direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C14×Q32, C28.46D8, C56.74D4, C56.74C23, C112.22C22, C4.8(C7×D8), C8.11(C7×D4), C4.9(D4×C14), C16.5(C2×C14), (C2×C16).4C14, C2.14(C14×D8), C14.86(C2×D8), (C2×C14).57D8, (C2×C112).10C2, C28.316(C2×D4), (C2×C28).428D4, C8.5(C22×C14), (C2×Q16).4C14, Q16.1(C2×C14), C22.16(C7×D8), (C14×Q16).11C2, (C2×C56).428C22, (C7×Q16).13C22, (C2×C4).84(C7×D4), (C2×C8).86(C2×C14), SmallGroup(448,915)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×Q32
G = < a,b,c | a14=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 146 in 82 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C14, C16, C2×C8, Q16, Q16, C2×Q8, C28, C28, C2×C14, C2×C16, Q32, C2×Q16, C56, C2×C28, C2×C28, C7×Q8, C2×Q32, C112, C2×C56, C7×Q16, C7×Q16, Q8×C14, C2×C112, C7×Q32, C14×Q16, C14×Q32
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C2×C14, Q32, C2×D8, C7×D4, C22×C14, C2×Q32, C7×D8, D4×C14, C7×Q32, C14×D8, C14×Q32
►Regular action on 448 points
Generators in S448
(1 298 269 205 308 45 340 185 90 249 157 18 365 370)(2 299 270 206 309 46 341 186 91 250 158 19 366 371)(3 300 271 207 310 47 342 187 92 251 159 20 367 372)(4 301 272 208 311 48 343 188 93 252 160 21 368 373)(5 302 257 193 312 33 344 189 94 253 145 22 353 374)(6 303 258 194 313 34 345 190 95 254 146 23 354 375)(7 304 259 195 314 35 346 191 96 255 147 24 355 376)(8 289 260 196 315 36 347 192 81 256 148 25 356 377)(9 290 261 197 316 37 348 177 82 241 149 26 357 378)(10 291 262 198 317 38 349 178 83 242 150 27 358 379)(11 292 263 199 318 39 350 179 84 243 151 28 359 380)(12 293 264 200 319 40 351 180 85 244 152 29 360 381)(13 294 265 201 320 41 352 181 86 245 153 30 361 382)(14 295 266 202 305 42 337 182 87 246 154 31 362 383)(15 296 267 203 306 43 338 183 88 247 155 32 363 384)(16 297 268 204 307 44 339 184 89 248 156 17 364 369)(49 121 112 419 280 238 133 210 167 67 406 433 393 336)(50 122 97 420 281 239 134 211 168 68 407 434 394 321)(51 123 98 421 282 240 135 212 169 69 408 435 395 322)(52 124 99 422 283 225 136 213 170 70 409 436 396 323)(53 125 100 423 284 226 137 214 171 71 410 437 397 324)(54 126 101 424 285 227 138 215 172 72 411 438 398 325)(55 127 102 425 286 228 139 216 173 73 412 439 399 326)(56 128 103 426 287 229 140 217 174 74 413 440 400 327)(57 113 104 427 288 230 141 218 175 75 414 441 385 328)(58 114 105 428 273 231 142 219 176 76 415 442 386 329)(59 115 106 429 274 232 143 220 161 77 416 443 387 330)(60 116 107 430 275 233 144 221 162 78 401 444 388 331)(61 117 108 431 276 234 129 222 163 79 402 445 389 332)(62 118 109 432 277 235 130 223 164 80 403 446 390 333)(63 119 110 417 278 236 131 224 165 65 404 447 391 334)(64 120 111 418 279 237 132 209 166 66 405 448 392 335)
(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)
(1 412 9 404)(2 411 10 403)(3 410 11 402)(4 409 12 401)(5 408 13 416)(6 407 14 415)(7 406 15 414)(8 405 16 413)(17 217 25 209)(18 216 26 224)(19 215 27 223)(20 214 28 222)(21 213 29 221)(22 212 30 220)(23 211 31 219)(24 210 32 218)(33 123 41 115)(34 122 42 114)(35 121 43 113)(36 120 44 128)(37 119 45 127)(38 118 46 126)(39 117 47 125)(40 116 48 124)(49 306 57 314)(50 305 58 313)(51 320 59 312)(52 319 60 311)(53 318 61 310)(54 317 62 309)(55 316 63 308)(56 315 64 307)(65 370 73 378)(66 369 74 377)(67 384 75 376)(68 383 76 375)(69 382 77 374)(70 381 78 373)(71 380 