direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C4⋊C8, C42.10C28, C4⋊2(C2×C56), (C2×C28)⋊8C8, C28⋊9(C2×C8), (C2×C4)⋊3C56, (C4×C28).25C4, C4.74(D4×C14), C28.69(C4⋊C4), (C2×C28).83Q8, C4.21(Q8×C14), (C2×C28).536D4, C28.479(C2×D4), C2.2(C22×C56), (C22×C8).6C14, C28.127(C2×Q8), C42.66(C2×C14), (C22×C28).21C4, C14.31(C22×C8), C23.38(C2×C28), (C22×C56).12C2, (C22×C4).14C28, (C2×C42).13C14, C22.11(C2×C56), C2.4(C14×M4(2)), (C4×C28).350C22, (C2×C56).359C22, (C2×C28).984C23, (C2×C14).31M4(2), C14.49(C2×M4(2)), C22.20(C22×C28), C22.10(C7×M4(2)), (C22×C28).608C22, C2.3(C14×C4⋊C4), C4.20(C7×C4⋊C4), (C2×C4×C28).36C2, C14.64(C2×C4⋊C4), (C2×C4).25(C7×Q8), (C2×C8).62(C2×C14), (C2×C4).72(C2×C28), (C2×C14).42(C2×C8), (C2×C4).146(C7×D4), C22.19(C7×C4⋊C4), (C2×C14).62(C4⋊C4), (C2×C28).334(C2×C4), (C2×C4).152(C22×C14), (C2×C14).234(C22×C4), (C22×C14).147(C2×C4), (C22×C4).136(C2×C14), SmallGroup(448,830)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C4⋊C8
G = < a,b,c | a14=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 162 in 138 conjugacy classes, 114 normal (30 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, C28, C28, C28, C2×C14, C2×C14, C4⋊C8, C2×C42, C22×C8, C56, C2×C28, C2×C28, C2×C28, C22×C14, C2×C4⋊C8, C4×C28, C2×C56, C2×C56, C22×C28, C7×C4⋊C8, C2×C4×C28, C22×C56, C14×C4⋊C8
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, D4, Q8, C23, C14, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C28, C2×C14, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C56, C2×C28, C7×D4, C7×Q8, C22×C14, C2×C4⋊C8, C7×C4⋊C4, C2×C56, C7×M4(2), C22×C28, D4×C14, Q8×C14, C7×C4⋊C8, C14×C4⋊C4, C22×C56, C14×M4(2), C14×C4⋊C8
►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)
(1 448 145 350)(2 435 146 337)(3 436 147 338)(4 437 148 339)(5 438 149 340)(6 439 150 341)(7 440 151 342)(8 441 152 343)(9 442 153 344)(10 443 154 345)(11 444 141 346)(12 445 142 347)(13 446 143 348)(14 447 144 349)(15 314 432 298)(16 315 433 299)(17 316 434 300)(18 317 421 301)(19 318 422 302)(20 319 423 303)(21 320 424 304)(22 321 425 305)(23 322 426 306)(24 309 427 307)(25 310 428 308)(26 311 429 295)(27 312 430 296)(28 313 431 297)(29 397 373 196)(30 398 374 183)(31 399 375 184)(32 400 376 185)(33 401 377 186)(34 402 378 187)(35 403 365 188)(36 404 366 189)(37 405 367 190)(38 406 368 191)(39 393 369 192)(40 394 370 193)(41 395 371 194)(42 396 372 195)(43 214 233 130)(44 215 234 131)(45 216 235 132)(46 217 236 133)(47 218 237 134)(48 219 238 135)(49 220 225 136)(50 221 226 137)(51 222 227 138)(52 223 228 139)(53 224 229 140)(54 211 230 127)(55 212 231 128)(56 213 232 129)(57 412 290 112)(58 413 291 99)(59 414 292 100)(60 415 293 101)(61 416 294 102)(62 417 281 103)(63 418 282 104)(64 419 283 105)(65 420 284 106)(66 407 285 107)(67 408 286 108)(68 409 287 109)(69 410 288 110)(70 411 289 111)(71 336 364 253)(72 323 351 254)(73 324 352 255)(74 325 353 256)(75 326 354 257)(76 327 355 258)(77 328 356 