metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic14⋊10Q8, C42.121D14, C14.642- (1+4), C4.49(Q8×D7), C7⋊1(Q8⋊3Q8), (C4×Q8).10D7, C4⋊C4.321D14, (Q8×C28).11C2, C28.107(C2×Q8), C4.17(C4○D28), (C2×Q8).175D14, C28⋊2Q8.24C2, Dic7.10(C2×Q8), Dic7.Q8.1C2, C28.115(C4○D4), C14.27(C22×Q8), (C2×C14).111C24, (C4×C28).164C22, (C2×C28).589C23, Dic7⋊Q8.7C2, (C4×Dic14).21C2, C4⋊Dic7.42C22, C28.6Q8.10C2, Dic7⋊3Q8.10C2, (Q8×C14).211C22, (C2×Dic7).50C23, (C4×Dic7).80C22, C22.136(C23×D7), Dic7⋊C4.114C22, C2.21(D4.10D14), (C2×Dic14).146C22, C2.11(C2×Q8×D7), C14.52(C2×C4○D4), C2.59(C2×C4○D28), (C7×C4⋊C4).339C22, (C2×C4).166(C22×D7), SmallGroup(448,1020)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 708 in 200 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C4 [×4], C4 [×15], C22, C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C14 [×3], C42, C42 [×2], C42 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×19], C2×Q8, C2×Q8 [×3], Dic7 [×4], Dic7 [×6], C28 [×4], C28 [×5], C2×C14, C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], Dic14 [×4], Dic14 [×4], C2×Dic7 [×8], C2×C28 [×3], C2×C28 [×4], C7×Q8 [×2], Q8⋊3Q8, C4×Dic7 [×6], Dic7⋊C4 [×14], C4⋊Dic7, C4⋊Dic7 [×4], C4×C28, C4×C28 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14, C2×Dic14 [×2], Q8×C14, C4×Dic14, C4×Dic14 [×2], C28⋊2Q8, C28.6Q8 [×2], Dic7⋊3Q8 [×2], Dic7.Q8 [×4], Dic7⋊Q8 [×2], Q8×C28, Dic14⋊10Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C4○D4 [×2], C24, D14 [×7], C22×Q8, C2×C4○D4, 2- (1+4), C22×D7 [×7], Q8⋊3Q8, C4○D28 [×2], Q8×D7 [×2], C23×D7, C2×C4○D28, C2×Q8×D7, D4.10D14, Dic14⋊10Q8
Generators and relations
G = < a,b,c,d | a28=c4=1, b2=a14, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a14b, bd=db, dcd-1=c-1 >
►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)
(1 185 15 171)(2 184 16 170)(3 183 17 169)(4 182 18 196)(5 181 19 195)(6 180 20 194)(7 179 21 193)(8 178 22 192)(9 177 23 191)(10 176 24 190)(11 175 25 189)(12 174 26 188)(13 173 27 187)(14 172 28 186)(29 111 43 97)(30 110 44 96)(31 109 45 95)(32 108 46 94)(33 107 47 93)(34 106 48 92)(35 105 49 91)(36 104 50 90)(37 103 51 89)(38 102 52 88)(39 101 53 87)(40 100 54 86)(41 99 55 85)(42 98 56 112)(57 252 71 238)(58 251 72 237)(59 250 73 236)(60 249 74 235)(61 248 75 234)(62 247 76 233)(63 246 77 232)(64 245 78 231)(65 244 79 230)(66 243 80 229)(67 242 81 228)(68 241 82 227)(69 240 83 226)(70 239 84 225)(113 302 127 288)(114 301 128 287)(115 300 129 286)(116 299 130 285)(117 298 131 284)(118 297 132 283)(119 296 133 282)(120 295 134 281)(121 294 135 308)(122 293 136 307)(123 292 137 306)(124 291 138 305)(125 290 139 304)(126 289 140 303)(141 387 155 373)(142 386 156 372)(143 385 157 371)(144 384 158 370)(145 383 159 369)(146 382 160 368)(147 381 161 367)(148 380 162 366)(149 379 163 365)(150 378 164 392)(151 377 165 391)(152 376 166 390)(153 375 167 389)(154 374 168 388)(197 413 211 399)(198 412 212 398)(199 411 213 397)(200 410 214 396)(201 409 215 395)(202 408 216 394)(203 407 217 393)(204 406 218 420)(205 405 219 419)(206 404 220 418)(207 403 221 417)(208 402 222 416)(209 401 223 415)(210 400 224 414)(253 364 267 350)(254 363 268 349)(255 362 269 348)(256 361 270 347)(257 360 271 346)(258 359 272 345)(259 358 273 344)(260 357 274 343)(261 356 275 342)(262 355 276 341)(263 354 277 340)(264 353 278 339)(265 352 279 338)(266 351 280 337)(309 437 323 423)(310 436 324 422)(311 435 325 421)(312 434 326 448)(313 433 327 447)(314 432 328 446)(315 431 329 445)(316 430 330 444)(317 429 331 443)(318 428 332 442)(319 427 333 441)(320 426 334 440)(321 425 335 439)(322 424 336 438)
(1 430 259 365)(2 431 260 366)(3 432 261 367)(4 433 262 368)(5 434 263 369)(6 435 264 370)(7 436 265 371)(8 437 266 372)(9 438 267 373)(10 439 268 374)(11 440 269 375)(12 441 270 376)(13 442 271 377)(14 443 272 378)(15 444 273 379)(16 445 274 380)(17 446 275 381)(18 447 276 382)(19 448 277 383)(20 421 278 384)(21 422 279 385)(22 423 280 386)(23 424 253 387)(24 425 254 388)(25 426 255 389)(26 427 256 390)(27 428 257 391)(28 429 258 392)(29 197 231 292)(30 198 232 293)(31 199 233 294)(32 200 234 295)(33 201 235 296)(34 202 236 297)(35 203 237 298)(36 204 238 299)(37 205 239 300)(38 206 240 301)(39 207 241 302)(40 208 242 303)(41 209 243 304)(42 210 244 305)(43 211 245 306)(44 212 246 307)(45 213 247 308)(46 214 248 281)(47 215 249 282)(48 216 250 283)(49 217 251 284)(50 218 252 285)(51 219 225 286)(52 220 226 287)(53 221 227 288)(54 222 228 289)(55 223 229 290)(56 224 230 291)(57 116 104 420)(58 117 105 393)(59 118 106 394)(60 119 107 395)(61 120 108 396)(62 121 109 397)(63 122 110 398)(64 123 111 399)(65 124 112 400)(66 125 85 401)(67 126 86 402)(68 127 87 403)(69 128 88 404)(70 129 89 405)(71 130 90 406)(72 131 91 407)(73 132 92 408)(74 133 93 409)(75 134 94 410)(76 135 95 411)(77 136 96 412)(78 137 97 413)(79 138 98 414)(80 139 99 415)(81 140 100 416)(82 113 101 417)(83 114 102 418)(84 115 103 419)(141 191 322 364)(142 192 323 337)(143 193 324 338)(144 194 325 339)(145 195 326 340)(146 196 327 341)(147 169 328 342)(148 170 329 343)(149 171 330 344)(150 172 331 345)(151 173 332 346)(152 174 333 347)(153 175 334 348)(154 176 335 349)(155 177 336 350)(156 178 309 351)(157 179 310 352)(158 180 311 353)(159 181 312 354)(160 182 313 355)(161 183 314 356)(162 184 315 357)(163 185 316 358)(164 186 317 359)(165 187 318 360)(166 188 319 361)(167 189 320 362)(168 190 321 363)
(1 414 259 138)(2 415 260 139)(3 416 261 140)(4 417 262 113)(5 418 263 114)(6 419 264 115)(7 420 265 116)(8 393 266 117)(9 394 267 118)(10 395 268 119)(11 396 269 120)(12 397 270 121)(13 398 271 122)(14 399 272 123)(15 400 273 124)(16 401 274 125)(17 402 275 126)(18 403 276 127)(19 404 277 128)(20 405 278 129)(21 406 279 130)(22 407 280 131)(23 408 253 132)(24 409 254 133)(25 410 255 134)(26 411 256 135)(27 412 257 136)(28 413 258 137)(29 150 231 331)(30 151 232 332)(31 152 233 333)(32 153 234 334)(33 154 235 335)(34 155 236 336)(35 156 237 309)(36 157 238 310)(37 158 239 311)(38 159 240 312)(39 160 241 313)(40 161 242 314)(41 162 243 315)(42 163 244 316)(43 164 245 317)(44 165 246 318)(45 166 247 319)(46 167 248 320)(47 168 249 321)(48 141 250 322)(49 142 251 323)(50 143 252 324)(51 144 225 325)(52 145 226 326)(53 146 227 327)(54 147 228 328)(55 148 229 329)(56 149 230 330)(57 436 104 371)(58 437 105 372)(59 438 106 373)(60 439 107 374)(61 440 108 375)(62 441 109 376)(63 442 110 377)(64 443 111 378)(65 444 112 379)(66 445 85 380)(67 446 86 381)(68 447 87 382)(69 448 88 383)(70 421 89 384)(71 422 90 385)(72 423 91 386)(73 424 92 387)(74 425 93 388)(75 426 94 389)(76 427 95 390)(77 428 96 391)(78 429 97 392)(79 430 98 365)(80 431 99 366)(81 432 100 367)(82 433 101 368)(83 434 102 369)(84 435 103 370)(169 222 342 289)(170 223 343 290)(171 224 344 291)(172 197 345 292)(173 198 346 293)(174 199 347 294)(175 200 348 295)(176 201 349 296)(177 202 350 297)(178 203 351 298)(179 204 352 299)(180 205 353 300)(181 206 354 301)(182 207 355 302)(183 208 356 303)(184 209 357 304)(185 210 358 305)(186 211 359 306)(187 212 360 307)(188 213 361 308)(189 214 362 281)(190 215 363 282)(191 216 364 283)(192 217 337 284)(193 218 338 285)(194 219 339 286)(195 220 340 287)(196 221 341 288)
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), (1,185,15,171)(2,184,16,170)(3,183,17,169)(4,182,18,196)(5,181,19,195)(6,180,20,194)(7,179,21,193)(8,178,22,192)(9,177,23,191)(10,176,24,190)(11,175,25,189)(12,174,26,188)(13,173,27,187)(14,172,28,186)(29,111,43,97)(30,110,44,96)(31,109,45,95)(32,108,46,94)(33,107,47,93)(34,106,48,92)(35,105,49,91)(36,104,50,90)(37,103,51,89)(38,102,52,88)(39,101,53,87)(40,100,54,86)(41,99,55,85)(42,98,56,112)(57,252,71,238)(58,251,72,237)(59,250,73,236)(60,249,74,235)(61,248,75,234)(62,247,76,233)(63,246,77,232)(64,245,78,231)(65,244,79,230)(66,243,80,229)(67,242,81,228)(68,241,82,227)(69,240,83,226)(70,239,84,225)(113,302,127,288)(114,301,128,287)(115,300,129,286)(116,299,130,285)(117,298,131,284)(118,297,132,283)(119,296,133,282)(120,295,134,281)(121,294,135,308)(122,293,136,307)(123,292,137,306)(124,291,138,305)(125,290,139,304)(126,289,140,303)(141,387,155,373)(142,386,156,372)(143,385,157,371)(144,384,158,370)(145,383,159,369)(146,382,160,368)(147,381,161,367)(148,380,162,366)(149,379,163,365)(150,378,164,392)(151,377,165,391)(152,376,166,390)(153,375,167,389)(154,374,168,388)(197,413,211,399)(198,412,212,398)(199,411,213,397)(200,410,214,396)(201,409,215,395)(202,408,216,394)(203,407,217,393)(204,406,218,420)(205,405,219,419)(206,404,220,418)(207,403,221,417)(208,402,222,416)(209,401,223,415)(210,400,224,414)(253,364,267,350)(254,363,268,349)(255,362,269,348)(256,361,270,347)(257,360,271,346)(258,359,272,345)(259,358,273,344)(260,357,274,343)(261,356,275,342)(262,355,276,341)(263,354,277,340)(264,353,278,339)(265,352,279,338)(266,351,280,337)(309,437,323,423)(310,436,324,422)(311,435,325,421)(312,434,326,448)(313,433,327,447)(314,432,328,446)(315,431,329,445)(316,430,330,444)(317,429,331,443)(318,428,332,442)(319,427,333,441)(320,426,334,440)(321,425,335,439)(322,424,336,438), (1,430,259,365)(2,431,260,366)(3,432,261,367)(4,433,262,368)(5,434,263,369)(6,435,264,370)(7,436,265,371)(8,437,266,372)(9,438,267,373)(10,439,268,374)(11,440,269,375)(12,441,270,376)(13,442,271,377)(14,443,272,378)(15,444,273,379)(16,445,274,380)(17,446,275,381)(18,447,276,382)(19,448,277,383)(20,421,278,384)(21,422,279,385)(22,423,280,386)(23,424,253,387)(24,425,254,388)(25,426,255,389)(26,427,256,390)(27,428,257,391)(28,429,258,392)(29,197,231,292)(30,198,232,293)(31,199,233,294)(32,200,234,295)(33,201,235,296)(34,202,236,297)(35,203,237,298)(36,204,238,299)(37,205,239,300)(38,206,240,301)(39,207,241,302)(40,208,242,303)(41,209,243,304)(42,210,244,305)(43,211,245,306)(44,212,246,307)(45,213,247,308)(46,214,248,281)(47,215,249,282)(48,216,250,283)(49,217,251,284)(50,218,252,285)(51,219,225,286)(52,220,226,287)(53,221,227,288)(54,222,228,289)(55,223,229,290)(56,224,230,291)(57,116,104,420)(58,117,105,393)(59,118,106,394)(60,119,107,395)(61,120,108,396)(62,121,109,397)(63,122,110,398)(64,123,111,399)(65,124,112,400)(66,125,85,401)(67,126,86,402)(68,127,87,403)(69,128,88,404)(70,129,89,405)(71,130,90,406)(72,131,91,407)(73,132,92,408)(74,133,93,409)(75,134,94,410)(76,135,95,411)(77,136,96,412)(78,137,97,413)(79,138,98,414)(80,139,99,415)(81,140,100,416)(82,113,101,417)(83,114,102,418)(84,115,103,419)(141,191,322,364)(142,192,323,337)(143,193,324,338)(144,194,325,339)(145,195,326,340)(146,196,327,341)(147,169,328,342)(148,170,329,343)(149,171,330,344)(150,172,331,345)(151,173,332,346)(152,174,333,347)(153,175,334,348)(154,176,335,349)(155,177,336,350)(156,178,309,351)(157,179,310,352)(158,180,311,353)(159,181,312,354)(160,182,313,355)(161,183,314,356)(162,184,315,357)(163,185,316,358)(164,186,317,359)(165,187,318,360)(166,188,319,361)(167,189,320,362)(168,190,321,363), (1,414,259,138)(2,415,260,139)(3,416,261,140)(4,417,262,113)(5,418,263,114)(6,419,264,115)(7,420,265,116)(8,393,266,117)(9,394,267,118)(10,395,268,119)(11,396,269,120)(12,397,270,121)(13,398,271,122)(14,399,272,123)(15,400,273,124)(16,401,274,125)(17,402,275,126)(18,403,276,127)(19,404,277,128)(20,405,278,129)(21,406,279,130)(22,407,280,131)(23,408,253,132)(24,409,254,133)(25,410,255,134)(26,411,256,135)(27,412,257,136)(28,413,258,137)(29,150,231,331)(30,151,232,332)(31,152,233,333)(32,153,234,334)(33,154,235,335)(34,155,236,336)(35,156,237,309)(36,157,238,310)(37,158,239,311)(38,159,240,312)(39,160,241,313)(40,161,242,314)(41,162,243,315)(42,163,244,316)(43,164,245,317)(44,165,246,318)(45,166,247,319)(46,167,248,320)(47,168,249,321)(48,141,250,322)(49,142,251,323)(50,143,252,324)(51,144,225,325)(52,145,226,326)(53,146,227,327)(54,147,228,328)(55,148,229,329)(56,149,230,330)(57,436,104,371)(58,437,105,372)(59,438,106,373)(60,439,107,374)(61,440,108,375)(62,441,109,376)(63,442,110,377)(64,443,111,378)(65,444,112,379)(66,445,85,380)(67,446,86,381)(68,447,87,382)(69,448,88,383)(70,421,89,384)(71,422,90,385)(72,423,91,386)(73,424,92,387)(74,425,93,388)(75,426,94,389)(76,427,95,390)(77,428,96,391)(78,429,97,392)(79,430,98,365)(80,431,99,366)(81,432,100,367)(82,433,101,368)(83,434,102,369)(84,435,103,370)(169,222,342,289)(170,223,343,290)(171,224,344,291)(172,197,345,292)(173,198,346,293)(174,199,347,294)(175,200,348,295)(176,201,349,296)(177,202,350,297)(178,203,351,298)(179,204,352,299)(180,205,353,300)(181,206,354,301)(182,207,355,302)(183,208,356,303)(184,209,357,304)(185,210,358,305)(186,211,359,306)(187,212,360,307)(188,213,361,308)(189,214,362,281)(190,215,363,282)(191,216,364,283)(192,217,337,284)(193,218,338,285)(194,219,339,286)(195,220,340,287)(196,221,341,288)>;