79 372)(72 379 80 371)(81 279 89 287)(82 278 90 286)(83 277 91 285)(84 276 92 284)(85 275 93 283)(86 274 94 282)(87 273 95 281)(88 288 96 280)(97 337 105 345)(98 352 106 344)(99 351 107 343)(100 350 108 342)(101 349 109 341)(102 348 110 340)(103 347 111 339)(104 346 112 338)(129 159 137 151)(130 158 138 150)(131 157 139 149)(132 156 140 148)(133 155 141 147)(134 154 142 146)(135 153 143 145)(136 152 144 160)(161 353 169 361)(162 368 170 360)(163 367 171 359)(164 366 172 358)(165 365 173 357)(166 364 174 356)(167 363 175 355)(168 362 176 354)(177 417 185 425)(178 432 186 424)(179 431 187 423)(180 430 188 422)(181 429 189 421)(182 428 190 420)(183 427 191 419)(184 426 192 418)(193 322 201 330)(194 321 202 329)(195 336 203 328)(196 335 204 327)(197 334 205 326)(198 333 206 325)(199 332 207 324)(200 331 208 323)(225 244 233 252)(226 243 234 251)(227 242 235 250)(228 241 236 249)(229 256 237 248)(230 255 238 247)(231 254 239 246)(232 253 240 245)(257 395 265 387)(258 394 266 386)(259 393 267 385)(260 392 268 400)(261 391 269 399)(262 390 270 398)(263 389 271 397)(264 388 272 396)(289 448 297 440)(290 447 298 439)(291 446 299 438)(292 445 300 437)(293 444 301 436)(294 443 302 435)(295 442 303 434)(296 441 304 433)
G:=sub<Sym(448)| (1,298,269,205,308,45,340,185,90,249,157,18,365,370)(2,299,270,206,309,46,341,186,91,250,158,19,366,371)(3,300,271,207,310,47,342,187,92,251,159,20,367,372)(4,301,272,208,311,48,343,188,93,252,160,21,368,373)(5,302,257,193,312,33,344,189,94,253,145,22,353,374)(6,303,258,194,313,34,345,190,95,254,146,23,354,375)(7,304,259,195,314,35,346,191,96,255,147,24,355,376)(8,289,260,196,315,36,347,192,81,256,148,25,356,377)(9,290,261,197,316,37,348,177,82,241,149,26,357,378)(10,291,262,198,317,38,349,178,83,242,150,27,358,379)(11,292,263,199,318,39,350,179,84,243,151,28,359,380)(12,293,264,200,319,40,351,180,85,244,152,29,360,381)(13,294,265,201,320,41,352,181,86,245,153,30,361,382)(14,295,266,202,305,42,337,182,87,246,154,31,362,383)(15,296,267,203,306,43,338,183,88,247,155,32,363,384)(16,297,268,204,307,44,339,184,89,248,156,17,364,369)(49,121,112,419,280,238,133,210,167,67,406,433,393,336)(50,122,97,420,281,239,134,211,168,68,407,434,394,321)(51,123,98,421,282,240,135,212,169,69,408,435,395,322)(52,124,99,422,283,225,136,213,170,70,409,436,396,323)(53,125,100,423,284,226,137,214,171,71,410,437,397,324)(54,126,101,424,285,227,138,215,172,72,411,438,398,325)(55,127,102,425,286,228,139,216,173,73,412,439,399,326)(56,128,103,426,287,229,140,217,174,74,413,440,400,327)(57,113,104,427,288,230,141,218,175,75,414,441,385,328)(58,114,105,428,273,231,142,219,176,76,415,442,386,329)(59,115,106,429,274,232,143,220,161,77,416,443,387,330)(60,116,107,430,275,233,144,221,162,78,401,444,388,331)(61,117,108,431,276,234,129,222,163,79,402,445,389,332)(62,118,109,432,277,235,130,223,164,80,403,446,390,333)(63,119,110,417,278,236,131,224,165,65,404,447,391,334)(64,120,111,418,279,237,132,209,166,66,405,448,392,335), (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), (1,412,9,404)(2,411,10,403)(3,410,11,402)(4,409,12,401)(5,408,13,416)(6,407,14,415)(7,406,15,414)(8,405,16,413)(17,217,25,209)(18,216,26,224)(19,215,27,223)(20,214,28,222)(21,213,29,221)(22,212,30,220)(23,211,31,219)(24,210,32,218)(33,123,41,115)(34,122,42,114)(35,121,43,113)(36,120,44,128)(37,119,45,127)(38,118,46,