259)(78 329 357 260)(79 330 358 261)(80 331 359 262)(81 332 360 263)(82 333 361 264)(83 334 362 265)(84 335 363 266)(85 239 168 384)(86 240 155 385)(87 241 156 386)(88 242 157 387)(89 243 158 388)(90 244 159 389)(91 245 160 390)(92 246 161 391)(93 247 162 392)(94 248 163 379)(95 249 164 380)(96 250 165 381)(97 251 166 382)(98 252 167 383)(113 172 200 273)(114 173 201 274)(115 174 202 275)(116 175 203 276)(117 176 204 277)(118 177 205 278)(119 178 206 279)(120 179 207 280)(121 180 208 267)(122 181 209 268)(123 182 210 269)(124 169 197 270)(125 170 198 271)(126 171 199 272)
(1 388 409 377 115 212 422 81)(2 389 410 378 116 213 423 82)(3 390 411 365 117 214 424 83)(4 391 412 366 118 215 425 84)(5 392 413 367 119 216 426 71)(6 379 414 368 120 217 427 72)(7 380 415 369 121 218 428 73)(8 381 416 370 122 219 429 74)(9 382 417 371 123 220 430 75)(10 383 418 372 124 221 431 76)(11 384 419 373 125 222 432 77)(12 385 420 374 126 223 433 78)(13 386 407 375 113 224 434 79)(14 387 408 376 114 211 421 80)(15 356 141 239 105 29 198 138)(16 357 142 240 106 30 199 139)(17 358 143 241 107 31 200 140)(18 359 144 242 108 32 201 127)(19 360 145 243 109 33 202 128)(20 361 146 244 110 34 203 129)(21 362 147 245 111 35 204 130)(22 363 148 246 112 36 205 131)(23 364 149 247 99 37 206 132)(24 351 150 248 100 38 207 133)(25 352 151 249 101 39 208 134)(26 353 152 250 102 40 209 135)(27 354 153 251 103 41 210 136)(28 355 154 252 104 42 197 137)(43 304 265 436 160 289 403 176)(44 305 266 437 161 290 404 177)(45 306 253 438 162 291 405 178)(46 307 254 439 163 292 406 179)(47 308 255 440 164 293 393 180)(48 295 256 441 165 294 394 181)(49 296 257 442 166 281 395 182)(50 297 258 443 167 282 396 169)(51 298 259 444 168 283 397 170)(52 299 260 445 155 284 398 171)(53 300 261 446 156 285 399 172)(54 301 262 447 157 286 400 173)(55 302 263 448 158 287 401 174)(56 303 264 435 159 288 402 175)(57 189 278 234 321 335 339 92)(58 190 279 235 322 336 340 93)(59 191 280 236 309 323 341 94)(60 192 267 237 310 324 342 95)(61 193 268 238 311 325 343 96)(62 194 269 225 312 326 344 97)(63 195 270 226 313 327 345 98)(64 196 271 227 314 328 346 85)(65 183 272 228 315 329 347 86)(66 184 273 229 316 330 348 87)(67 185 274 230 317 331 349 88)(68 186 275 231 318 332 350 89)(69 187 276 232 319 333 337 90)(70 188 277 233 320 334 338 91)
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), (1,448,145,350)(2,435,146,337)(3,436,147,338)(4,437,148,339)(5,438,149,340)(6,439,150,341)(7,440,151,342)(8,441,152,343)(9,442,153,344)(10,443,154,345)(11,444,141,346)(12,445,142,347)(13,446,143,348)(14,447,144,349)(15,314,432,298)(16,315,433,299)(17,316,434,300)(18,317,421,301)(19,318,422,302)(20,319,423,303)(21,320,424,304)(22,321,425,305)(23,322,426,306)(24,309,427,307)(25,310,428,308)(26,311,429,295)(27,312,430,296)(28,313,431,297)(29,397,373,196)(30,398,374,183)(31,399,375,184)(32,400,376,185)(33,401,377,186)(34,402,378,187)(35,403,365