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), (1,185,15,171)(2,184,16,170)(3,183,17,169)(4,182,18,196)(5,181,19,195)(6,180,20,194)(7,179,21,193)(8,178,22,192)(9,177,23,191)(10,176,24,190)(11,175,25,189)(12,174,26,188)(13,173,27,187)(14,172,28,186)(29,111,43,97)(30,110,44,96)(31,109,45,95)(32,108,46,94)(33,107,47,93)(34,106,48,92)(35,105,49,91)(36,104,50,90)(37,103,51,89)(38,102,52,88)(39,101,53,87)(40,100,54,86)(41,99,55,85)(42,98,56,112)(57,252,71,238)(58,251,72,237)(59,250,73,236)(60,249,74,235)(61,248,75,234)(62,247,76,233)(63,246,77,232)(64,245,78,231)(65,244,79,230)(66,243,80,229)(67,242,81,228)(68,241,82,227)(69,240,83,226)(70,239,84,225)(113,302,127,288)(114,301,128,287)(115,300,129,286)(116,299,130,285)(117,298,131,284)(118,297,132,283)(119,296,133,282)(120,295,134,281)(121,294,135,308)(122,293,136,307)(123,292,137,306)(124,291,138,305)(125,290,139,304)(126,289,140,303)(141,387,155,373)(142,386,156,372)(143,385,157,371)(144,384,158,370)(145,383,159,369)(146,382,160,368)(147,381,161,367)(148,380,162,366)(149,379,163,365)(150,378,164,392)(151,377,165,391)(152,376,166,390)(153,375,167,389)(154,374,168,388)(197,413,211,399)(198,412,212,398)(199,411,213,397)(200,410,214,396)(201,409,215,395)(202,408,216,394)(203,407,217,393)(204,406,218,420)(205,405,219,419)(206,404,220,418)(207,403,221,417)(208,402,222,416)(209,401,223,415)(210,400,224,414)(253,364,267,350)(254,363,268,349)(255,362,269,348)(256,361,270,347)(257,360,271,346)(258,359,272,345)(259,358,273,344)(260,357,274,343)(261,356,275,342)(262,355,276,341)(263,354,277,340)(264,353,278,339)(265,352,279,338)(266,351,280,337)(309,437,323,423)(310,436,324,422)(311,435,325,421)(312,434,326,448)(313,433,327,447)(314,432,328,446)(315,431,329,445)(316,430,330,444)(317,429,331,443)(318,428,332,442)(319,427,333,441)(320,426,334,440)(321,425,335,439)(322,424,336,438), (1,430,259,365)(2,431,260,366)(3,432,261,367)(4,433,262,368)(5,434,263,369)(6,435,264,370)(7,436,265,371)(8,437,266,372)(9,438,267,373)(10,439,268,374)(11,440,269,375)(12,441,270,376)(13,442,271,377)(14,443,272,378)(15,444,273,379)(16,445,274,380)(17,446,275,381)(18,447,276,382)(19,448,277,383)(20,421,278,384)(21,422,279,385)(22,423,280,386)(23,424,253,387)(24,425,254,388)(25,426,255,389)(26,427,256,390)(27,428,257,391)(28,429,258,392)(29,197,231,292)(30,198,232,293)(31,199,233,294)(32,200,234,295)(33,201,235,296)(34,202,236,297)(35,203,237,298)(36,204,238,299)(37,205,239,300)(38,206,240,301)(39,207,241,302)(40,208,242,303)(41,209,243,304)(42,210,244,305)(43,211,245,306)(44,212,246,307)(45,213,247,308)(46,214,248,281)(47,215,249,282)(48,216,250,283)(49,217,251,284)(50,218,252,285)(51,219,225,286)(52,220,226,287)(53,221,227,288)(54,222,228,289)(55,223,229,290)(56,224,230,291)(57,116,104,420)(58,117,105,393)(59,118,106,394)(60,119,107,395)(61,120,108,396)(62,121,109,397)(63,122,110,398)(64,123,111,399)(65,124,112,400)(66,125,85,401)(67,126,86,402)(68,127,87,403)(69,128,88,404)(70,129,89,405)(71,130,90,406)(72,131,91,407)(73,132,92,408)(74,133,93,409)(75,134,94,410)(76,135,95,411)(77,136,96,412)(78,137,97,413)(79,138,98,414)(80,139,99,415)(81,140,100,416)(82,113,101,417)(83,114,102,418)(84,115,103,419)(141,191,322,364)(142,192,323,337)(143,193,324,338)(144,194,325,339)(145,195,326,340)(146,196,327,341)(147,169,328,342)(148,170,329,343)(149,171,330,344)(150,172,331,345)(151,173,332,346)(152,174,333,347)(153,175,334,348)(154,176,335,349)(155,177,336,350)(156,178,309,351)(157,179,310,352)(158,180,311,353)(159,181,312,354)(160,182,313,355)(161,183,314,356)(162,184,315,357)(163,185,316,358)(164,186,317,359)(165,187,318,360)(166,188,319,361)(167,189,320,362)(168,190,321,363), (1,414,259,138)(2,415,260,139)(3,416,261,140)(4,417,262,113)(5,418,263,114)(6,419,264,115)(7,420,265,116)(8,393,266,117)(9,394,267,118)(10,395,268,119)(11,396,269,120)(12,397,270,121)(13,398,271,122)(14,399,272,123)(15,400,273,124)(16,401,274,125)(17,402,275,126)(18,403,276,127)(19,404,277,128)(20,405,278,129)(21,406,279,130)(22,407,280,131)(23,408,253,132)(24,409,254,133)(25,410,255,134)(26,411,256,135)(27,412,257,136)(28,413,258,137)(29,150,231,331)(30,151,232,332)(31,152,233,333)(32,153,234,334)(33,154,235,335)(34,155,236,336)(35,156,237,309)(36,157,238,310)(37,158,239,311)(38,159,240,312)(39,160,241,313)(40,161,242,314)(41,162,243,315)(42,163,244,316)(43,164,245,317)(44,165,246,318)(45,166,247,319)(46,167,248,320)(47,168,249,321)(48,141,250,322)(49,142,251,323)(50,143,252,324)(51,144,225,325)(52,145,226,326)(53,146,227,327)(54,147,228,328)(55,148,229,329)(56,149,230,330)(57,436,104,371)(58,437,105,372)(59,438,106,373)(60,439,107,374)(61,440,108,375)(62,441,109,376)(63,442,110,377)(64,443,111,378)(65,444,112,379)(66,445,85,380)(67,446,86,381)(68,447,87,382)(69,448,88,383)(70,421,89,384)(71,422,90,385)(72,423,91,386)(73,424,92,387)(74,425,93,388)(75,426,94,389)(76,427,95,390)(77,428,96,391)(78,429,97,392)(79,430,98,365)(80,431,99,366)(81,432,100,367)(82,433,101,368)(83,434,102,369)(84,435,103,370)(169,222,342,289)(170,223,343,290)(171,224,344,291)(172,197,345,292)(173,198,346,293)(174,199,347,294)(175,200,348,295)(176,201,349,296)(177,202,350,297)(178,203,351,298)(179,204,352,299)(180,205,353,300)(181,206,354,301)(182,207,355,302)(183,208,356,303)(184,209,357,304)(185,210,358,305)(186,211,359,306)(187,212,360,307)(188,213,361,308)(189,214,362,281)(190,215,363,282)(191,216,364,283)(192,217,337,284)(193,218,338,285)(194,219,339,286)(195,220,340,287)(196,221,341,288) );
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)], [(1,185,15,171),(2,184,16,170),(3,183,17,169),(4,182,18,196),(5,181,19,195),(6,180,20,194),(7,179,21,193),(8,178,22,192),(9,177,23,191),(10,176,24,190),(11,175,25,189),(12,174,26,188),(13,173,27,187),(14,172,28,186),(29,111,43,97),(30,110,44,96),(31,109,45,95),(32,108,46,94),(33,107,47,93),(34,106,48,92),(35,105,49,91),(36,104,50,90),(37,103,51,89),(38,102,52,88),(39,101,53,87),(40,100,54,86),(41,99,55,85),(42,98,56,112),(57,252,71,238),(58,251,72,237),(59,250,73,236),(60,249,74,235),(61,248,75,234),(62,247,76,233),(63,246,77,232),(64,245,78,231),(65,244,79,230),(66,243,80,229),(67,242,81,228),(68,241,82,227),(69,240,83,226),(70,239,84,225),(113,302,127,288),(114,301,128,287),(115,300,129,286),(116,299,130,285),(117,298,131,284),(118,297,132,283),(119,296,133,282),(120,295,134,281),(121,294,135,308),(122,293,136,307),(123,292,137,306),(124,291,138,305),(125,290,139,304),(126,289,140,303),(141,387,155,373),(142,386,156,372),(143,385,157,371),(144,384,158,370),(145,383,159,369),(146,382,160,368),(147,381,161,367),(148,380,162,366),(149,379,163,365),(150,378,164,392),(151,377,165,391),(152,376,166,390),(153,375,167,389),(154,374,168,388),(197,413,211,399),(198,412,212,398),(199,411,213,397),(200,410,214,396),(201,409,215,395),(202,408,216,394),(203,407,217,393),(204,406,218,420),(205,405,219,419),(206,404,220,418),(207,403,221,417),(208,402,222,416),(209,401,223,415),(210,400,224,414),(253,364,267,350),(254,363,268,349),(255,362,269,348),(256,361,270,347),(257,360,271,346),(258,359,272,345),(259,358,273,344),(260,357,274,343),(261,356,275,342),(262,355,276,341),(263,354,277,340),(264,353,278,339),(265,352,279,338),(266,351,280,337),(309,437,323,423),(310,436,324,422),(311,435,325,421),(312,434,326,448),(313,433,327,447),(314,432,328,446),(315,431,329,445),(316,430,330,444),(317,429,331,443),(318,428,332,442),(319,427,333,441),(320,426,334,440),(321,425,335,439),(322,424,336,438)], [(1,430,259,365),(2,431,260,366),(3,432,261,367),(4,433,262,368),(5,434,263,369),(6,435,264,370),(7,436,265,371),(8,437,266,372),(9,438,267,373),(10,439,268,374),(11,440,269,375),(12,441,270,376),(13,442,271,377),(14,443,272,378),(15,444,273,379),(16,445,274,380),(17,446,275,381),(18,447,276,382),(19,448,277,383),(20,421,278,384),(21,422,279,385),(22,423,280,386),(23,424,253,387),(24,425,254,388),(25,426,255,389),(26,427,256,390),(27,428,257,391),(28,429,258,392),(29,197,231,292),(30,198,232,293),(31,199,233,294),(32,200,234,295),(33,201,235,296),(34,202,236,297),(35,203,237,298),(36,204,238,299),(37,205,239,300),(38,206,240,301),(39,207,241,302),(40,208,242,303),(41,209,243,304),(42,210,244,305),(43,211,245,306),(44,212,246,307),(45,213,247,308),(46,214,248,281),(47,215,249,282),(48,216,250,283),(49,217,251,284),(50,218,252,285),(51,219,225,286),(52,220,226,287),(53,221,227,288),(54,222,228,289),(55,223,229,290),(56,224,230,291),(57,116,104,420),(58,117,105,393),(59,118,106,394),(60,119,107,395),(61,120,108,396),(62,121,109,397),(63,122,110,398),(64,123,111,399),(65,124,112,400),(66,125,85,401),(67,126,86,402),(68,127,87,403),(69,128,88,404),(70,129,89,405),(71,130,90,406),(72,131,91,407),(73,132,92,408),(74,133,93,409),(75,134,94,410),(76,135,95,411),(77,136,96,412),(78,137,97,413),(79,138,98,414),(80,139,99,415),(81,140,100,416),(82,113,101,417),(83,114,102,418),(84,115,103,419),(141,191,322,364),(142,192,323,337),(143,193,324,338),(144,194,325,339),(145,195,326,340),(146,196,327,341),(147,169,328,342),(148,170,329,343),(149,171,330,344),(150,172,331,345),(151,173,332,346),(152,174,333,347),(153,175,334,348),(154,176,335,349),(155,177,336,350),(156,178,309,351),(157,179,310,352),(158,180,311,353),(159,181,312,354),(160,182,313,355),(161,183,314,356),(162,184,315,357),(163,185,316,358),(164,186,317,359),(165,187,318,360),(166,188,319,361),(167,189,320,362),(168,190,321,363)], [(1,414,259,138),(2,415,260,139),(3,416,261,140),(4,417,262,113),(5,418,263,114),(6,419,264,115),(7,420,265,116),(8,393,266,117),(9,394,267,118),(10,395,268,119),(11,396,269,120),(12,397,270,121),(13,398,271,122),(14,399,272,123),(15,400,273,124),(16,401,274,125),(17,402,275,126),(18,403,276,127),(19,404,277,128),(20,405,278,129),(21,406,279,130),(22,407,280,131),(23,408,253,132),(24,409,254,133),(25,410,255,134),(26,411,256,135),(27,412,257,136),(28,413,258,137),(29,150,231,331),(30,151,232,332),(31,152,233,333),(32,153,234,334),(33,154,235,335),(34,155,236,336),(35,156,237,309),(36,157,238,310),(37,158,239,311),(38,159,240,312),(39,160,241,313),(40,161,242,314),(41,162,243,315),(42,163,244,316),(43,164,245,317),(44,165,246,318),(45,166,247,319),(46,167,248,320),(47,168,249,321),(48,141,250,322),(49,142,251,323),(50,143,252,324),(51,144,225,325),(52,145,226,326),(53,146,227,327),(54,147,228,328),(55,148,229,329),(56,149,230,330),(57,436,104,371),(58,437,105,372),(59,438,106,373),(60,439,107,374),(61,440,108,375),(62,441,109,376),(63,442,110,377),(64,443,111,378),(65,444,112,379),(66,445,85,380),(67,446,86,381),(68,447,87,382),(69,448,88,383),(70,421,89,384),(71,422,90,385),(72,423,91,386),(73,424,92,387),(74,425,93,388),(75,426,94,389),(76,427,95,390),(77,428,96,391),(78,429,97,392),(79,430,98,365),(80,431,99,366),(81,432,100,367),(82,433,101,368),(83,434,102,369),(84,435,103,370),(169,222,342,289),(170,223,343,290),(171,224,344,291),(172,197,345,292),(173,198,346,293),(174,199,347,294),(175,200,348,295),(176,201,349,296),(177,202,350,297),(178,203,351,298),(179,204,352,299),(180,205,353,300),(181,206,354,301),(182,207,355,302),(183,208,356,303),(184,209,357,304),(185,210,358,305),(186,211,359,306),(187,212,360,307),(188,213,361,308),(189,214,362,281),(190,215,363,282),(191,216,364,283),(192,217,337,284),(193,218,338,285),(194,219,339,286),(195,220,340,287),(196,221,341,288)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 2 | 21 | 0 | 0 |
| 14 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 21 | 17 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 24 | 14 | 0 | 0 |
| 19 | 5 | 0 | 0 |
| 0 | 0 | 1 | 27 |
| 0 | 0 | 1 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 7 | 16 |
| 0 | 0 | 15 | 22 |
G:=sub<GL(4,GF(29))| [2,14,0,0,21,17,0,0,0,0,1,0,0,0,0,1],[12,21,0,0,0,17,0,0,0,0,28,0,0,0,0,28],[24,19,0,0,14,5,0,0,0,0,1,1,0,0,27,28],[28,0,0,0,0,28,0,0,0,0,7,15,0,0,16,22] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | ··· | 4U | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | 2- (1+4) | Q8×D7 | D4.10D14 |
| kernel | Dic14⋊10Q8 | C4×Dic14 | C28⋊2Q8 | C28.6Q8 | Dic7⋊3Q8 | Dic7.Q8 | Dic7⋊Q8 | Q8×C28 | Dic14 | C4×Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | C4 | C14 | C4 | C2 |
| # reps | 1 | 3 | 1 | 2 | 2 | 4 | 2 | 1 | 4 | 3 | 4 | 9 | 9 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
Dic_{14}\rtimes_{10}Q_8 % in TeX
G:=Group("Dic14:10Q8"); // GroupNames label
G:=SmallGroup(448,1020);
// by ID
G=gap.SmallGroup(448,1020);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,477,232,100,185,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=c^4=1,b^2=a^14,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^14*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