direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic7⋊C8, C14⋊2(C4⋊C8), Dic7⋊4(C2×C8), (C2×Dic7)⋊4C8, C28.46(C4⋊C4), C28.85(C2×Q8), (C2×C28).68Q8, (C22×C8).5D7, (C2×C28).497D4, C28.432(C2×D4), (C2×C8).290D14, (C22×C56).8C2, C22.16(C8×D7), C23.61(C4×D7), C14.18(C22×C8), (C4×Dic7).20C4, (C2×C4).58Dic14, C4.50(C2×Dic14), C4.33(Dic7⋊C4), (C2×C56).350C22, (C2×C28).854C23, (C22×C4).461D14, C22.9(C8⋊D7), C14.10(C2×M4(2)), (C2×C14).11M4(2), (C22×Dic7).11C4, C22.25(Dic7⋊C4), (C22×C28).557C22, (C4×Dic7).279C22, C7⋊3(C2×C4⋊C8), C2.18(D7×C2×C8), C14.42(C2×C4⋊C4), C2.3(C2×C8⋊D7), C22.57(C2×C4×D7), (C2×C14).18(C2×C8), (C2×C4).182(C4×D7), C4.122(C2×C7⋊D4), (C22×C7⋊C8).19C2, C2.2(C2×Dic7⋊C4), (C2×C4×Dic7).37C2, (C2×C14).47(C4⋊C4), (C2×C28).252(C2×C4), (C2×C7⋊C8).317C22, (C2×C4).275(C7⋊D4), (C22×C14).89(C2×C4), (C2×Dic7).97(C2×C4), (C2×C4).796(C22×D7), (C2×C14).124(C22×C4), SmallGroup(448,633)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic7⋊C8
G = < a,b,c,d | a2=b14=d8=1, c2=b7, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b7c >
Subgroups: 420 in 138 conjugacy classes, 87 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C2×C8, C22×C4, C22×C4, Dic7, Dic7, C28, C28, C2×C14, C2×C14, C4⋊C8, C2×C42, C22×C8, C22×C8, C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C2×C4⋊C8, C2×C7⋊C8, C2×C7⋊C8, C4×Dic7, C2×C56, C2×C56, C22×Dic7, C22×C28, Dic7⋊C8, C22×C7⋊C8, C2×C4×Dic7, C22×C56, C2×Dic7⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, D7, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, D14, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), Dic14, C4×D7, C7⋊D4, C22×D7, C2×C4⋊C8, C8×D7, C8⋊D7, Dic7⋊C4, C2×Dic14, C2×C4×D7, C2×C7⋊D4, Dic7⋊C8, D7×C2×C8, C2×C8⋊D7, C2×Dic7⋊C4, C2×Dic7⋊C8
►Regular action on 448 points
Generators in S448
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 350)(16 337)(17 338)(18 339)(19 340)(20 341)(21 342)(22 343)(23 344)(24 345)(25 346)(26 347)(27 348)(28 349)(29 389)(30 390)(31 391)(32 392)(33 379)(34 380)(35 381)(36 382)(37 383)(38 384)(39 385)(40 386)(41 387)(42 388)(43 173)(44 174)(45 175)(46 176)(47 177)(48 178)(49 179)(50 180)(51 181)(52 182)(53 169)(54 170)(55 171)(56 172)(57 333)(58 334)(59 335)(60 336)(61 323)(62 324)(63 325)(64 326)(65 327)(66 328)(67 329)(68 330)(69 331)(70 332)(71 302)(72 303)(73 304)(74 305)(75 306)(76 307)(77 308)(78 295)(79 296)(80 297)(81 298)(82 299)(83 300)(84 301)(99 426)(100 427)(101 428)(102 429)(103 430)(104 431)(105 432)(106 433)(107 434)(108 421)(109 422)(110 423)(111 424)(112 425)(113 164)(114 165)(115 166)(116 167)(117 168)(118 155)(119 156)(120 157)(121 158)(122 159)(123 160)(124 161)(125 162)(126 163)(127 366)(128 367)(129 368)(130 369)(131 370)(132 371)(133 372)(134 373)(135 374)(136 375)(137 376)(138 377)(139 378)(140 365)(141 393)(142 394)(143 395)(144 396)(145 397)(146 398)(147 399)(148 400)(149 401)(150 402)(151 403)(152 404)(153 405)(154 406)(183 224)(184 211)(185 212)(186 213)(187 214)(188 215)(189 216)(190 217)(191 218)(192 219)(193 220)(194 221)(195 222)(196 223)(197 275)(198 276)(199 277)(200 278)(201 279)(202 280)(203 267)(204 268)(205 269)(206 270)(207 271)(208 272)(209 273)(210 274)(225 281)(226 282)(227 283)(228 284)(229 285)(230 286)(231 287)(232 288)(233 289)(234 290)(235 291)(236 292)(237 293)(238 294)(239 315)(240 316)(241 317)(242 318)(243 319)(244 320)(245 321)(246 322)(247 309)(248 310)(249 311)(250 312)(251 313)(252 314)(253 418)(254 419)(255 420)(256 407)(257 408)(258 409)(259 410)(260 411)(261 412)(262 413)(263 414)(264 415)(265 416)(266 417)(351 435)(352 436)(353 437)(354 438)(355 439)(356 440)(357 441)(358 442)(359 443)(360 444)(361 445)(362 446)(363 447)(364 448)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154)(155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182)(183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224)(225 226 227 228 229 230 231 232 233 234 235 236 237 238)(239 240 241 242 243 244 245 246 247 248 249 250 251 252)(253 254 255 256 257 258 259 260 261 262 263 264 265 266)(267 268 269 270 271 272 273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288 289 290 291 292 293 294)(295 296 297 298 299 300 301 302 303 304 305 306 307 308)(309 310 311 312 313 314 315 316 317 318 319 320 321 322)(323 324 325 326 327 328 329 330 331 332 333 334 335 336)(337 338 339 340 341 342 343 344 345 346 347 348 349 350)(351 352 353 354 355 356 357 358 359 360 361 362 363 364)(365 366 367 368 369 370 371 372 373 374 375 376 377 378)(379 380 381 382 383 384 385 386 387 388 389 390 391 392)(393 394 395 396 397 398 399 400 401 402 403 404 405 406)(407 408 409 410 411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430 431 432 433 434)(435 436 437 438 439 440 441 442 443 444 445 446 447 448)
(1 196 8 189)(2 195 9 188)(3 194 10 187)(4 193 11 186)(5 192 12 185)(6 191 13 184)(7 190 14 183)(15 125 22 118)(16 124 23 117)(17 123 24 116)(18 122 25 115)(19 121 26 114)(20 120 27 113)(21 119 28 126)(29 246 36 239)(30 245 37 252)(31 244 38 251)(32 243 39 250)(33 242 40 249)(34 241 41 248)(35 240 42 247)(43 326 50 333)(44 325 51 332)(45 324 52 331)(46 323 53 330)(47 336 54 329)(48 335 55 328)(49 334 56 327)(57 173 64 180)(58 172 65 179)(59 171 66 178)(60 170 67 177)(61 169 68 176)(62 182 69 175)(63 181 70 174)(71 228 78 235)(72 227 79 234)(73 226 80 233)(74 225 81 232)(75 238 82 231)(76 237 83 230)(77 236 84 229)(85 216 92 223)(86 215 93 222)(87 214 94 221)(88 213 95 220)(89 212 96 219)(90 211 97 218)(91 224 98 217)(99 351 106 358)(100 364 107 357)(101 363 108 356)(102 362 109 355)(103 361 110 354)(104 360 111 353)(105 359 112 352)(127 401 134 394)(128 400 135 393)(129 399 136 406)(130 398 137 405)(131 397 138 404)(132 396 139 403)(133 395 140 402)(141 367 148 374)(142 366 149 373)(143 365 150 372)(144 378 151 371)(145 377 152 370)(146 376 153 369)(147 375 154 368)(155 350 162 343)(156 349 163 342)(157 348 164 341)(158 347 165 340)(159 346 166 339)(160 345 167 338)(161 344 168 337)(197 255 204 262)(198 254 205 261)(199 253 206 260)(200 266 207 259)(201 265 208 258)(202 264 209 257)(203 263 210 256)(267 414 274 407)(268 413 275 420)(269 412 276 419)(270 411 277 418)(271 410 278 417)(272 409 279 416)(273 408 280 415)(281 298 288 305)(282 297 289 304)(283 296 290 303)(284 295 291 302)(285 308 292 301)(286 307 293 300)(287 306 294 299)(309 381 316 388)(310 380 317 387)(311 379 318 386)(312 392 319 385)(313 391 320 384)(314 390 321 383)(315 389 322 382)(421 440 428 447)(422 439 429 446)(423 438 430 445)(424 437 431 444)(425 436 432 443)(426 435 433 442)(427 448 434 441)
(1 199 400 114 379 182 102 236)(2 200 401 115 380 169 103 237)(3 201 402 116 381 170 104 238)(4 202 403 117 382 171 105 225)(5 203 404 118 383 172 106 226)(6 204 405 119 384 173 107 227)(7 205 406 120 385 174 108 228)(8 206 393 121 386 175 109 229)(9 207 394 122 387 176 110 230)(10 208 395 123 388 177 111 231)(11 209 396 124 389 178 112 232)(12 210 397 125 390 179 99 233)(13 197 398 126 391 180 100 234)(14 198 399 113 392 181 101 235)(15 321 65 351 80 185 263 138)(16 322 66 352 81 186 264 139)(17 309 67 353 82 187 265 140)(18 310 68 354 83 188 266 127)(19 311 69 355 84 189 253 128)(20 312 70 356 71 190 254 129)(21 313 57 357 72 191 255 130)(22 314 58 358 73 192 256 131)(23 315 59 359 74 193 257 132)(24 316 60 360 75 194 258 133)(25 317 61 361 76 195 259 134)(26 318 62 362 77 196 260 135)(27 319 63 363 78 183 261 136)(28 320 64 364 79 184 262 137)(29 48 425 288 88 273 144 161)(30 49 426 289 89 274 145 162)(31 50 427 290 90 275 146 163)(32 51 428 291 91 276 147 164)(33 52 429 292 92 277 148 165)(34 53 430 293 93 278 149 166)(35 54 431 294 94 279 150 167)(36 55 432 281 95 280 151 168)(37 56 433 282 96 267 152 155)(38 43 434 283 97 268 153 156)(39 44 421 284 98 269 154 157)(40 45 422 285 85 270 141 158)(41 46 423 286 86 271 142 159)(42 47 424 287 87 272 143 160)(211 413 376 349 244 326 448 296)(212 414 377 350 245 327 435 297)(213 415 378 337 246 328 436 298)(214 416 365 338 247 329 437 299)(215 417 366 339 248 330 438 300)(216 418 367 340 249 331 439 301)(217 419 368 341 250 332 440 302)(218 420 369 342 251 333 441 303)(219 407 370 343 252 334 442 304)(220 408 371 344 239 335 443 305)(221 409 372 345 240 336 444 306)(222 410 373 346 241 323 445 307)(223 411 374 347 242 324 446 308)(224 412 375 348 243 325 447 295)
G:=sub<Sym(448)| (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,350)(16,337)(17,338)(18,339)(19,340)(20,341)(21,342)(22,343)(23,344)(24,345)(25,346)(26,347)(27,348)(28,349)(29,389)(30,390)(31,391)(32,392)(33,379)(34,380)(35,381)(36,382)(37,383)(38,384)(39,385)(40,386)(41,387)(42,388)(43,173)(44,174)(45,175)(46,176)(47,177)(48,178)(49,179)(50,180)(51,181)(52,182)(53,169)(54,170)(55,171)(56,172)(57,333)(58,334)(59,335)(60,336)(61,323)(62,324)(63,325)(64,326)(65,327)(66,328)(67,329)(68,330)(69,331)(70,332)(71,302)(72,303)(73,304)(74,305)(75,306)(76,307)(77,308)(78,295)(79,296)(80,297)(81,298)(82,299)(83,300)(84,301)(99,426)(100,427)(101,428)(102,429)(103,430)(104,431)(105,432)(106,433)(107,434)(108,421)(109,422)(110,423)(111,424)(112,425)(113,164)(114,165)(115,166)(116,167)(117,168)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,161)(125,162)(126,163)(127,366)(128,367)(129,368)(130,369)(131,370)(132,371)(133,372)(134,373)(135,374)(136,375)(137,376)(138,377)(139,378)(140,365)(141,393)(142,394)(143,395)(144,396)(145,397)(146,398)(147,399)(148,400)(149,401)(150,402)(151,403)(152,404)(153,405)(154,406)(183,224)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216)(190,217)(191,218)(192,219)(193,220)(194,221)(195,222)(196,223)(197,275)(198,276)(199,277)(200,278)(201,279)(202,280)(203,267)(204,268)(205,269)(206,270)(207,271)(208,272)(209,273)(210,274)(225,281)(226,282)(227,283)(228,284)(229,285)(230,286)(231,287)(232,288)(233,289)(234,290)(235,291)(236,292)(237,293)(238,294)(239,315)(240,316)(241,317)(242,318)(243,319)(244,320)(245,321)(246,322)(247,309)(248,310)(249,311)(250,312)(251,313)(252,314)(253,418)(254,419)(255,420)(256,407)(257,408)(258,409)(259,410)(260,411)(261,412)(262,413)(263,414)(264,415)(265,416)(266,417)(351,435)(352,436)(353,437)(354,438)(355,439)(356,440)(357,441)(358,442)(359,443)(360,444)(361,445)(362,446)(363,447)(364,448), (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232,233,234,235,236,237,238)(239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266)(267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294)(295,296,297,298,299,300,301,302,303,304,305,306,307,308)(309,310,311,312,313,314,315,316,317,318,319,320,321,322)(323,324,325,326,327,328,329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344,345,346,347,348,349,350)(351,352,353,354,355,356,357,358,359,360,361,362,363,364)(365,366,367,368,369,370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387,388,389,390,391,392)(393,394,395,396,397,398,399,400,401,402,403,404,405,406)(407,408,409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432,433,434)(435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,196,8,189)(2,195,9,188)(3,194,10,187)(4,193,11,186)(5,192,12,185)(6,191,13,184)(7,190,14,183)(15,125,22,118)(16,124,23,117)(17,123,24,116)(18,122,25,115)(19,121,26,114)(20,120,27,113)(21,119,28,126)(29,246,36,239)(30,245,37,252)(31,244,38,251)(32,243,39,250)(33,242,40,249)(34,241,41,248)(35,240,42,247)(43,326,50,333)(44,325,51,332)(45,324,52,331)(46,323,53,330)(47,336,54,329)(48,335,55,328)(49,334,56,327)(57,173,64,180)(58,172,65,179)(59,171,66,178)(60,170,67,177)(61,169,68,176)(62,182,69,175)(63,181,70,174)(71,228,78,235)(72,227,79,234)(73,226,80,233)(74,225,81,232)(75,238,82,231)(76,237,83,230)(77,236,84,229)(85,216,92,223)(86,215,93,222)(87,214,94,221)(88,213,95,220)(89,212,96,219)(90,211,97,218)(91,224,98,217)(99,351,106,358)(100,364,107,357)(101,363,108,356)(102,362,109,355)(103,361,110,354)(104,360,111,353)(105,359,112,352)(127,401,134,394)(128,400,135,393)(129,399,136,406)(130,398,137,405)(131,397,138,404)(132,396,139,403)(133,395,140,402)(141,367,148,374)(142,366,149,373)(143,365,150,372)(144,378,151,371)(145,377,152,370)(146,376,153,369)(147,375,154,368)(155,350,162,343)(156,349,163,342)(157,348,164,341)(158,347,165,340)(159,346,166,339)(160,345,167,338)(161,344,168,337)(197,255,204,262)(198,254,205,261)(199,253,206,260)(200,266,207,259)(201,265,208,258)(202,264,209,257)(203,263,210,256)(267,414,274,407)(268,413,275,420)(269,412,276,419)(270,411,277,418)(271,410,278,417)(272,409,279,416)(273,408,280,415)(281,298,288,305)(282,297,289,304)(283,296,290,303)(284,295,291,302)(285,308,292,301)(286,307,293,300)(287,306,294,299)(309,381,316,388)(310,380,317,387)(311,379,318,386)(312,392,319,385)(313,391,320,384)(314,390,321,383)(315,389,322,382)(421,440,428,447)(422,439,429,446)(423,438,430,445)(424,437,431,444)(425,436,432,443)(426,435,433,442)(427,448,434,441), (1,199,400,114,379,182,102,236)(2,200,401,115,380,169,103,237)(3,201,402,116,381,170,104,238)(4,202,403,117,382,171,105,225)(5,203,404,118,383,172,106,226)(6,204,405,119,384,173,107,227)(7,205,406,120,385,174,108,228)(8,206,393,121,386,175,109,229)(9,207,394,122,387,176,110,230)(10,208,395,123,388,177,111,231)(11,209,396,124,389,178,112,232)(12,210,397,125,390,179,99,233)(13,197,398,126,391,180,100,234)(14,198,399,113,392,181,101,235)(15,321,65,351,80,185,263,138)(16,322,66,352,81,186,264,139)(17,309,67,353,82,187,265,140)(18,310,68,354,83,188,266,127)(19,311,69,355,84,189,253,128)(20,312,70,356,71,190,254,129)(21,313,57,357,72,191,255,130)(22,314,58,358,73,192,256,131)(23,315,59,359,74,193,257,132)(24,316,60,360,75,194,258,133)(25,317,61,361,76,195,259,134)(26,318,62,362,77,196,260,135)(27,319,63,363,78,183,261,136)(28,320,64,364,79,184,262,137)(29,48,425,288,88,273,144,161)(30,49,426,289,89,274,145,162)(31,50,427,290,90,275,146,163)(32,51,428,291,91,276,147,164)(33,52,429,292,92,277,148,165)(34,53,430,293,93,278,149,166)(35,54,431,294,94,279,150,167)(36,55,432,281,95,280,151,168)(37,56,433,282,96,267,152,155)(38,43,434,283,97,268,153,156)(39,44,421,284,98,269,154,157)(40,45,422,285,85,270,141,158)(41,46,423,286,86,271,142,159)(42,47,424,287,87,272,143,160)(211,413,376,349,244,326,448,296)(212,414,377,350,245,327,435,297)(213,415,378,337,246,328,436,298)(214,416,365,338,247,329,437,299)(215,417,366,339,248,330,438,300)(216,418,367,340,249,331,439,301)(217,419,368,341,250,332,440,302)(218,420,369,342,251,333,441,303)(219,407,370,343,252,334,442,304)(220,408,371,344,239,335,443,305)(221,409,372,345,240,336,444,306)(222,410,373,346,241,323,445,307)(223,411,374,347,242,324,446,308)(224,412,375,348,243,325,447,295)>;
G:=Group( (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,350)(16,337)(17,338)(18,339)(19,340)(20,341)(21,342)(22,343)(23,344)(24,345)(25,346)(26,347)(27,348)(28,349)(29,389)(30,390)(31,391)(32,392)(33,379)(34,380)(35,381)(36,382)(37,383)(38,384)(39,385)(40,386)(41,387)(42,388)(43,173)(44,174)(45,175)(46,176)(47,177)(48,178)(49,179)(50,180)(51,181)(52,182)(53,169)(54,170)(55,171)(56,172)(57,333)(58,334)(59,335)(60,336)(61,323)(62,324)(63,325)(64,326)(65,327)(66,328)(67,329)(68,330)(69,331)(70,332)(71,302)(72,303)(73,304)(74,305)(75,306)(76,307)(77,308)(78,295)(79,296)(80,297)(81,298)(82,299)(83,300)(84,301)(99,426)(100,427)(101,428)(102,429)(103,430)(104,431)(105,432)(106,433)(107,434)(108,421)(109,422)(110,423)(111,424)(112,425)(113,164)(114,165)(115,166)(116,167)(117,168)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,161)(125,162)(126,163)(127,366)(128,367)(129,368)(130,369)(131,370)(132,371)(133,372)(134,373)(135,374)(136,375)(137,376)(138,377)(139,378)(140,365)(141,393)(142,394)(143,395)(144,396)(145,397)(146,398)(147,399)(148,400)(149,401)(150,402)(151,403)(152,404)(153,405)(154,406)(183,224)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216)(190,217)(191,218)(192,219)(193,220)(194,221)(195,222)(196,223)(197,275)(198,276)(199,277)(200,278)(201,279)(202,280)(203,267)(204,268)(205,269)(206,270)(207,271)(208,272)(209,273)(210,274)(225,281)(226,282)(227,283)(228,284)(229,285)(230,286)(231,287)(232,288)(233,289)(234,290)(235,291)(236,292)(237,293)(238,294)(239,315)(240,316)(241,317)(242,318)(243,319)(244,320)(245,321)(246,322)(247,309)(248,310)(249,311)(250,312)(251,313)(252,314)(253,418)(254,419)(255,420)(256,407)(257,408)(258,409)(259,410)(260,411)(261,412)(262,413)(263,414)(264,415)(265,416)(266,417)(351,435)(352,436)(353,437)(354,438)(355,439)(356,440)(357,441)(358,442)(359,443)(360,444)(361,445)(362,446)(363,447)(364,448), (1,2,3,4,5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232,233,234,235,236,237,238)(239,240,241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264,265,266)(267,268,269,270,271,272,273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288,289,290,291,292,293,294)(295,296,297,298,299,300,301,302,303,304,305,306,307,308)(309,310,311,312,313,314,315,316,317,318,319,320,321,322)(323,324,325,326,327,328,329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344,345,346,347,348,349,350)(351,352,353,354,355,356,357,358,359,360,361,362,363,364)(365,366,367,368,369,370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387,388,389,390,391,392)(393,394,395,396,397,398,399,400,401,402,403,404,405,406)(407,408,409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432,433,434)(435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,196,8,189)(2,195,9,188)(3,194,10,187)(4,193,11,186)(5,192,12,185)(6,191,13,184)(7,190,14,183)(15,125,22,118)(16,124,23,117)(17,123,24,116)(18,122,25,115)(19,121,26,114)(20,120,27,113)(21,119,28,126)(29,246,36,239)(30,245,37,252)(31,244,38,251)(32,243,39,250)(33,242,40,249)(34,241,41,248)(35,240,42,247)(43,326,50,333)(44,325,51,332)(45,324,52,331)(46,323,53,330)(47,336,54,329)(48,335,55,328)(49,334,56,327)(57,173,64,180)(58,172,65,179)(59,171,66,178)(60,170,67,177)(61,169,68,176)(62,182,69,175)(63,181,70,174)(71,228,78,235)(72,227,79,234)(73,226,80,233)(74,225,81,232)(75,238,82,231)(76,237,83,230)(77,236,84,229)(85,216,92,223)(86,215,93,222)(87,214,94,221)(88,213,95,220)(89,212,96,219)(90,211,97,218)(91,224,98,217)(99,351,106,358)(100,364,107,357)(101,363,108,356)(102,362,109,355)(103,361,110,354)(104,360,111,353)(105,359,112,352)(127,401,134,394)(128,400,135,393)(129,399,136,406)(130,398,137,405)(131,397,138,404)(132,396,139,403)(133,395,140,402)(141,367,148,374)(142,366,149,373)(143,365,150,372)(144,378,151,371)(145,377,152,370)(146,376,153,369)(147,375,154,368)(155,350,162,343)(156,349,163,342)(157,348,164,341)(158,347,165,340)(159,346,166,339)(160,345,167,338)(161,344,168,337)(197,255,204,262)(198,254,205,261)(199,253,206,260)(200,266,207,259)(201,265,208,258)(202,264,209,257)(203,263,210,256)(267,414,274,407)(268,413,275,420)(269,412,276,419)(270,411,277,418)(271,410,278,417)(272,409,279,416)(273,408,280,415)(281,298,288,305)(282,297,289,304)(283,296,290,303)(284,295,291,302)(285,308,292,301)(286,307,293,300)(287,306,294,299)(309,381,316,388)(310,380,317,387)(311,379,318,386)(312,392,319,385)(313,391,320,384)(314,390,321,383)(315,389,322,382)(421,440,428,447)(422,439,429,446)(423,438,430,445)(424,437,431,444)(425,436,432,443)(426,435,433,442)(427,448,434,441), (1,199,400,114,379,182,102,236)(2,200,401,115,380,169,103,237)(3,201,402,116,381,170,104,238)(4,202,403,117,382,171,105,225)(5,203,404,118,383,172,106,226)(6,204,405,119,384,173,107,227)(7,205,406,120,385,174,108,228)(8,206,393,121,386,175,109,229)(9,207,394,122,387,176,110,230)(10,208,395,123,388,177,111,231)(11,209,396,124,389,178,112,232)(12,210,397,125,390,179,99,233)(13,197,398,126,391,180,100,234)(14,198,399,113,392,181,101,235)(15,321,65,351,80,185,263,138)(16,322,66,352,81,186,264,139)(17,309,67,353,82,187,265,140)(18,310,68,354,83,188,266,127)(19,311,69,355,84,189,253,128)(20,312,70,356,71,190,254,129)(21,313,57,357,72,191,255,130)(22,314,58,358,73,192,256,131)(23,315,59,359,74,193,257,132)(24,316,60,360,75,194,258,133)(25,317,61,361,76,195,259,134)(26,318,62,362,77,196,260,135)(27,319,63,363,78,183,261,136)(28,320,64,364,79,184,262,137)(29,48,425,288,88,273,144,161)(30,49,426,289,89,274,145,162)(31,50,427,290,90,275,146,163)(32,51,428,291,91,276,147,164)(33,52,429,292,92,277,148,165)(34,53,430,293,93,278,149,166)(35,54,431,294,94,279,150,167)(36,55,432,281,95,280,151,168)(37,56,433,282,96,267,152,155)(38,43,434,283,97,268,153,156)(39,44,421,284,98,269,154,157)(40,45,422,285,85,270,141,158)(41,46,423,286,86,271,142,159)(42,47,424,287,87,272,143,160)(211,413,376,349,244,326,448,296)(212,414,377,350,245,327,435,297)(213,415,378,337,246,328,436,298)(214,416,365,338,247,329,437,299)(215,417,366,339,248,330,438,300)(216,418,367,340,249,331,439,301)(217,419,368,341,250,332,440,302)(218,420,369,342,251,333,441,303)(219,407,370,343,252,334,442,304)(220,408,371,344,239,335,443,305)(221,409,372,345,240,336,444,306)(222,410,373,346,241,323,445,307)(223,411,374,347,242,324,446,308)(224,412,375,348,243,325,447,295) );
G=PermutationGroup([[(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,85),(9,86),(10,87),(11,88),(12,89),(13,90),(14,91),(15,350),(16,337),(17,338),(18,339),(19,340),(20,341),(21,342),(22,343),(23,344),(24,345),(25,346),(26,347),(27,348),(28,349),(29,389),(30,390),(31,391),(32,392),(33,379),(34,380),(35,381),(36,382),(37,383),(38,384),(39,385),(40,386),(41,387),(42,388),(43,173),(44,174),(45,175),(46,176),(47,177),(48,178),(49,179),(50,180),(51,181),(52,182),(53,169),(54,170),(55,171),(56,172),(57,333),(58,334),(59,335),(60,336),(61,323),(62,324),(63,325),(64,326),(65,327),(66,328),(67,329),(68,330),(69,331),(70,332),(71,302),(72,303),(73,304),(74,305),(75,306),(76,307),(77,308),(78,295),(79,296),(80,297),(81,298),(82,299),(83,300),(84,301),(99,426),(100,427),(101,428),(102,429),(103,430),(104,431),(105,432),(106,433),(107,434),(108,421),(109,422),(110,423),(111,424),(112,425),(113,164),(114,165),(115,166),(116,167),(117,168),(118,155),(119,156),(120,157),(121,158),(122,159),(123,160),(124,161),(125,162),(126,163),(127,366),(128,367),(129,368),(130,369),(131,370),(132,371),(133,372),(134,373),(135,374),(136,375),(137,376),(138,377),(139,378),(140,365),(141,393),(142,394),(143,395),(144,396),(145,397),(146,398),(147,399),(148,400),(149,401),(150,402),(151,403),(152,404),(153,405),(154,406),(183,224),(184,211),(185,212),(186,213),(187,214),(188,215),(189,216),(190,217),(191,218),(192,219),(193,220),(194,221),(195,222),(196,223),(197,275),(198,276),(199,277),(200,278),(201,279),(202,280),(203,267),(204,268),(205,269),(206,270),(207,271),(208,272),(209,273),(210,274),(225,281),(226,282),(227,283),(228,284),(229,285),(230,286),(231,287),(232,288),(233,289),(234,290),(235,291),(236,292),(237,293),(238,294),(239,315),(240,316),(241,317),(242,318),(243,319),(244,320),(245,321),(246,322),(247,309),(248,310),(249,311),(250,312),(251,313),(252,314),(253,418),(254,419),(255,420),(256,407),(257,408),(258,409),(259,410),(260,411),(261,412),(262,413),(263,414),(264,415),(265,416),(266,417),(351,435),(352,436),(353,437),(354,438),(355,439),(356,440),(357,441),(358,442),(359,443),(360,444),(361,445),(362,446),(363,447),(364,448)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154),(155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182),(183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220,221,222,223,224),(225,226,227,228,229,230,231,232,233,234,235,236,237,238),(239,240,241,242,243,244,245,246,247,248,249,250,251,252),(253,254,255,256,257,258,259,260,261,262,263,264,265,266),(267,268,269,270,271,272,273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288,289,290,291,292,293,294),(295,296,297,298,299,300,301,302,303,304,305,306,307,308),(309,310,311,312,313,314,315,316,317,318,319,320,321,322),(323,324,325,326,327,328,329,330,331,332,333,334,335,336),(337,338,339,340,341,342,343,344,345,346,347,348,349,350),(351,352,353,354,355,356,357,358,359,360,361,362,363,364),(365,366,367,368,369,370,371,372,373,374,375,376,377,378),(379,380,381,382,383,384,385,386,387,388,389,390,391,392),(393,394,395,396,397,398,399,400,401,402,403,404,405,406),(407,408,409,410,411,412,413,414,415,416,417,418,419,420),(421,422,423,424,425,426,427,428,429,430,431,432,433,434),(435,436,437,438,439,440,441,442,443,444,445,446,447,448)], [(1,196,8,189),(2,195,9,188),(3,194,10,187),(4,193,11,186),(5,192,12,185),(6,191,13,184),(7,190,14,183),(15,125,22,118),(16,124,23,117),(17,123,24,116),(18,122,25,115),(19,121,26,114),(20,120,27,113),(21,119,28,126),(29,246,36,239),(30,245,37,252),(31,244,38,251),(32,243,39,250),(33,242,40,249),(34,241,41,248),(35,240,42,247),(43,326,50,333),(44,325,51,332),(45,324,52,331),(46,323,53,330),(47,336,54,329),(48,335,55,328),(49,334,56,327),(57,173,64,180),(58,172,65,179),(59,171,66,178),(60,170,67,177),(61,169,68,176),(62,182,69,175),(63,181,70,174),(71,228,78,235),(72,227,79,234),(73,226,80,233),(74,225,81,232),(75,238,82,231),(76,237,83,230),(77,236,84,229),(85,216,92,223),(86,215,93,222),(87,214,94,221),(88,213,95,220),(89,212,96,219),(90,211,97,218),(91,224,98,217),(99,351,106,358),(100,364,107,357),(101,363,108,356),(102,362,109,355),(103,361,110,354),(104,360,111,353),(105,359,112,352),(127,401,134,394),(128,400,135,393),(129,399,136,406),(130,398,137,405),(131,397,138,404),(132,396,139,403),(133,395,140,402),(141,367,148,374),(142,366,149,373),(143,365,150,372),(144,378,151,371),(145,377,152,370),(146,376,153,369),(147,375,154,368),(155,350,162,343),(156,349,163,342),(157,348,164,341),(158,347,165,340),(159,346,166,339),(160,345,167,338),(161,344,168,337),(197,255,204,262),(198,254,205,261),(199,253,206,260),(200,266,207,259),(201,265,208,258),(202,264,209,257),(203,263,210,256),(267,414,274,407),(268,413,275,420),(269,412,276,419),(270,411,277,418),(271,410,278,417),(272,409,279,416),(273,408,280,415),(281,298,288,305),(282,297,289,304),(283,296,290,303),(284,295,291,302),(285,308,292,301),(286,307,293,300),(287,306,294,299),(309,381,316,388),(310,380,317,387),(311,379,318,386),(312,392,319,385),(313,391,320,384),(314,390,321,383),(315,389,322,382),(421,440,428,447),(422,439,429,446),(423,438,430,445),(424,437,431,444),(425,436,432,443),(426,435,433,442),(427,448,434,441)], [(1,199,400,114,379,182,102,236),(2,200,401,115,380,169,103,237),(3,201,402,116,381,170,104,238),(4,202,403,117,382,171,105,225),(5,203,404,118,383,172,106,226),(6,204,405,119,384,173,107,227),(7,205,406,120,385,174,108,228),(8,206,393,121,386,175,109,229),(9,207,394,122,387,176,110,230),(10,208,395,123,388,177,111,231),(11,209,396,124,389,178,112,232),(12,210,397,125,390,179,99,233),(13,197,398,126,391,180,100,234),(14,198,399,113,392,181,101,235),(15,321,65,351,80,185,263,138),(16,322,66,352,81,186,264,139),(17,309,67,353,82,187,265,140),(18,310,68,354,83,188,266,127),(19,311,69,355,84,189,253,128),(20,312,70,356,71,190,254,129),(21,313,57,357,72,191,255,130),(22,314,58,358,73,192,256,131),(23,315,59,359,74,193,257,132),(24,316,60,360,75,194,258,133),(25,317,61,361,76,195,259,134),(26,318,62,362,77,196,260,135),(27,319,63,363,78,183,261,136),(28,320,64,364,79,184,262,137),(29,48,425,288,88,273,144,161),(30,49,426,289,89,274,145,162),(31,50,427,290,90,275,146,163),(32,51,428,291,91,276,147,164),(33,52,429,292,92,277,148,165),(34,53,430,293,93,278,149,166),(35,54,431,294,94,279,150,167),(36,55,432,281,95,280,151,168),(37,56,433,282,96,267,152,155),(38,43,434,283,97,268,153,156),(39,44,421,284,98,269,154,157),(40,45,422,285,85,270,141,158),(41,46,423,286,86,271,142,159),(42,47,424,287,87,272,143,160),(211,413,376,349,244,326,448,296),(212,414,377,350,245,327,435,297),(213,415,378,337,246,328,436,298),(214,416,365,338,247,329,437,299),(215,417,366,339,248,330,438,300),(216,418,367,340,249,331,439,301),(217,419,368,341,250,332,440,302),(218,420,369,342,251,333,441,303),(219,407,370,343,252,334,442,304),(220,408,371,344,239,335,443,305),(221,409,372,345,240,336,444,306),(222,410,373,346,241,323,445,307),(223,411,374,347,242,324,446,308),(224,412,375,348,243,325,447,295)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D7 | M4(2) | D14 | D14 | Dic14 | C4×D7 | C7⋊D4 | C4×D7 | C8×D7 | C8⋊D7 |
| kernel | C2×Dic7⋊C8 | Dic7⋊C8 | C22×C7⋊C8 | C2×C4×Dic7 | C22×C56 | C4×Dic7 | C22×Dic7 | C2×Dic7 | C2×C28 | C2×C28 | C22×C8 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 16 | 2 | 2 | 3 | 4 | 6 | 3 | 12 | 6 | 12 | 6 | 24 | 24 |
Matrix representation of C2×Dic7⋊C8 ►in GL5(𝔽113)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 112 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 104 | 1 |
| 0 | 0 | 0 | 112 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 82 | 47 | 0 | 0 |
| 0 | 107 | 31 | 0 | 0 |
| 0 | 0 | 0 | 70 | 70 |
| 0 | 0 | 0 | 22 | 43 |
| 18 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 55 | 38 |
| 0 | 0 | 0 | 75 | 58 |
G:=sub<GL(5,GF(113))| [1,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,1,0,0,0,112,0,0,0,0,0,0,104,112,0,0,0,1,0],[1,0,0,0,0,0,82,107,0,0,0,47,31,0,0,0,0,0,70,22,0,0,0,70,43],[18,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,55,75,0,0,0,38,58] >;
C2×Dic7⋊C8 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_7\rtimes C_8 % in TeX
G:=Group("C2xDic7:C8"); // GroupNames label
G:=SmallGroup(448,633);
// by ID
G=gap.SmallGroup(448,633);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,422,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^14=d^8=1,c^2=b^7,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^7*c>;
// generators/relations