direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C16×Dic7, C112⋊8C4, C28.16C42, C7⋊C16⋊9C4, C7⋊2(C4×C16), C7⋊C8.5C8, C14.7(C4×C8), C4.20(C8×D7), C8.40(C4×D7), C2.2(D7×C16), C14.3(C2×C16), C56.57(C2×C4), C28.25(C2×C8), (C2×C16).10D7, C2.2(C8×Dic7), C22.8(C8×D7), (C2×C112).14C2, (C2×C8).332D14, (C2×Dic7).7C8, C4.15(C4×Dic7), C8.23(C2×Dic7), (C4×Dic7).22C4, (C8×Dic7).17C2, (C2×C56).398C22, (C2×C7⋊C8).17C4, (C2×C7⋊C16).12C2, (C2×C14).9(C2×C8), (C2×C4).166(C4×D7), (C2×C28).240(C2×C4), SmallGroup(448,57)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C16×Dic7 |
Generators and relations for C16×Dic7
G = < a,b,c | a16=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C16×Dic7
►Regular action on 448 points
Generators in S448
(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)
(1 278 154 24 375 330 102 187 58 265 294 122 211 46)(2 279 155 25 376 331 103 188 59 266 295 123 212 47)(3 280 156 26 377 332 104 189 60 267 296 124 213 48)(4 281 157 27 378 333 105 190 61 268 297 125 214 33)(5 282 158 28 379 334 106 191 62 269 298 126 215 34)(6 283 159 29 380 335 107 192 63 270 299 127 216 35)(7 284 160 30 381 336 108 177 64 271 300 128 217 36)(8 285 145 31 382 321 109 178 49 272 301 113 218 37)(9 286 146 32 383 322 110 179 50 257 302 114 219 38)(10 287 147 17 384 323 111 180 51 258 303 115 220 39)(11 288 148 18 369 324 112 181 52 259 304 116 221 40)(12 273 149 19 370 325 97 182 53 260 289 117 222 41)(13 274 150 20 371 326 98 183 54 261 290 118 223 42)(14 275 151 21 372 327 99 184 55 262 291 119 224 43)(15 276 152 22 373 328 100 185 56 263 292 120 209 44)(16 277 153 23 374 329 101 186 57 264 293 121 210 45)(65 84 390 233 363 200 341 170 432 306 251 442 143 414)(66 85 391 234 364 201 342 171 417 307 252 443 144 415)(67 86 392 235 365 202 343 172 418 308 253 444 129 416)(68 87 393 236 366 203 344 173 419 309 254 445 130 401)(69 88 394 237 367 204 345 174 420 310 255 446 131 402)(70 89 395 238 368 205 346 175 421 311 256 447 132 403)(71 90 396 239 353 206 347 176 422 312 241 448 133 404)(72 91 397 240 354 207 348 161 423 313 242 433 134 405)(73 92 398 225 355 208 349 162 424 314 243 434 135 406)(74 93 399 226 356 193 350 163 425 315 244 435 136 407)(75 94 400 227 357 194 351 164 426 316 245 436 137 408)(76 95 385 228 358 195 352 165 427 317 246 437 138 409)(77 96 386 229 359 196 337 166 428 318 247 438 139 410)(78 81 387 230 360 197 338 167 429 319 248 439 140 411)(79 82 388 231 361 198 339 168 430 320 249 440 141 412)(80 83 389 232 362 199 340 169 431 305 250 441 142 413)
(1 416 187 343)(2 401 188 344)(3 402 189 345)(4 403 190 346)(5 404 191 347)(6 405 192 348)(7 406 177 349)(8 407 178 350)(9 408 179 351)(10 409 180 352)(11 410 181 337)(12 411 182 338)(13 412 183 339)(14 413 184 340)(15 414 185 341)(16 415 186 342)(17 246 303 228)(18 247 304 229)(19 248 289 230)(20 249 290 231)(21 250 291 232)(22 251 292 233)(23 252 293 234)(24 253 294 235)(25 254 295 236)(26 255 296 237)(27 256 297 238)(28 241 298 239)(29 242 299 240)(30 243 300 225)(31 244 301 226)(32 245 302 227)(33 70 105 175)(34 71 106 176)(35 72 107 161)(36 73 108 162)(37 74 109 163)(38 75 110 164)(39 76 111 165)(40 77 112 166)(41 78 97 167)(42 79 98 168)(43 80 99 169)(44 65 100 170)(45 66 101 171)(46 67 102 172)(47 68 103 173)(48 69 104 174)(49 193 285 136)(50 194 286 137)(51 195 287 138)(52 196 288 139)(53 197 273 140)(54 198 274 141)(55 199 275 142)(56 200 276 143)(57 201 277 144)(58 202 278 129)(59 203 279 130)(60 204 280 131)(61 205 281 132)(62 206 282 133)(63 207 283 134)(64 208 284 135)(81 325 429 222)(82 326 430 223)(83 327 431 224)(84 328 432 209)(85 329 417 210)(86 330 418 211)(87 331 419 212)(88 332 420 213)(89 333 421 214)(90 334 422 215)(91 335 423 216)(92 336 424 217)(93 321 425 218)(94 322 426 219)(95 323 427 220)(96 324 428 221)(113 399 382 315)(114 400 383 316)(115 385 384 317)(116 386 369 318)(117 387 370 319)(118 388 371 320)(119 389 372 305)(120 390 373 306)(121 391 374 307)(122 392 375 308)(123 393 376 309)(124 394 377 310)(125 395 378 311)(126 396 379 312)(127 397 380 313)(128 398 381 314)(145 435 272 356)(146 436 257 357)(147 437 258 358)(148 438 259 359)(149 439 260 360)(150 440 261 361)(151 441 262 362)(152 442 263 363)(153 443 264 364)(154 444 265 365)(155 445 266 366)(156 446 267 367)(157 447 268 368)(158 448 269 353)(159 433 270 354)(160 434 271 355)
G:=sub<Sym(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), (1,278,154,24,375,330,102,187,58,265,294,122,211,46)(2,279,155,25,376,331,103,188,59,266,295,123,212,47)(3,280,156,26,377,332,104,189,60,267,296,124,213,48)(4,281,157,27,378,333,105,190,61,268,297,125,214,33)(5,282,158,28,379,334,106,191,62,269,298,126,215,34)(6,283,159,29,380,335,107,192,63,270,299,127,216,35)(7,284,160,30,381,336,108,177,64,271,300,128,217,36)(8,285,145,31,382,321,109,178,49,272,301,113,218,37)(9,286,146,32,383,322,110,179,50,257,302,114,219,38)(10,287,147,17,384,323,111,180,51,258,303,115,220,39)(11,288,148,18,369,324,112,181,52,259,304,116,221,40)(12,273,149,19,370,325,97,182,53,260,289,117,222,41)(13,274,150,20,371,326,98,183,54,261,290,118,223,42)(14,275,151,21,372,327,99,184,55,262,291,119,224,43)(15,276,152,22,373,328,100,185,56,263,292,120,209,44)(16,277,153,23,374,329,101,186,57,264,293,121,210,45)(65,84,390,233,363,200,341,170,432,306,251,442,143,414)(66,85,391,234,364,201,342,171,417,307,252,443,144,415)(67,86,392,235,365,202,343,172,418,308,253,444,129,416)(68,87,393,236,366,203,344,173,419,309,254,445,130,401)(69,88,394,237,367,204,345,174,420,310,255,446,131,402)(70,89,395,238,368,205,346,175,421,311,256,447,132,403)(71,90,396,239,353,206,347,176,422,312,241,448,133,404)(72,91,397,240,354,207,348,161,423,313,242,433,134,405)(73,92,398,225,355,208,349,162,424,314,243,434,135,406)(74,93,399,226,356,193,350,163,425,315,244,435,136,407)(75,94,400,227,357,194,351,164,426,316,245,436,137,408)(76,95,385,228,358,195,352,165,427,317,246,437,138,409)(77,96,386,229,359,196,337,166,428,318,247,438,139,410)(78,81,387,230,360,197,338,167,429,319,248,439,140,411)(79,82,388,231,361,198,339,168,430,320,249,440,141,412)(80,83,389,232,362,199,340,169,431,305,250,441,142,413), (1,416,187,343)(2,401,188,344)(3,402,189,345)(4,403,190,346)(5,404,191,347)(6,405,192,348)(7,406,177,349)(8,407,178,350)(9,408,179,351)(10,409,180,352)(11,410,181,337)(12,411,182,338)(13,412,183,339)(14,413,184,340)(15,414,185,341)(16,415,186,342)(17,246,303,228)(18,247,304,229)(19,248,289,230)(20,249,290,231)(21,250,291,232)(22,251,292,233)(23,252,293,234)(24,253,294,235)(25,254,295,236)(26,255,296,237)(27,256,297,238)(28,241,298,239)(29,242,299,240)(30,243,300,225)(31,244,301,226)(32,245,302,227)(33,70,105,175)(34,71,106,176)(35,72,107,161)(36,73,108,162)(37,74,109,163)(38,75,110,164)(39,76,111,165)(40,77,112,166)(41,78,97,167)(42,79,98,168)(43,80,99,169)(44,65,100,170)(45,66,101,171)(46,67,102,172)(47,68,103,173)(48,69,104,174)(49,193,285,136)(50,194,286,137)(51,195,287,138)(52,196,288,139)(53,197,273,140)(54,198,274,141)(55,199,275,142)(56,200,276,143)(57,201,277,144)(58,202,278,129)(59,203,279,130)(60,204,280,131)(61,205,281,132)(62,206,282,133)(63,207,283,134)(64,208,284,135)(81,325,429,222)(82,326,430,223)(83,327,431,224)(84,328,432,209)(85,329,417,210)(86,330,418,211)(87,331,419,212)(88,332,420,213)(89,333,421,214)(90,334,422,215)(91,335,423,216)(92,336,424,217)(93,321,425,218)(94,322,426,219)(95,323,427,220)(96,324,428,221)(113,399,382,315)(114,400,383,316)(115,385,384,317)(116,386,369,318)(117,387,370,319)(118,388,371,320)(119,389,372,305)(120,390,373,306)(121,391,374,307)(122,392,375,308)(123,393,376,309)(124,394,377,310)(125,395,378,311)(126,396,379,312)(127,397,380,313)(128,398,381,314)(145,435,272,356)(146,436,257,357)(147,437,258,358)(148,438,259,359)(149,439,260,360)(150,440,261,361)(151,441,262,362)(152,442,263,363)(153,443,264,364)(154,444,265,365)(155,445,266,366)(156,446,267,367)(157,447,268,368)(158,448,269,353)(159,433,270,354)(160,434,271,355)>;
G:=Group( (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), (1,278,154,24,375,330,102,187,58,265,294,122,211,46)(2,279,155,25,376,331,103,188,59,266,295,123,212,47)(3,280,156,26,377,332,104,189,60,267,296,124,213,48)(4,281,157,27,378,333,105,190,61,268,297,125,214,33)(5,282,158,28,379,334,106,191,62,269,298,126,215,34)(6,283,159,29,380,335,107,192,63,270,299,127,216,35)(7,284,160,30,381,336,108,177,64,271,300,128,217,36)(8,285,145,31,382,321,109,178,49,272,301,113,218,37)(9,286,146,32,383,322,110,179,50,257,302,114,219,38)(10,287,147,17,384,323,111,180,51,258,303,115,220,39)(11,288,148,18,369,324,112,181,52,259,304,116,221,40)(12,273,149,19,370,325,97,182,53,260,289,117,222,41)(13,274,150,20,371,326,98,183,54,261,290,118,223,42)(14,275,151,21,372,327,99,184,55,262,291,119,224,43)(15,276,152,22,373,328,100,185,56,263,292,120,209,44)(16,277,153,23,374,329,101,186,57,264,293,121,210,45)(65,84,390,233,363,200,341,170,432,306,251,442,143,414)(66,85,391,234,364,201,342,171,417,307,252,443,144,415)(67,86,392,235,365,202,343,172,418,308,253,444,129,416)(68,87,393,236,366,203,344,173,419,309,254,445,130,401)(69,88,394,237,367,204,345,174,420,310,255,446,131,402)(70,89,395,238,368,205,346,175,421,311,256,447,132,403)(71,90,396,239,353,206,347,176,422,312,241,448,133,404)(72,91,397,240,354,207,348,161,423,313,242,433,134,405)(73,92,398,225,355,208,349,162,424,314,243,434,135,406)(74,93,399,226,356,193,350,163,425,315,244,435,136,407)(75,94,400,227,357,194,351,164,426,316,245,436,137,408)(76,95,385,228,358,195,352,165,427,317,246,437,138,409)(77,96,386,229,359,196,337,166,428,318,247,438,139,410)(78,81,387,230,360,197,338,167,429,319,248,439,140,411)(79,82,388,231,361,198,339,168,430,320,249,440,141,412)(80,83,389,232,362,199,340,169,431,305,250,441,142,413), (1,416,187,343)(2,401,188,344)(3,402,189,345)(4,403,190,346)(5,404,191,347)(6,405,192,348)(7,406,177,349)(8,407,178,350)(9,408,179,351)(10,409,180,352)