,188)(36,404,366,189)(37,405,367,190)(38,406,368,191)(39,393,369,192)(40,394,370,193)(41,395,371,194)(42,396,372,195)(43,214,233,130)(44,215,234,131)(45,216,235,132)(46,217,236,133)(47,218,237,134)(48,219,238,135)(49,220,225,136)(50,221,226,137)(51,222,227,138)(52,223,228,139)(53,224,229,140)(54,211,230,127)(55,212,231,128)(56,213,232,129)(57,412,290,112)(58,413,291,99)(59,414,292,100)(60,415,293,101)(61,416,294,102)(62,417,281,103)(63,418,282,104)(64,419,283,105)(65,420,284,106)(66,407,285,107)(67,408,286,108)(68,409,287,109)(69,410,288,110)(70,411,289,111)(71,336,364,253)(72,323,351,254)(73,324,352,255)(74,325,353,256)(75,326,354,257)(76,327,355,258)(77,328,356,259)(78,329,357,260)(79,330,358,261)(80,331,359,262)(81,332,360,263)(82,333,361,264)(83,334,362,265)(84,335,363,266)(85,239,168,384)(86,240,155,385)(87,241,156,386)(88,242,157,387)(89,243,158,388)(90,244,159,389)(91,245,160,390)(92,246,161,391)(93,247,162,392)(94,248,163,379)(95,249,164,380)(96,250,165,381)(97,251,166,382)(98,252,167,383)(113,172,200,273)(114,173,201,274)(115,174,202,275)(116,175,203,276)(117,176,204,277)(118,177,205,278)(119,178,206,279)(120,179,207,280)(121,180,208,267)(122,181,209,268)(123,182,210,269)(124,169,197,270)(125,170,198,271)(126,171,199,272), (1,388,409,377,115,212,422,81)(2,389,410,378,116,213,423,82)(3,390,411,365,117,214,424,83)(4,391,412,366,118,215,425,84)(5,392,413,367,119,216,426,71)(6,379,414,368,120,217,427,72)(7,380,415,369,121,218,428,73)(8,381,416,370,122,219,429,74)(9,382,417,371,123,220,430,75)(10,383,418,372,124,221,431,76)(11,384,419,373,125,222,432,77)(12,385,420,374,126,223,433,78)(13,386,407,375,113,224,434,79)(14,387,408,376,114,211,421,80)(15,356,141,239,105,29,198,138)(16,357,142,240,106,30,199,139)(17,358,143,241,107,31,200,140)(18,359,144,242,108,32,201,127)(19,360,145,243,109,33,202,128)(20,361,146,244,110,34,203,129)(21,362,147,245,111,35,204,130)(22,363,148,246,112,36,205,131)(23,364,149,247,99,37,206,132)(24,351,150,248,100,38,207,133)(25,352,151,249,101,39,208,134)(26,353,152,250,102,40,209,135)(27,354,153,251,103,41,210,136)(28,355,154,252,104,42,197,137)(43,304,265,436,160,289,403,176)(44,305,266,437,161,290,404,177)(45,306,253,438,162,291,405,178)(46,307,254,439,163,292,406,179)(47,308,255,440,164,293,393,180)(48,295,256,441,165,294,394,181)(49,296,257,442,166,281,395,182)(50,297,258,443,167,282,396,169)(51,298,259,444,168,283,397,170)(52,299,260,445,155,284,398,171)(53,300,261,446,156,285,399,172)(54,301,262,447,157,286,400,173)(55,302,263,448,158,287,401,174)(56,303,264,435,159,288,402,175)(57,189,278,234,321,335,339,92)(58,190,279,235,322,336,340,93)(59,191,280,236,309,323,341,94)(60,192,267,237,310,324,342,95)(61,193,268,238,311,325,343,96)(62,194,269,225,312,326,344,97)(63,195,270,226,313,327,345,98)(64,196,271,227,314,328,346,85)(65,183,272,228,315,329,347,86)(66,184,273,229,316,330,348,87)(67,185,274,230,317,331,349,88)(68,186,275,231,318,332,350,89)(69,187,276,232,319,333,337,90)(70,188,277,233,320,334,338,91)>;
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), (1,448,145,350)(2,435,146,337)(3,436,147,338)(4,437,148,339)(5,438,149,340)(6,439,150,341)(7,440,151,342)(8,441,152,343)(9,442,153,344)(10,443,154,345)(11,444,141,346)(12,445,142,347)(13,446,143,348)(14,447,144,349)(15,314,432,298)(16,315,433,299)(17,316,434,300)(18,317,421,301)(19,318,422,302)(20,319,423,303)(21,320,424,304)(22,321,425,305)(23,322,426,306)(24,309,427,307)(25,310,428,308)(26,311,429,295)(27,312,430,296)(28,313,431,297)(29,397,373,196)(30,398,374,183)(31,399,375,184)(32,400,376,185)(33,401,377,186)(34,402,378,187)(35,403,365,188)(36,404,366,189)(37,405,367,190)(38,406,368,191)(39,393,369,192)(40,394,370,193)(41,395,371,194)(42,396,372,195)(43,214,233,130)(44,215,234,131)(45,216,235,132)(46,217,236,133)(47,218,237,134)(48,219,238,135)(49,220,225,136)(50,221,226,137)(51,222,227,138)(52,223,228,139)(53,224,229,140)(54,211,230,127)(55,212,231,128)(56,213,232,129)(57,412,290,112)(58,413,291,99)(59,414,292,100)(60,415,293,101)(61,416,294,102)(62,417,281,103)(63,418,282,104)(64,419,283,105)(65,420,284,106)(66,407,285,107)(67,408,286,108)(68,409,287,109)(69,410,288,110)(70,411,289,111)(71,336,364,253)(72,323,351,254)(73,324,352,255)(74,325,353,256)(75,326,354,257)(76,327,355,258)(77,328,356,259)(78,329,357,260)(79,330,358,261)(80,331,359,262)(81,332,360,263)(82,333,361,264)(83,334,362,265)(84,335,363,266)(85,239,168,384)(86,240,155,385)(87,241,156,386)(88,242,157,387)(89,243,158,388)(90,244,159,389)(91,245,160,390)(92,246,161,391)(93,247,162,392)(94,248,163,379)(95,249,164,380)(96,250,165,381)(97,251,166,382)(98,252,167,383)(113,172,200,273)(114,173,201,274)(115,174,202,275)(116,175,203,276)(117,176,204,277)(118,177,205,278)(119,178,206,279)(120,179,207,280)(121,180,208,267)(122,181,209,268)(123,182,210,269)(124,169,197,270)(125,170,198,271)(126,171,199,272), (1,388,409,377,115,212,422,81)(2,389,410,378,116,213,423,82)(3,390,411,365,117,214,424,83)(4,391,412,366,118,215,425,84)(5,392,413,367,119,216,426,71)(6,379,414,368,120,217,427,72)(7,380,415,369,121,218,428,73)(8,381,416,370,122,219,429,74)(9,382,417,371,123,220,430,75)(10,383,418,372,124,221,431,76)(11,384,419,373,125,222,432,77)(12,385,420,374,126,223,433,78)(13,386,407,375,113,224,434,79)(14,387,408,376,114,211,421,80)(15,356,141,239,105,29,198,138)(16,357,142,240,106,30,199,139)(17,358,143,241,107,31,200,140)(18,359,144,242,108,32,201,127)(19,360,145,243,109,33,202,128)(20,361,146,244,110,34,203,129)(21,362,147,245,111,35,204,130)(22,363,148,246,112,36,205,131)(23,364,149,247,99,37,206,132)(24,351,150,248,100,38,207,133)(25,352,151,249,101,39,208,134)(26,353,152,250,102,40,209,135)(27,354,153,251,103,41,210,136)(28,355,154,252,104,42,197,137)(43,304,265,436,160,289,403,176)(44,305,266,437,161,290,404,177)(45,306,253,438,162,291,405,178)(46,307,254,439,163,292,406,179)(47,308,255,440,164,293,393,180)(48,295,256,441,165,294,394,181)(49,296,257,442,166,281,395,182)(50,297,258,443,167,282,396,169)(51,298,259,444,168,283,397,170)(52,299,260,445,155,284,398,171)(53,300,261,446,156,285,399,172)(54,301,262,447,157,286,400,173)(55,302,263,448,158,287,401,174)(56,303,264,435,159,288,402,175)(57,189,278,234,321,335,339,92)(58,190,279,235,322,336,340,93)(59,191,280,236,309,323,341,94)(60,192,267,237,310,324,342,95)(61,193,268,238,311,325,343,96)(62,194,269,225,312,326,344,97)(63,195,270,226,313,327,345,98)(64,196,271,227,314,328,346,85)(65,183,272,228,315,329,347,86)(66,184,273,229,316,330,348,87)(67,185,274,230,317,331,349,88)(68,186,275,231,318,332,350,89)(69,187,276,232,319,333,337,90)(70,188,277,233,320,334,338,91) );
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)], [(1,448,145,350),(2,435,146,337),(3,436,147,338),(4,437,148,339),(5,438,149,340),(6,439,150,341),(7,440,151,342),(8,441,152,343),(9,442,153,344),(10,443,154,345),(11,444,141,346),(12,445,142,347),(13,446,143,348),(14,447,144,349),(15,314,432,298),(16,315,433,299),(17,316,434,300),(18,317,421,301),(19,318,422,302),(20,319,423,303),(21,320,424,304),(22,321,425,305),(23,322,426,306),(24,309,427,307),(25,310,428,308),(26,311,429,295),(27,312,430,296),(28,313,431,297),(29,397,373,196),(30,398,374,183),(31,399,375,184),(32,400,376,185),(33,401,377,186),(34,402,378,187),(35,403,365,188),(36,404,366,189),(37,405,367,190),(38,406,368,191),(39,393,369,192),(40,394,370,193),(41,395,371,194),(42,396,372,195),(43,214,233,130),(44,215,234,131),(45,216,235,132),(46,217,236,133),(47,218,237,134),(48,219,238,135),(49,220,225,136),(50,221,226,137),(51,222,227,138),(52,223,228,139),(53,224,229,140),(54,211,230,127),(55,212,231,128),(56,213,232,129),(57,412,290,112),(58,413,291,99),(59,414,292,100),(60,415,293,101),(61,416,294,102),(62,417,281,103),(63,418,282,104),(64,419,283,105),(65,420,284,106),(66,407,285,107),(67,408,286,108),(68,409,287,109),(69,410,288,110),(70,411,289,111),(71,336,364,253),(72,323,351,254),(73,324,352,255),(74,325,353,256),(75,326,354,257),(76,327,355,258),(77,328,356,259),(78,329,357,260),(79,330,358,261),(80,331,359,262),(81,332,360,263),(82,333,361,264),(83,334,362,265),(84,335,363,266),(85,239,168,384),(86,240,155,385),(87,241,156,386),(88,242,157,387),(89,243,158,388),(90,244,159,389),(91,245,160,390),(92,246,161,391),(93,247,162,392),(94,248,163,379),(95,249,164,380),(96,250,165,381),(97,251,166,382),(98,252,167,383),(113,172,200,273),(114,173,201,274),(115,174,202,275),(116,175,203,276),(117,176,204,277),(118,177,205,278),(119,178,206,279),(120,179,207,280),(121,180,208,267),(122,181,209,268),(123,182,210,269),(124,169,197,270),(125,170,198,271),(126,171,199,272)], [(1,388,409,377,115,212,422,81),(2,389,410,378,116,213,423,82),(3,390,411,365,117,214,424,83),(4,391,412,366,118,215,425,84),(5,392,413,367,119,216,426,71),(6,379,414,368,120,217,427,72),(7,380,415,369,121,218,428,73),(8,381,416,370,122,219,429,74),(9,382,417,371,123,220,430,75),(10,383,418,372,124,221,431,76),(11,384,419,373,125,222,432,77),(12,385,420,374,126,223,433,78),(13,386,407,375,113,224,434,79),(14,387,408,376,114,211,421,80),(15,356,141,239,105,29,198,138),(16,357,142,240,106,30,199,139),(17,358,143,241,107,31,200,140),(18,359,144,242,108,32,201,127),(19,360,145,243,109,33,202,128),(20,361,146,244,110,34,203,129),(21,362,147,245,111,35,204,130),(22,363,148,246,112,36,205,131),(23,364,149,247,99,37,206,132),(24,351,150,248,100,38,207,133),(25,352,151,249,101,39,208,134),(26,353,152,250,102,40,209,135),(27,354,153,251,103,41,210,136),(28,355,154,252,104,42,197,137),(43,304,265,436,160,289,403,176),(44,305,266,437,161,290,404,177),(45,306,253,438,162,291,405,178),(46,307,254,439,163,292,406,179),(47,308,255,440,164,293,393,180),(48,295,256,441,165,294,394,181),(49,296,257,442,166,281,395,182),(50,297,258,443,167,282,396,169),(51,298,259,444,168,283,397,170),(52,299,260,445,155,284,398,171),(53,300,261,446,156,285,399,172),(54,301,262,447,157,286,400,173),(55,302,263,448,158,287,401,174),(56,303,264,435,159,288,402,175),(57,189,278,234,321,335,339,92),(58,190,279,235,322,336,340,93),(59,191,280,236,309,323,341,94),(60,192,267,237,310,324,342,95),(61,193,268,238,311,325,343,96),(62,194,269,225,312,326,344,97),(63,195,270,226,313,327,345,98),(64,196,271,227,314,328,346,85),(65,183,272,228,315,329,347,86),(66,184,273,229,316,330,348,87),(67,185,274,230,317,331,349,88),(68,186,275,231,318,332,350,89),(69,187,276,232,319,333,337,90),(70,188,277,233,320,334,338,91)]])
280 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 7A | ··· | 7F | 8A | ··· | 8P | 14A | ··· | 14AP | 28A | ··· | 28AV | 28AW | ··· | 28CR | 56A | ··· | 56CR |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C8 | C14 | C14 | C14 | C28 | C28 | C56 | D4 | Q8 | M4(2) | C7×D4 | C7×Q8 | C7×M4(2) |
| kernel | C14×C4⋊C8 | C7×C4⋊C8 | C2×C4×C28 | C22×C56 | C4×C28 | C22×C28 | C2×C4⋊C8 | C2×C28 | C4⋊C8 | C2×C42 | C22×C8 | C42 | C22×C4 | C2×C4 | C2×C28 | C2×C28 | C2×C14 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 2 | 4 | 4 | 6 | 16 | 24 | 6 | 12 | 24 | 24 | 96 | 2 | 2 | 4 | 12 | 12 | 24 |
Matrix representation of C14×C4⋊C8 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 106 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 18 | 15 |
| 18 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 30 |
| 0 | 0 | 1 | 95 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,106,0,0,0,0,106],[112,0,0,0,0,1,0,0,0,0,98,18,0,0,0,15],[18,0,0,0,0,1,0,0,0,0,18,1,0,0,30,95] >;
C14×C4⋊C8 in GAP, Magma, Sage, TeX
C_{14}\times C_4\rtimes C_8 % in TeX
G:=Group("C14xC4:C8"); // GroupNames label
G:=SmallGroup(448,830);
// by ID
G=gap.SmallGroup(448,830);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,400,124]);
// Polycyclic
G:=Group<a,b,c|a^14=b^4=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations