,329,96,192,357,383,229,263,207)(18,220,264,242,384,342,193,109,330,428,42,412,306,158,70,115)(19,128,71,143,307,397,43,441,331,94,194,355,385,227,265,205)(20,218,266,240,386,340,195,107,332,426,44,410,308,156,72,113)(21,126,73,141,281,395,45,439,333,92,196,353,387,225,267,203)(22,216,268,238,388,338,169,105,334,424,46,408,282,154,74,139)(23,124,75,167,283,393,47,437,335,90,170,351,389,251,269,201)(24,214,270,236,390,364,171,103,336,422,48,406,284,152,76,137)(25,122,77,165,285,419,49,435,309,88,172,349,391,249,271,199)(26,212,272,234,392,362,173,101,310,448,50,404,286,150,78,135)(27,120,79,163,287,417,51,433,311,86,174,347,365,247,273,197)(28,210,274,232,366,360,175,99,312,446,52,402,288,148,80,133) );
G=PermutationGroup([[(1,67,289,39,313,190,367,261),(2,262,368,191,314,40,290,68),(3,69,291,41,315,192,369,263),(4,264,370,193,316,42,292,70),(5,71,293,43,317,194,371,265),(6,266,372,195,318,44,294,72),(7,73,295,45,319,196,373,267),(8,268,374,169,320,46,296,74),(9,75,297,47,321,170,375,269),(10,270,376,171,322,48,298,76),(11,77,299,49,323,172,377,271),(12,272,378,173,324,50,300,78),(13,79,301,51,325,174,379,273),(14,274,380,175,326,52,302,80),(15,81,303,53,327,176,381,275),(16,276,382,177,328,54,304,82),(17,83,305,55,329,178,383,277),(18,278,384,179,330,56,306,84),(19,57,307,29,331,180,385,279),(20,280,386,181,332,30,308,58),(21,59,281,31,333,182,387,253),(22,254,388,183,334,32,282,60),(23,61,283,33,335,184,389,255),(24,256,390,185,336,34,284,62),(25,63,285,35,309,186,391,257),(26,258,392,187,310,36,286,64),(27,65,287,37,311,188,365,259),(28,260,366,189,312,38,288,66),(85,446,416,148,119,210,246,360),(86,361,247,211,120,149,417,447),(87,448,418,150,121,212,248,362),(88,363,249,213,122,151,419,421),(89,422,420,152,123,214,250,364),(90,337,251,215,124,153,393,423),(91,424,394,154,125,216,252,338),(92,339,225,217,126,155,395,425),(93,426,396,156,127,218,226,340),(94,341,227,219,128,157,397,427),(95,428,398,158,129,220,228,342),(96,343,229,221,130,159,399,429),(97,430,400,160,131,222,230,344),(98,345,231,223,132,161,401,431),(99,432,402,162,133,224,232,346),(100,347,233,197,134,163,403,433),(101,434,404,164,135,198,234,348),(102,349,235,199,136,165,405,435),(103,436,406,166,137,200,236,350),(104,351,237,201,138,167,407,437),(105,438,408,168,139,202,238,352),(106,353,239,203,140,141,409,439),(107,440,410,142,113,204,240,354),(108,355,241,205,114,143,411,441),(109,442,412,144,115,206,242,356),(110,357,243,207,116,145,413,443),(111,444,414,146,117,208,244,358),(112,359,245,209,118,147,415,445)], [(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)], [(1,118,81,161,289,415,53,431,313,112,176,345,367,245,275,223),(2,208,276,230,368,358,177,97,314,444,54,400,290,146,82,131),(3,116,83,159,291,413,55,429,315,110,178,343,369,243,277,221),(4,206,278,228,370,356,179,95,316,442,56,398,292,144,84,129),(5,114,57,157,293,411,29,427,317,108,180,341,371,241,279,219),(6,204,280,226,372,354,181,93,318,440,30,396,294,142,58,127),(7,140,59,155,295,409,31,425,319,106,182,339,373,239,253,217),(8,202,254,252,374,352,183,91,320,438,32,394,296,168,60,125),(9,138,61,153,297,407,33,423,321,104,184,337,375,237,255,215),(10,200,256,250,376,350,185,89,322,436,34,420,298,166,62,123),(11,136,63,151,299,405,35,421,323,102,186,363,377,235,257,213),(12,198,258,248,378,348,187,87,324,434,36,418,300,164,64,121),(13,134,65,149,301,403,37,447,325,100,188,361,379,233,259,211),(14,224,260,246,380,346,189,85,326,432,38,416,302,162,66,119),(15,132,67,147,303,401,39,445,327,98,190,359,381,231,261,209),(16,222,262,244,382,344,191,111,328,430,40,414,304,160,68,117),(17,130,69,145,305,399,41,443,329,96,192,357,383,229,263,207),(18,220,264,242,384,342,193,109,330,428,42,412,306,158,70,115),(19,128,71,143,307,397,43,441,331,94,194,355,385,227,265,205),(20,218,266,240,386,340,195,107,332,426,44,410,308,156,72,113),(21,126,73,141,281,395,45,439,333,92,196,353,387,225,267,203),(22,216,268,238,388,338,169,105,334,424,46,408,282,154,74,139),(23,124,75,167,283,393,47,437,335,90,170,351,389,251,269,201),(24,214,270,236,390,364,171,103,336,422,48,406,284,152,76,137),(25,122,77,165,285,419,49,435,309,88,172,349,391,249,271,199),(26,212,272,234,392,362,173,101,310,448,50,404,286,150,78,135),(27,120,79,163,287,417,51,433,311,86,174,347,365,247,273,197),(28,210,274,232,366,360,175,99,312,446,52,402,288,148,80,133)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 16A | ··· | 16H | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | + | + | + | - | - | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D7 | Q16 | D8 | D14 | D16 | Q32 | Dic14 | C4×D7 | C7⋊D4 | C7⋊Q16 | D4⋊D7 | C7⋊D16 | C7⋊Q32 |
| kernel | C8.4Dic14 | C2×C7⋊C16 | C56⋊1C4 | C7×C2.D8 | C7⋊C16 | C56 | C2×C28 | C2.D8 | C28 | C2×C14 | C2×C8 | C14 | C14 | C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 3 | 3 | 6 | 6 |
Matrix representation of C8.4Dic14 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 51 | 51 |
| 0 | 0 | 31 | 0 |
| 0 | 98 | 0 | 0 |
| 15 | 92 | 0 | 0 |
| 0 | 0 | 103 | 14 |
| 0 | 0 | 17 | 10 |
| 52 | 44 | 0 | 0 |
| 49 | 61 | 0 | 0 |
| 0 | 0 | 22 | 36 |
| 0 | 0 | 95 | 99 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,51,31,0,0,51,0],[0,15,0,0,98,92,0,0,0,0,103,17,0,0,14,10],[52,49,0,0,44,61,0,0,0,0,22,95,0,0,36,99] >;
C8.4Dic14 in GAP, Magma, Sage, TeX
C_8._4{\rm Dic}_{14} % in TeX
G:=Group("C8.4Dic14"); // GroupNames label
G:=SmallGroup(448,46);
// by ID
G=gap.SmallGroup(448,46);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,141,36,346,192,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=1,c^2=a*b^14,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations
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