(11,410,181,337)(12,411,182,338)(13,412,183,339)(14,413,184,340)(15,414,185,341)(16,415,186,342)(17,246,303,228)(18,247,304,229)(19,248,289,230)(20,249,290,231)(21,250,291,232)(22,251,292,233)(23,252,293,234)(24,253,294,235)(25,254,295,236)(26,255,296,237)(27,256,297,238)(28,241,298,239)(29,242,299,240)(30,243,300,225)(31,244,301,226)(32,245,302,227)(33,70,105,175)(34,71,106,176)(35,72,107,161)(36,73,108,162)(37,74,109,163)(38,75,110,164)(39,76,111,165)(40,77,112,166)(41,78,97,167)(42,79,98,168)(43,80,99,169)(44,65,100,170)(45,66,101,171)(46,67,102,172)(47,68,103,173)(48,69,104,174)(49,193,285,136)(50,194,286,137)(51,195,287,138)(52,196,288,139)(53,197,273,140)(54,198,274,141)(55,199,275,142)(56,200,276,143)(57,201,277,144)(58,202,278,129)(59,203,279,130)(60,204,280,131)(61,205,281,132)(62,206,282,133)(63,207,283,134)(64,208,284,135)(81,325,429,222)(82,326,430,223)(83,327,431,224)(84,328,432,209)(85,329,417,210)(86,330,418,211)(87,331,419,212)(88,332,420,213)(89,333,421,214)(90,334,422,215)(91,335,423,216)(92,336,424,217)(93,321,425,218)(94,322,426,219)(95,323,427,220)(96,324,428,221)(113,399,382,315)(114,400,383,316)(115,385,384,317)(116,386,369,318)(117,387,370,319)(118,388,371,320)(119,389,372,305)(120,390,373,306)(121,391,374,307)(122,392,375,308)(123,393,376,309)(124,394,377,310)(125,395,378,311)(126,396,379,312)(127,397,380,313)(128,398,381,314)(145,435,272,356)(146,436,257,357)(147,437,258,358)(148,438,259,359)(149,439,260,360)(150,440,261,361)(151,441,262,362)(152,442,263,363)(153,443,264,364)(154,444,265,365)(155,445,266,366)(156,446,267,367)(157,447,268,368)(158,448,269,353)(159,433,270,354)(160,434,271,355) );
G=PermutationGroup([[(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)], [(1,278,154,24,375,330,102,187,58,265,294,122,211,46),(2,279,155,25,376,331,103,188,59,266,295,123,212,47),(3,280,156,26,377,332,104,189,60,267,296,124,213,48),(4,281,157,27,378,333,105,190,61,268,297,125,214,33),(5,282,158,28,379,334,106,191,62,269,298,126,215,34),(6,283,159,29,380,335,107,192,63,270,299,127,216,35),(7,284,160,30,381,336,108,177,64,271,300,128,217,36),(8,285,145,31,382,321,109,178,49,272,301,113,218,37),(9,286,146,32,383,322,110,179,50,257,302,114,219,38),(10,287,147,17,384,323,111,180,51,258,303,115,220,39),(11,288,148,18,369,324,112,181,52,259,304,116,221,40),(12,273,149,19,370,325,97,182,53,260,289,117,222,41),(13,274,150,20,371,326,98,183,54,261,290,118,223,42),(14,275,151,21,372,327,99,184,55,262,291,119,224,43),(15,276,152,22,373,328,100,185,56,263,292,120,209,44),(16,277,153,23,374,329,101,186,57,264,293,121,210,45),(65,84,390,233,363,200,341,170,432,306,251,442,143,414),(66,85,391,234,364,201,342,171,417,307,252,443,144,415),(67,86,392,235,365,202,343,172,418,308,253,444,129,416),(68,87,393,236,366,203,344,173,419,309,254,445,130,401),(69,88,394,237,367,204,345,174,420,310,255,446,131,402),(70,89,395,238,368,205,346,175,421,311,256,447,132,403),(71,90,396,239,353,206,347,176,422,312,241,448,133,404),(72,91,397,240,354,207,348,161,423,313,242,433,134,405),(73,92,398,225,355,208,349,162,424,314,243,434,135,406),(74,93,399,226,356,193,350,163,425,315,244,435,136,407),(75,94,400,227,357,194,351,164,426,316,245,436,137,408),(76,95,385,228,358,195,352,165,427,317,246,437,138,409),(77,96,386,229,359,196,337,166,428,318,247,438,139,410),(78,81,387,230,360,197,338,167,429,319,248,439,140,411),(79,82,388,231,361,198,339,168,430,320,249,440,141,412),(80,83,389,232,362,199,340,169,431,305,250,441,142,413)], [(1,416,187,343),(2,401,188,344),(3,402,189,345),(4,403,190,346),(5,404,191,347),(6,405,192,348),(7,406,177,349),(8,407,178,350),(9,408,179,351),(10,409,180,352),(11,410,181,337),(12,411,182,338),(13,412,183,339),(14,413,184,340),(15,414,185,341),(16,415,186,342),(17,246,303,228),(18,247,304,229),(19,248,289,230),(20,249,290,231),(21,250,291,232),(22,251,292,233),(23,252,293,234),(24,253,294,235),(25,254,295,236),(26,255,296,237),(27,256,297,238),(28,241,298,239),(29,242,299,240),(30,243,300,225),(31,244,301,226),(32,245,302,227),(33,70,105,175),(34,71,106,176),(35,72,107,161),(36,73,108,162),(37,74,109,163),(38,75,110,164),(39,76,111,165),(40,77,112,166),(41,78,97,167),(42,79,98,168),(43,80,99,169),(44,65,100,170),(45,66,101,171),(46,67,102,172),(47,68,103,173),(48,69,104,174),(49,193,285,136),(50,194,286,137),(51,195,287,138),(52,196,288,139),(53,197,273,140),(54,198,274,141),(55,199,275,142),(56,200,276,143),(57,201,277,144),(58,202,278,129),(59,203,279,130),(60,204,280,131),(61,205,281,132),(62,206,282,133),(63,207,283,134),(64,208,284,135),(81,325,429,222),(82,326,430,223),(83,327,431,224),(84,328,432,209),(85,329,417,210),(86,330,418,211),(87,331,419,212),(88,332,420,213),(89,333,421,214),(90,334,422,215),(91,335,423,216),(92,336,424,217),(93,321,425,218),(94,322,426,219),(95,323,427,220),(96,324,428,221),(113,399,382,315),(114,400,383,316),(115,385,384,317),(116,386,369,318),(117,387,370,319),(118,388,371,320),(119,389,372,305),(120,390,373,306),(121,391,374,307),(122,392,375,308),(123,393,376,309),(124,394,377,310),(125,395,378,311),(126,396,379,312),(127,397,380,313),(128,398,381,314),(145,435,272,356),(146,436,257,357),(147,437,258,358),(148,438,259,359),(149,439,260,360),(150,440,261,361),(151,441,262,362),(152,442,263,363),(153,443,264,364),(154,444,265,365),(155,445,266,366),(156,446,267,367),(157,447,268,368),(158,448,269,353),(159,433,270,354),(160,434,271,355)]])
160 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14I | 16A | ··· | 16P | 16Q | ··· | 16AF | 28A | ··· | 28L | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8 | C16 | D7 | Dic7 | D14 | C4×D7 | C4×D7 | C8×D7 | C8×D7 | D7×C16 |
| kernel | C16×Dic7 | C2×C7⋊C16 | C8×Dic7 | C2×C112 | C7⋊C16 | C112 | C2×C7⋊C8 | C4×Dic7 | C7⋊C8 | C2×Dic7 | Dic7 | C2×C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 32 | 3 | 6 | 3 | 6 | 6 | 12 | 12 | 48 |
Matrix representation of C16×Dic7 ►in GL3(𝔽113) generated by
| 98 | 0 | 0 |
| 0 | 71 | 0 |
| 0 | 0 | 71 |
| 112 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 112 | 9 |
| 98 | 0 | 0 |
| 0 | 88 | 90 |
| 0 | 91 | 25 |
G:=sub<GL(3,GF(113))| [98,0,0,0,71,0,0,0,71],[112,0,0,0,0,112,0,1,9],[98,0,0,0,88,91,0,90,25] >;
C16×Dic7 in GAP, Magma, Sage, TeX
C_{16}\times {\rm Dic}_7 % in TeX
G:=Group("C16xDic7"); // GroupNames label
G:=SmallGroup(448,57);
// by ID
G=gap.SmallGroup(448,57);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,64,80,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^16=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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