126)(39,117,47,125)(40,116,48,124)(49,306,57,314)(50,305,58,313)(51,320,59,312)(52,319,60,311)(53,318,61,310)(54,317,62,309)(55,316,63,308)(56,315,64,307)(65,370,73,378)(66,369,74,377)(67,384,75,376)(68,383,76,375)(69,382,77,374)(70,381,78,373)(71,380,79,372)(72,379,80,371)(81,279,89,287)(82,278,90,286)(83,277,91,285)(84,276,92,284)(85,275,93,283)(86,274,94,282)(87,273,95,281)(88,288,96,280)(97,337,105,345)(98,352,106,344)(99,351,107,343)(100,350,108,342)(101,349,109,341)(102,348,110,340)(103,347,111,339)(104,346,112,338)(129,159,137,151)(130,158,138,150)(131,157,139,149)(132,156,140,148)(133,155,141,147)(134,154,142,146)(135,153,143,145)(136,152,144,160)(161,353,169,361)(162,368,170,360)(163,367,171,359)(164,366,172,358)(165,365,173,357)(166,364,174,356)(167,363,175,355)(168,362,176,354)(177,417,185,425)(178,432,186,424)(179,431,187,423)(180,430,188,422)(181,429,189,421)(182,428,190,420)(183,427,191,419)(184,426,192,418)(193,322,201,330)(194,321,202,329)(195,336,203,328)(196,335,204,327)(197,334,205,326)(198,333,206,325)(199,332,207,324)(200,331,208,323)(225,244,233,252)(226,243,234,251)(227,242,235,250)(228,241,236,249)(229,256,237,248)(230,255,238,247)(231,254,239,246)(232,253,240,245)(257,395,265,387)(258,394,266,386)(259,393,267,385)(260,392,268,400)(261,391,269,399)(262,390,270,398)(263,389,271,397)(264,388,272,396)(289,448,297,440)(290,447,298,439)(291,446,299,438)(292,445,300,437)(293,444,301,436)(294,443,302,435)(295,442,303,434)(296,441,304,433)>;
G:=Group( (1,298,269,205,308,45,340,185,90,249,157,18,365,370)(2,299,270,206,309,46,341,186,91,250,158,19,366,371)(3,300,271,207,310,47,342,187,92,251,159,20,367,372)(4,301,272,208,311,48,343,188,93,252,160,21,368,373)(5,302,257,193,312,33,344,189,94,253,145,22,353,374)(6,303,258,194,313,34,345,190,95,254,146,23,354,375)(7,304,259,195,314,35,346,191,96,255,147,24,355,376)(8,289,260,196,315,36,347,192,81,256,148,25,356,377)(9,290,261,197,316,37,348,177,82,241,149,26,357,378)(10,291,262,198,317,38,349,178,83,242,150,27,358,379)(11,292,263,199,318,39,350,179,84,243,151,28,359,380)(12,293,264,200,319,40,351,180,85,244,152,29,360,381)(13,294,265,201,320,41,352,181,86,245,153,30,361,382)(14,295,266,202,305,42,337,182,87,246,154,31,362,383)(15,296,267,203,306,43,338,183,88,247,155,32,363,384)(16,297,268,204,307,44,339,184,89,248,156,17,364,369)(49,121,112,419,280,238,133,210,167,67,406,433,393,336)(50,122,97,420,281,239,134,211,168,68,407,434,394,321)(51,123,98,421,282,240,135,212,169,69,408,435,395,322)(52,124,99,422,283,225,136,213,170,70,409,436,396,323)(53,125,100,423,284,226,137,214,171,71,410,437,397,324)(54,126,101,424,285,227,138,215,172,72,411,438,398,325)(55,127,102,425,286,228,139,216,173,73,412,439,399,326)(56,128,103,426,287,229,140,217,174,74,413,440,400,327)(57,113,104,427,288,230,141,218,175,75,414,441,385,328)(58,114,105,428,273,231,142,219,176,76,415,442,386,329)(59,115,106,429,274,232,143,220,161,77,416,443,387,330)(60,116,107,430,275,233,144,221,162,78,401,444,388,331)(61,117,108,431,276,234,129,222,163,79,402,445,389,332)(62,118,109,432,277,235,130,223,164,80,403,446,390,333)(63,119,110,417,278,236,131,224,165,65,404,447,391,334)(64,120,111,418,279,237,132,209,166,66,405,448,392,335), (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), (1,412,9,404)(2,411,10,403)(3,410,11,402)(4,409,12,401)(5,408,13,416)(6,407,14,415)(7,406,15,414)(8,405,16,413)(17,217,25,209)(18,216,26,224)(19,215,27,223)(20,214,28,222)(21,213,29,221)(22,212,30,220)(23,211,31,219)(24,210,32,218)(33,123,41,115)(34,122,42,114)(35,121,43,113)(36,120,44,128)(37,119,45,127)(38,118,46,126)(39,117,47,125)(40,116,48,124)(49,306,57,314)(50,305,58,313)(51,320,59,312)(52,319,60,311)(53,318,61,310)(54,317,62,309)(55,316,63,308)(56,315,64,307)(65,370,73,378)(66,369,74,377)(67,384,75,376)(68,383,76,375)(69,382,77,374)(70,381,78,373)(71,380,79,372)(72,379,80,371)(81,279,89,287)(82,278,90,286)(83,277,91,285)(84,276,92,284)(85,275,93,283)(86,274,94,282)(87,273,95,281)(88,288,96,280)(97,337,105,345)(98,352,106,344)(99,351,107,343)(100,350,108,342)(101,349,109,341)(102,348,110,340)(103,347,111,339)(104,346,112,338)(129,159,137,151)(130,158,138,150)(131,157,139,149)(132,156,140,148)(133,155,141,147)(134,154,142,146)(135,153,143,145)(136,152,144,160)(161,353,169,361)(162,368,170,360)(163,367,171,359)(164,366,172,358)(165,365,173,357)(166,364,174,356)(167,363,175,355)(168,362,176,354)(177,417,185,425)(178,432,186,424)(179,431,187,423)(180,430,188,422)(181,429,189,421)(182,428,190,420)(183,427,191,419)(184,426,192,418)(193,322,201,330)(194,321,202,329)(195,336,203,328)(196,335,204,327)(197,334,205,326)(198,333,206,325)(199,332,207,324)(200,331,208,323)(225,244,233,252)(226,243,234,251)(227,242,235,250)(228,241,236,249)(229,256,237,248)(230,255,238,247)(231,254,239,246)(232,253,240,245)(257,395,265,387)(258,394,266,386)(259,393,267,385)(260,392,268,400)(261,391,269,399)(262,390,270,398)(263,389,271,397)(264,388,272,396)(289,448,297,440)(290,447,298,439)(291,446,299,438)(292,445,300,437)(293,444,301,436)(294,443,302,435)(295,442,303,434)(296,441,304,433) );
G=PermutationGroup([[(1,298,269,205,308,45,340,185,90,249,157,18,365,370),(2,299,270,206,309,46,341,186,91,250,158,19,366,371),(3,300,271,207,310,47,342,187,92,251,159,20,367,372),(4,301,272,208,311,48,343,188,93,252,160,21,368,373),(5,302,257,193,312,33,344,189,94,253,145,22,353,374),(6,303,258,194,313,34,345,190,95,254,146,23,354,375),(7,304,259,195,314,35,346,191,96,255,147,24,355,376),(8,289,260,196,315,36,347,192,81,256,148,25,356,377),(9,290,261,197,316,37,348,177,82,241,149,26,357,378),(10,291,262,198,317,38,349,178,83,242,150,27,358,379),(11,292,263,199,318,39,350,179,84,243,151,28,359,380),(12,293,264,200,319,40,351,180,85,244,152,29,360,381),(13,294,265,201,320,41,352,181,86,245,153,30,361,382),(14,295,266,202,305,42,337,182,87,246,154,31,362,383),(15,296,267,203,306,43,338,183,88,247,155,32,363,384),(16,297,268,204,307,44,339,184,89,248,156,17,364,369),(49,121,112,419,280,238,133,210,167,67,406,433,393,336),(50,122,97,420,281,239,134,211,168,68,407,434,394,321),(51,123,98,421,282,240,135,212,169,69,408,435,395,322),(52,124,99,422,283,225,136,213,170,70,409,436,396,323),(53,125,100,423,284,226,137,214,171,71,410,437,397,324),(54,126,101,424,285,227,138,215,172,72,411,438,398,325),(55,127,102,425,286,228,139,216,173,73,412,439,399,326),(56,128,103,426,287,229,140,217,174,74,413,440,400,327),(57,113,104,427,288,230,141,218,175,75,414,441,385,328),(58,114,105,428,273,231,142,219,176,76,415,442,386,329),(59,115,106,429,274,232,143,220,161,77,416,443,387,330),(60,116,107,430,275,233,144,221,162,78,401,444,388,331),(61,117,108,431,276,234,129,222,163,79,402,445,389,332),(62,118,109,432,277,235,130,223,164,80,403,446,390,333),(63,119,110,417,278,236,131,224,165,65,404,447,391,334),(64,120,111,418,279,237,132,209,166,66,405,448,392,335)], [(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)], [(1,412,9,404),(2,411,10,403),(3,410,11,402),(4,409,12,401),(5,408,13,416),(6,407,14,415),(7,406,15,414),(8,405,16,413),(17,217,25,209),(18,216,26,224),(19,215,27,223),(20,214,28,222),(21,213,29,221),(22,212,30,220),(23,211,31,219),(24,210,32,218),(33,123,41,115),(34,122,42,114),(35,121,43,113),(36,120,44,128),(37,119,45,127),(38,118,46,126),(39,117,47,125),(40,116,48,124),(49,306,57,314),(50,305,58,313),(51,320,59,312),(52,319,60,311),(53,318,61,310),(54,317,62,309),(55,316,63,308),(56,315,64,307),(65,370,73,378),(66,369,74,377),(67,384,75,376),(68,383,76,375),(69,382,77,374),(70,381,78,373),(71,380,79,372),(72,379,80,371),(81,279,89,287),(82,278,90,286),(83,277,91,285),(84,276,92,284),(85,275,93,283),(86,274,94,282),(87,273,95,281),(88,288,96,280),(97,337,105,345),(98,352,106,344),(99,351,107,343),(100,350,108,342),(101,349,109,341),(102,348,110,340),(103,347,111,339),(104,346,112,338),(129,159,137,151),(130,158,138,150),(131,157,139,149),(132,156,140,148),(133,155,141,147),(134,154,142,146),(135,153,143,145),(136,152,144,160),(161,353,169,361),(162,368,170,360),(163,367,171,359),(164,366,172,358),(165,365,173,357),(166,364,174,356),(167,363,175,355),(168,362,176,354),(177,417,185,425),(178,432,186,424),(179,431,187,423),(180,430,188,422),(181,429,189,421),(182,428,190,420),(183,427,191,419),(184,426,192,418),(193,322,201,330),(194,321,202,329),(195,336,203,328),(196,335,204,327),(197,334,205,326),(198,333,206,325),(199,332,207,324),(200,331,208,323),(225,244,233,252),(226,243,234,251),(227,242,235,250),(228,241,236,249),(229,256,237,248),(230,255,238,247),(231,254,239,246),(232,253,240,245),(257,395,265,387),(258,394,266,386),(259,393,267,385),(260,392,268,400),(261,391,269,399),(262,390,270,398),(263,389,271,397),(264,388,272,396),(289,448,297,440),(290,447,298,439),(291,446,299,438),(292,445,300,437),(293,444,301,436),(294,443,302,435),(295,442,303,434),(296,441,304,433)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 16A | ··· | 16H | 28A | ··· | 28L | 28M | ··· | 28AJ | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | D4 | D8 | D8 | Q32 | C7×D4 | C7×D4 | C7×D8 | C7×D8 | C7×Q32 |
| kernel | C14×Q32 | C2×C112 | C7×Q32 | C14×Q16 | C2×Q32 | C2×C16 | Q32 | C2×Q16 | C56 | C2×C28 | C28 | C2×C14 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 6 | 6 | 24 | 12 | 1 | 1 | 2 | 2 | 8 | 6 | 6 | 12 | 12 | 48 |
Matrix representation of C14×Q32 ►in GL4(𝔽113) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 109 |
| 31 | 82 | 0 | 0 |
| 31 | 31 | 0 | 0 |
| 0 | 0 | 4 | 18 |
| 0 | 0 | 95 | 4 |
| 31 | 82 | 0 | 0 |
| 82 | 82 | 0 | 0 |
| 0 | 0 | 9 | 12 |
| 0 | 0 | 12 | 104 |
G:=sub<GL(4,GF(113))| [7,0,0,0,0,7,0,0,0,0,109,0,0,0,0,109],[31,31,0,0,82,31,0,0,0,0,4,95,0,0,18,4],[31,82,0,0,82,82,0,0,0,0,9,12,0,0,12,104] >;
C14×Q32 in GAP, Magma, Sage, TeX
C_{14}\times Q_{32} % in TeX
G:=Group("C14xQ32"); // GroupNames label
G:=SmallGroup(448,915);
// by ID
G=gap.SmallGroup(448,915);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1576,5884,2951,242,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c|a^14=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations