direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C28.3Q8, (C2×C28).30Q8, C28.66(C2×Q8), C4⋊C4.260D14, C14.8(C22×Q8), (C2×C14).43C24, C14⋊3(C42.C2), C4.31(C2×Dic14), (C2×C4).37Dic14, (C2×C28).134C23, (C22×C4).358D14, C22.81(C23×D7), C4⋊Dic7.357C22, (C22×C28).73C22, C2.10(C22×Dic14), C22.37(C2×Dic14), C23.323(C22×D7), C22.72(D4⋊2D7), Dic7⋊C4.104C22, (C22×C14).392C23, C22.32(Q8⋊2D7), (C4×Dic7).248C22, (C2×Dic7).183C23, (C22×Dic7).208C22, C7⋊3(C2×C42.C2), (C2×C4⋊C4).26D7, (C14×C4⋊C4).19C2, C14.70(C2×C4○D4), C2.5(C2×Q8⋊2D7), (C2×C14).51(C2×Q8), (C2×C4×Dic7).13C2, C2.14(C2×D4⋊2D7), (C2×C4⋊Dic7).43C2, (C7×C4⋊C4).292C22, (C2×Dic7⋊C4).22C2, (C2×C4).139(C22×D7), (C2×C14).170(C4○D4), SmallGroup(448,952)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C28.3Q8
G = < a,b,c,d | a2=b4=c28=1, d2=c14, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b2c-1 >
Subgroups: 772 in 226 conjugacy classes, 127 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C2×C42.C2, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C28.3Q8, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C2×C4⋊Dic7, C14×C4⋊C4, C2×C28.3Q8
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C4○D4, C24, D14, C42.C2, C22×Q8, C2×C4○D4, Dic14, C22×D7, C2×C42.C2, C2×Dic14, D4⋊2D7, Q8⋊2D7, C23×D7, C28.3Q8, C22×Dic14, C2×D4⋊2D7, C2×Q8⋊2D7, C2×C28.3Q8
►Regular action on 448 points
Generators in S448
(1 221)(2 222)(3 223)(4 224)(5 197)(6 198)(7 199)(8 200)(9 201)(10 202)(11 203)(12 204)(13 205)(14 206)(15 207)(16 208)(17 209)(18 210)(19 211)(20 212)(21 213)(22 214)(23 215)(24 216)(25 217)(26 218)(27 219)(28 220)(29 118)(30 119)(31 120)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 140)(52 113)(53 114)(54 115)(55 116)(56 117)(57 349)(58 350)(59 351)(60 352)(61 353)(62 354)(63 355)(64 356)(65 357)(66 358)(67 359)(68 360)(69 361)(70 362)(71 363)(72 364)(73 337)(74 338)(75 339)(76 340)(77 341)(78 342)(79 343)(80 344)(81 345)(82 346)(83 347)(84 348)(85 229)(86 230)(87 231)(88 232)(89 233)(90 234)(91 235)(92 236)(93 237)(94 238)(95 239)(96 240)(97 241)(98 242)(99 243)(100 244)(101 245)(102 246)(103 247)(104 248)(105 249)(106 250)(107 251)(108 252)(109 225)(110 226)(111 227)(112 228)(141 180)(142 181)(143 182)(144 183)(145 184)(146 185)(147 186)(148 187)(149 188)(150 189)(151 190)(152 191)(153 192)(154 193)(155 194)(156 195)(157 196)(158 169)(159 170)(160 171)(161 172)(162 173)(163 174)(164 175)(165 176)(166 177)(167 178)(168 179)(253 326)(254 327)(255 328)(256 329)(257 330)(258 331)(259 332)(260 333)(261 334)(262 335)(263 336)(264 309)(265 310)(266 311)(267 312)(268 313)(269 314)(270 315)(271 316)(272 317)(273 318)(274 319)(275 320)(276 321)(277 322)(278 323)(279 324)(280 325)(281 423)(282 424)(283 425)(284 426)(285 427)(286 428)(287 429)(288 430)(289 431)(290 432)(291 433)(292 434)(293 435)(294 436)(295 437)(296 438)(297 439)(298 440)(299 441)(300 442)(301 443)(302 444)(303 445)(304 446)(305 447)(306 448)(307 421)(308 422)(365 405)(366 406)(367 407)(368 408)(369 409)(370 410)(371 411)(372 412)(373 413)(374 414)(375 415)(376 416)(377 417)(378 418)(379 419)(380 420)(381 393)(382 394)(383 395)(384 396)(385 397)(386 398)(387 399)(388 400)(389 401)(390 402)(391 403)(392 404)
(1 46 359 417)(2 418 360 47)(3 48 361 419)(4 420 362 49)(5 50 363 393)(6 394 364 51)(7 52 337 395)(8 396 338 53)(9 54 339 397)(10 398 340 55)(11 56 341 399)(12 400 342 29)(13 30 343 401)(14 402 344 31)(15 32 345 403)(16 404 346 33)(17 34 347 405)(18 406 348 35)(19 36 349 407)(20 408 350 37)(21 38 351 409)(22 410 352 39)(23 40 353 411)(24 412 354 41)(25 42 355 413)(26 414 356 43)(27 44 357 415)(28 416 358 45)(57 367 211 125)(58 126 212 368)(59 369 213 127)(60 128 214 370)(61 371 215 129)(62 130 216 372)(63 373 217 131)(64 132 218 374)(65 375 219 133)(66 134 220 376)(67 377 221 135)(68 136 222 378)(69 379 223 137)(70 138 224 380)(71 381 197 139)(72 140 198 382)(73 383 199 113)(74 114 200 384)(75 385 201 115)(76 116 202 386)(77 387 203 117)(78 118 204 388)(79 389 205 119)(80 120 206 390)(81 391 207 121)(82 122 208 392)(83 365 209 123)(84 124 210 366)(85 256 171 307)(86 308 172 257)(87 258 173 281)(88 282 174 259)(89 260 175 283)(90 284 176 261)(91 262 177 285)(92 286 178 263)(93 264 179 287)(94 288 180 265)(95 266 181 289)(96 290 182 267)(97 268 183 291)(98 292 184 269)(99 270 185 293)(100 294 186 271)(101 272 187 295)(102 296 188 273)(103 274 189 297)(104 298 190 275)(105 276 191 299)(106 300 192 277)(107 278 193 301)(108 302 194 279)(109 280 195 303)(110 304 196 253)(111 254 169 305)(112 306 170 255)(141 310 238 430)(142 431 239 311)(143 312 240 432)(144 433 241 313)(145 314 242 434)(146 435 243 315)(147 316 244 436)(148 437 245 317)(149 318 246 438)(150 439 247 319)(151 320 248 440)(152 441 249 321)(153 322 250 442)(154 443 251 323)(155 324 252 444)(156 445 225 325)(157 326 226 446)(158 447 227 327)(159 328 228 448)(160 421 229 329)(161 330 230 422)(162 423 231 331)(163 332 232 424)(164 425 233 333)(165 334 234 426)(166 427 235 335)(167 336 236 428)(168 429 237 309)
(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 15 434)(2 327 16 313)(3 446 17 432)(4 325 18 311)(5 444 19 430)(6 323 20 309)(7 442 21 428)(8 321 22 335)(9 440 23 426)(10 319 24 333)(11 438 25 424)(12 317 26 331)(13 436 27 422)(14 315 28 329)(29 245 43 231)(30 147 44 161)(31 243 45 229)(32 145 46 159)(33 241 47 227)(34 143 48 157)(35 239 49 225)(36 141 50 155)(37 237 51 251)(38 167 52 153)(39 235 53 249)(40 165 54 151)(41 233 55 247)(42 163 56 149)(57 265 71 279)(58 287 72 301)(59 263 73 277)(60 285 74 299)(61 261 75 275)(62 283 76 297)(63 259 77 273)(64 281 78 295)(65 257 79 271)(66 307 80 293)(67 255 81 269)(68 305 82 291)(69 253 83 267)(70 303 84 289)(85 120 99 134)(86 389 100 375)(87 118 101 132)(88 387 102 373)(89 116 103 130)(90 385 104 371)(91 114 105 128)(92 383 106 369)(93 140 107 126)(94 381 108 367)(95 138 109 124)(96 379 110 365)(97 136 111 122)(98 377 112 391)(113 192 127 178)(115 190 129 176)(117 188 131 174)(119 186 133 172)(121 184 135 170)(123 182 137 196)(125 180 139 194)(142 420 156 406)(144 418 158 404)(146 416 160 402)(148 414 162 400)(150 412 164 398)(152 410 166 396)(154 408 168 394)(169 392 183 378)(171 390 185 376)(173 388 187 374)(175 386 189 372)(177 384 191 370)(179 382 193 368)(181 380 195 366)(197 302 211 288)(198 278 212 264)(199 300 213 286)(200 276 214 262)(201 298 215 284)(202 274 216 260)(203 296 217 282)(204 272 218 258)(205 294 219 308)(206 270 220 256)(207 292 221 306)(208 268 222 254)(209 290 223 304)(210 266 224 280)(226 405 240 419)(228 403 242 417)(230 401 244 415)(232 399 246 413)(234 397 248 411)(236 395 250 409)(238 393 252 407)(310 363 324 349)(312 361 326 347)(314 359 328 345)(316 357 330 343)(318 355 332 341)(320 353 334 339)(322 351 336 337)(338 441 352 427)(340 439 354 425)(342 437 356 423)(344 435 358 421)(346 433 360 447)(348 431 362 445)(350 429 364 443)
G:=sub<Sym(448)| (1,221)(2,222)(3,223)(4,224)(5,197)(6,198)(7,199)(8,200)(9,201)(10,202)(11,203)(12,204)(13,205)(14,206)(15,207)(16,208)(17,209)(18,210)(19,211)(20,212)(21,213)(22,214)(23,215)(24,216)(25,217)(26,218)(27,219)(28,220)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,113)(53,114)(54,115)(55,116)(56,117)(57,349)(58,350)(59,351)(60,352)(61,353)(62,354)(63,355)(64,356)(65,357)(66,358)(67,359)(68,360)(69,361)(70,362)(71,363)(72,364)(73,337)(74,338)(75,339)(76,340)(77,341)(78,342)(79,343)(80,344)(81,345)(82,346)(83,347)(84,348)(85,229)(86,230)(87,231)(88,232)(89,233)(90,234)(91,235)(92,236)(93,237)(94,238)(95,239)(96,240)(97,241)(98,242)(99,243)(100,244)(101,245)(102,246)(103,247)(104,248)(105,249)(106,250)(107,251)(108,252)(109,225)(110,226)(111,227)(112,228)(141,180)(142,181)(143,182)(144,183)(145,184)(146,185)(147,186)(148,187)(149,188)(150,189)(151,190)(152,191)(153,192)(154,193)(155,194)(156,195)(157,196)(158,169)(159,170)(160,171)(161,172)(162,173)(163,174)(164,175)(165,176)(166,177)(167,178)(168,179)(253,326)(254,327)(255,328)(256,329)(257,330)(258,331)(259,332)(260,333)(261,334)(262,335)(263,336)(264,309)(265,310)(266,311)(267,312)(268,313)(269,314)(270,315)(271,316)(272,317)(273,318)(274,319)(275,320)(276,321)(277,322)(278,323)(279,324)(280,325)(281,423)(282,424)(283,425)(284,426)(285,427)(286,428)(287,429)(288,430)(289,431)(290,432)(291,433)(292,434)(293,435)(294,436)(295,437)(296,438)(297,439)(298,440)(299,441)(300,442)(301,443)(302,444)(303,445)(304,446)(305,447)(306,448)(307,421)(308,422)(365,405)(366,406)(367,407)(368,408)(369,409)(370,410)(371,411)(372,412)(373,413)(374,414)(375,415)(376,416)(377,417)(378,418)(379,419)(380,420)(381,393)(382,394)(383,395)(384,396)(385,397)(386,398)(387,399)(388,400)(389,401)(390,402)(391,403)(392,404), (1,46,359,417)(2,418,360,47)(3,48,361,419)(4,420,362,49)(5,50,363,393)(6,394,364,51)(7,52,337,395)(8,396,338,53)(9,54,339,397)(10,398,340,55)(11,56,341,399)(12,400,342,29)(13,30,343,401)(14,402,344,31)(15,32,345,403)(16,404,346,33)(17,34,347,405)(18,406,348,35)(19,36,349,407)(20,408,350,37)(21,38,351,409)(22,410,352,39)(23,40,353,411)(24,412,354,41)(25,42,355,413)(26,414,356,43)(27,44,357,415)(28,416,358,45)(57,367,211,125)(58,126,212,368)(59,369,213,127)(60,128,214,370)(61,371,215,129)(62,130,216,372)(63,373,217,131)(64,132,218,374)(65,375,219,133)(66,134,220,376)(67,377,221,135)(68,136,222,378)(69,379,223,137)(70,138,224,380)(71,381,197,139)(72,140,198,382)(73,383,199,113)(74,114,200,384)(75,385,201,115)(76,116,202,386)(77,387,203,117)(78,118,204,388)(79,389,205,119)(80,120,206,390)(81,391,207,121)(82,122,208,392)(83,365,209,123)(84,124,210,366)(85,256,171,307)(86,308,172,257)(87,258,173,281)(88,282,174,259)(89,260,175,283)(90,284,176,261)(91,262,177,285)(92,286,178,263)(93,264,179,287)(94,288,180,265)(95,266,181,289)(96,290,182,267)(97,268,183,291)(98,292,184,269)(99,270,185,293)(100,294,186,271)(101,272,187,295)(102,296,188,273)(103,274,189,297)(104,298,190,275)(105,276,191,299)(106,300,192,277)(107,278,193,301)(108,302,194,279)(109,280,195,303)(110,304,196,253)(111,254,169,305)(112,306,170,255)(141,310,238,430)(142,431,239,311)(143,312,240,432)(144,433,241,313)(145,314,242,434)(146,435,243,315)(147,316,244,436)(148,437,245,317)(149,318,246,438)(150,439,247,319)(151,320,248,440)(152,441,249,321)(153,322,250,442)(154,443,251,323)(155,324,252,444)(156,445,225,325)(157,326,226,446)(158,447,227,327)(159,328,228,448)(160,421,229,329)(161,330,230,422)(162,423,231,331)(163,332,232,424)(164,425,233,333)(165,334,234,426)(166,427,235,335)(167,336,236,428)(168,429,237,309), (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), (1,448,15,434)(2,327,16,313)(3,446,17,432)(4,325,18,311)(5,444,19,430)(6,323,20,309)(7,442,21,428)(8,321,22,335)(9,440,23,426)(10,319,24,333)(11,438,25,424)(12,317,26,331)(13,436,27,422)(14,315,28,329)(29,245,43,231)(30,147,44,161)(31,243,45,229)(32,145,46,159)(33,241,47,227)(34,143,48,157)(35,239,49,225)(36,141,50,155)(37,237,51,251)(38,167,52,153)(39,235,53,249)(40,165,54,151)(41,233,55,247)(42,163,56,149)(57,265,71,279)(58,287,72,301)(59,263,73,277)(60,285,74,299)(61,261,75,275)(62,283,76,297)(63,259,77,273)(64,281,78,295)(65,257,79,271)(66,307,80,293)(67,255,81,269)(68,305,82,291)(69,253,83,267)(70,303,84,289)(85,120,99,134)(86,389,100,375)(87,118,101,132)(88,387,102,373)(89,116,103,130)(90,385,104,371)(91,114,105,128)(92,383,106,369)(93,140,107,126)(94,381,108,367)(95,138,109,124)(96,379,110,365)(97,136,111,122)(98,377,112,391)(113,192,127,178)(115,190,129,176)(117,188,131,174)(119,186,133,172)(121,184,135,170)(123,182,137,196)(125,180,139,194)(142,420,156,406)(144,418,158,404)(146,416,160,402)(148,414,162,400)(150,412,164,398)(152,410,166,396)(154,408,168,394)(169,392,183,378)(171,390,185,376)(173,388,187,374)(175,386,189,372)(177,384,191,370)(179,382,193,368)(181,380,195,366)(197,302,211,288)(198,278,212,264)(199,300,213,286)(200,276,214,262)(201,298,215,284)(202,274,216,260)(203,296,217,282)(204,272,218,258)(205,294,219,308)(206,270,220,256)(207,292,221,306)(208,268,222,254)(209,290,223,304)(210,266,224,280)(226,405,240,419)(228,403,242,417)(230,401,244,415)(232,399,246,413)(234,397,248,411)(236,395,250,409)(238,393,252,407)(310,363,324,349)(312,361,326,347)(314,359,328,345)(316,357,330,343)(318,355,332,341)(320,353,334,339)(322,351,336,337)(338,441,352,427)(340,439,354,425)(342,437,356,423)(344,435,358,421)(346,433,360,447)(348,431,362,445)(350,429,364,443)>;
G:=Group( (1,221)(2,222)(3,223)(4,224)(5,197)(6,198)(7,199)(8,200)(9,201)(10,202)(11,203)(12,204)(13,205)(14,206)(15,207)(16,208)(17,209)(18,210)(19,211)(20,212)(21,213)(22,214)(23,215)(24,216)(25,217)(26,218)(27,219)(28,220)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,113)(53,114)(54,115)(55,116)(56,117)(57,349)(58,350)(59,351)(60,352)(61,353)(62,354)(63,355)(64,356)(65,357)(66,358)(67,359)(68,360)(69,361)(70,362)(71,363)(72,364)(73,337)(74,338)(75,339)(76,340)(77,341)(78,342)(79,343)(80,344)(81,345)(82,346)(83,347)(84,348)(85,229)(86,230)(87,231)(88,232)(89,233)(90,234)(91,235)(92,236)(93,237)(94,238)(95,239)(96,240)(97,241)(98,242)(99,243)(100,244)(101,245)(102,246)(103,247)(104,248)(105,249)(106,250)(107,251)(108,252)(109,225)(110,226)(111,227)(112,228)(141,180)(142,181)(143,182)(144,183)(145,184)(146,185)(147,186)(148,187)(149,188)(150,189)(151,190)(152,191)(153,192)(154,193)(155,194)(156,195)(157,196)(158,169)(159,170)(160,171)(161,172)(162,173)(163,174)(164,175)(165,176)(166,177)(167,178)(168,179)(253,326)(254,327)(255,328)(256,329)(257,330)(258,331)(259,332)(260,333)(261,334)(262,335)(263,336)(264,309)(265,310)(266,311)(267,312)(268,313)(269,314)(270,315)(271,316)(272,317)(273,318)(274,319)(275,320)(276,321)(277,322)(278,323)(279,324)(280,325)(281,423)(282,424)(283,425)(284,426)(285,427)(286,428)(287,429)(288,430)(289,431)(290,432)(291,433)(292,434)(293,435)(294,436)(295,437)(296,438)(297,439)(298,440)(299,441)(300,442)(301,443)(302,444)(303,445)(304,446)(305,447)(306,448)(307,421)(308,422)(365,405)(366,406)(367,407)(368,408)(369,409)(370,410)(371,411)(372,412)(373,413)(374,414)(375,415)(376,416)(377,417)(378,418)(379,419)(380,420)(381,393)(382,394)(383,395)(384,396)(385,397)(386,398)(387,399)(388,400)(389,401)(390,402)(391,403)(392,404), (1,46,359,417)(2,418,360,47)(3,48,361,419)(4,420,362,49)(5,50,363,393)(6,394,364,51)(7,52,337,395)(8,396,338,53)(9,54,339,397)(10,398,340,55)(11,56,341,399)(12,400,342,29)(13,30,343,401)(14,402,344,31)(15,32,345,403)(16,404,346,33)(17,34,347,405)(18,406,348,35)(19,36,349,407)(20,408,350,37)(21,38,351,409)(22,410,352,39)(23,40,353,411)(24,412,354,41)(25,42,355,413)(26,414,356,43)(27,44,357,415)(28,416,358,45)(57,367,211,125)(58,126,212,368)(59,369,213,127)(60,128,214,370)(61,371,215,129)(62,130,216,372)(63,373,217,131)(64,132,218,374)(65,375,219,133)(66,134,220,376)(67,377,221,135)(68,136,222,378)(69,379,223,137)(70,138,224,380)(71,381,197,139)(72,140,198,382)(73,383,199,113)(74,114,200,384)(75,385,201,115)(76,116,202,386)(77,387,203,117)(78,118,204,388)(79,389,205,119)(80,120,206,390)(81,391,207,121)(82,122,208,392)(83,365,209,123)(84,124,210,366)(85,256,171,307)(86,308,172,257)(87,258,173,281)(88,282,174,259)(89,260,175,283)(90,284,176,261)(91,262,177,285)(92,286,178,263)(93,264,179,287)(94,288,180,265)(95,266,181,289)(96,290,182,267)(97,268,183,291)(98,292,184,269)(99,270,185,293)(100,294,186,271)(101,272,187,295)(102,296,188,273)(103,274,189,297)(104,298,190,275)(105,276,191,299)(106,300,192,277)(107,278,193,301)(108,302,194,279)(109,280,195,303)(110,304,196,253)(111,254,169,305)(112,306,170,255)(141,310,238,430)(142,431,239,311)(143,312,240,432)(144,433,241,313)(145,314,242,434)(146,435,243,315)(147,316,244,436)(148,437,245,317)(149,318,246,438)(150,439,247,319)(151,320,248,440)(152,441,249,321)(153,322,250,442)(154,443,251,323)(155,324,252,444)(156,445,225,325)(157,326,226,446)(158,447,227,327)(159,328,228,448)(160,421,229,329)(161,330,230,422)(162,423,231,331)(163,332,232,424)(164,425,233,333)(165,334,234,426)(166,427,235,335)(167,336,236,428)(168,429,237,309), (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), (1,448,15,434)(2,327,16,313)(3,446,17,432)(4,325,18,311)(5,444,19,430)(6,323,20,309)(7,442,21,428)(8,321,22,335)(9,440,23,426)(10,319,24,333)(11,438,25,424)(12,317,26,331)(13,436,27,422)(14,315,28,329)(29,245,43,231)(30,147,44,161)(31,243,45,229)(32,145,46,159)(33,241,47,227)(34,143,48,157)(35,239,49,225)(36,141,50,155)(37,237,51,251)(38,167,52,153)(39,235,53,249)(40,165,54,151)(41,233,55,247)(42,163,56,149)(57,265,71,279)(58,287,72,301)(59,263,73,277)(60,285,74,299)(61,261,75,275)(62,283,76,297)(63,259,77,273)(64,281,78,295)(65,257,79,271)(66,307,80,293)(67,255,81,269)(68,305,82,291)(69,253,83,267)(70,303,84,289)(85,120,99,134)(86,389,100,375)(87,118,101,132)(88,387,102,373)(89,116,103,130)(90,385,104,371)(91,114,105,128)(92,383,106,369)(93,140,107,126)(94,381,108,367)(95,138,109,124)(96,379,110,365)(97,136,111,122)(98,377,112,391)(113,192,127,178)(115,190,129,176)(117,188,131,174)(119,186,133,172)(121,184,135,170)(123,182,137,196)(125,180,139,194)(142,420,156,406)(144,418,158,404)(146,416,160,402)(148,414,162,400)(150,412,164,398)(152,410,166,396)(154,408,168,394)(169,392,183,378)(171,390,185,376)(173,388,187,374)(175,386,189,372)(177,384,191,370)(179,382,193,368)(181,380,195,366)(197,302,211,288)(198,278,212,264)(199,300,213,286)(200,276,214,262)(201,298,215,284)(202,274,216,260)(203,296,217,282)(204,272,218,258)(205,294,219,308)(206,270,220,256)(207,292,221,306)(208,268,222,254)(209,290,223,304)(210,266,224,280)(226,405,240,419)(228,403,242,417)(230,401,244,415)(232,399,246,413)(234,397,248,411)(236,395,250,409)(238,393,252,407)(310,363,324,349)(312,361,326,347)(314,359,328,345)(316,357,330,343)(318,355,332,341)(320,353,334,339)(322,351,336,337)(338,441,352,427)(340,439,354,425)(342,437,356,423)(344,435,358,421)(346,433,360,447)(348,431,362,445)(350,429,364,443) );
G=PermutationGroup([[(1,221),(2,222),(3,223),(4,224),(5,197),(6,198),(7,199),(8,200),(9,201),(10,202),(11,203),(12,204),(13,205),(14,206),(15,207),(16,208),(17,209),(18,210),(19,211),(20,212),(21,213),(22,214),(23,215),(24,216),(25,217),(26,218),(27,219),(28,220),(29,118),(30,119),(31,120),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,127),(39,128),(40,129),(41,130),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,137),(49,138),(50,139),(51,140),(52,113),(53,114),(54,115),(55,116),(56,117),(57,349),(58,350),(59,351),(60,352),(61,353),(62,354),(63,355),(64,356),(65,357),(66,358),(67,359),(68,360),(69,361),(70,362),(71,363),(72,364),(73,337),(74,338),(75,339),(76,340),(77,341),(78,342),(79,343),(80,344),(81,345),(82,346),(83,347),(84,348),(85,229),(86,230),(87,231),(88,232),(89,233),(90,234),(91,235),(92,236),(93,237),(94,238),(95,239),(96,240),(97,241),(98,242),(99,243),(100,244),(101,245),(102,246),(103,247),(104,248),(105,249),(106,250),(107,251),(108,252),(109,225),(110,226),(111,227),(112,228),(141,180),(142,181),(143,182),(144,183),(145,184),(146,185),(147,186),(148,187),(149,188),(150,189),(151,190),(152,191),(153,192),(154,193),(155,194),(156,195),(157,196),(158,169),(159,170),(160,171),(161,172),(162,173),(163,174),(164,175),(165,176),(166,177),(167,178),(168,179),(253,326),(254,327),(255,328),(256,329),(257,330),(258,331),(259,332),(260,333),(261,334),(262,335),(263,336),(264,309),(265,310),(266,311),(267,312),(268,313),(269,314),(270,315),(271,316),(272,317),(273,318),(274,319),(275,320),(276,321),(277,322),(278,323),(279,324),(280,325),(281,423),(282,424),(283,425),(284,426),(285,427),(286,428),(287,429),(288,430),(289,431),(290,432),(291,433),(292,434),(293,435),(294,436),(295,437),(296,438),(297,439),(298,440),(299,441),(300,442),(301,443),(302,444),(303,445),(304,446),(305,447),(306,448),(307,421),(308,422),(365,405),(366,406),(367,407),(368,408),(369,409),(370,410),(371,411),(372,412),(373,413),(374,414),(375,415),(376,416),(377,417),(378,418),(379,419),(380,420),(381,393),(382,394),(383,395),(384,396),(385,397),(386,398),(387,399),(388,400),(389,401),(390,402),(391,403),(392,404)], [(1,46,359,417),(2,418,360,47),(3,48,361,419),(4,420,362,49),(5,50,363,393),(6,394,364,51),(7,52,337,395),(8,396,338,53),(9,54,339,397),(10,398,340,55),(11,56,341,399),(12,400,342,29),(13,30,343,401),(14,402,344,31),(15,32,345,403),(16,404,346,33),(17,34,347,405),(18,406,348,35),(19,36,349,407),(20,408,350,37),(21,38,351,409),(22,410,352,39),(23,40,353,411),(24,412,354,41),(25,42,355,413),(26,414,356,43),(27,44,357,415),(28,416,358,45),(57,367,211,125),(58,126,212,368),(59,369,213,127),(60,128,214,370),(61,371,215,129),(62,130,216,372),(63,373,217,131),(64,132,218,374),(65,375,219,133),(66,134,220,376),(67,377,221,135),(68,136,222,378),(69,379,223,137),(70,138,224,380),(71,381,197,139),(72,140,198,382),(73,383,199,113),(74,114,200,384),(75,385,201,115),(76,116,202,386),(77,387,203,117),(78,118,204,388),(79,389,205,119),(80,120,206,390),(81,391,207,121),(82,122,208,392),(83,365,209,123),(84,124,210,366),(85,256,171,307),(86,308,172,257),(87,258,173,281),(88,282,174,259),(89,260,175,283),(90,284,176,261),(91,262,177,285),(92,286,178,263),(93,264,179,287),(94,288,180,265),(95,266,181,289),(96,290,182,267),(97,268,183,291),(98,292,184,269),(99,270,185,293),(100,294,186,271),(101,272,187,295),(102,296,188,273),(103,274,189,297),(104,298,190,275),(105,276,191,299),(106,300,192,277),(107,278,193,301),(108,302,194,279),(109,280,195,303),(110,304,196,253),(111,254,169,305),(112,306,170,255),(141,310,238,430),(142,431,239,311),(143,312,240,432),(144,433,241,313),(145,314,242,434),(146,435,243,315),(147,316,244,436),(148,437,245,317),(149,318,246,438),(150,439,247,319),(151,320,248,440),(152,441,249,321),(153,322,250,442),(154,443,251,323),(155,324,252,444),(156,445,225,325),(157,326,226,446),(158,447,227,327),(159,328,228,448),(160,421,229,329),(161,330,230,422),(162,423,231,331),(163,332,232,424),(164,425,233,333),(165,334,234,426),(166,427,235,335),(167,336,236,428),(168,429,237,309)], [(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)], [(1,448,15,434),(2,327,16,313),(3,446,17,432),(4,325,18,311),(5,444,19,430),(6,323,20,309),(7,442,21,428),(8,321,22,335),(9,440,23,426),(10,319,24,333),(11,438,25,424),(12,317,26,331),(13,436,27,422),(14,315,28,329),(29,245,43,231),(30,147,44,161),(31,243,45,229),(32,145,46,159),(33,241,47,227),(34,143,48,157),(35,239,49,225),(36,141,50,155),(37,237,51,251),(38,167,52,153),(39,235,53,249),(40,165,54,151),(41,233,55,247),(42,163,56,149),(57,265,71,279),(58,287,72,301),(59,263,73,277),(60,285,74,299),(61,261,75,275),(62,283,76,297),(63,259,77,273),(64,281,78,295),(65,257,79,271),(66,307,80,293),(67,255,81,269),(68,305,82,291),(69,253,83,267),(70,303,84,289),(85,120,99,134),(86,389,100,375),(87,118,101,132),(88,387,102,373),(89,116,103,130),(90,385,104,371),(91,114,105,128),(92,383,106,369),(93,140,107,126),(94,381,108,367),(95,138,109,124),(96,379,110,365),(97,136,111,122),(98,377,112,391),(113,192,127,178),(115,190,129,176),(117,188,131,174),(119,186,133,172),(121,184,135,170),(123,182,137,196),(125,180,139,194),(142,420,156,406),(144,418,158,404),(146,416,160,402),(148,414,162,400),(150,412,164,398),(152,410,166,396),(154,408,168,394),(169,392,183,378),(171,390,185,376),(173,388,187,374),(175,386,189,372),(177,384,191,370),(179,382,193,368),(181,380,195,366),(197,302,211,288),(198,278,212,264),(199,300,213,286),(200,276,214,262),(201,298,215,284),(202,274,216,260),(203,296,217,282),(204,272,218,258),(205,294,219,308),(206,270,220,256),(207,292,221,306),(208,268,222,254),(209,290,223,304),(210,266,224,280),(226,405,240,419),(228,403,242,417),(230,401,244,415),(232,399,246,413),(234,397,248,411),(236,395,250,409),(238,393,252,407),(310,363,324,349),(312,361,326,347),(314,359,328,345),(316,357,330,343),(318,355,332,341),(320,353,334,339),(322,351,336,337),(338,441,352,427),(340,439,354,425),(342,437,356,423),(344,435,358,421),(346,433,360,447),(348,431,362,445),(350,429,364,443)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | C4○D4 | D14 | D14 | Dic14 | D4⋊2D7 | Q8⋊2D7 |
| kernel | C2×C28.3Q8 | C28.3Q8 | C2×C4×Dic7 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C14×C4⋊C4 | C2×C28 | C2×C4⋊C4 | C2×C14 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 3 | 1 | 4 | 3 | 8 | 12 | 9 | 24 | 6 | 6 |
Matrix representation of C2×C28.3Q8 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 25 |
| 0 | 0 | 0 | 0 | 27 | 7 |
| 15 | 1 | 0 | 0 | 0 | 0 |
| 26 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 0 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 14 |
| 0 | 0 | 0 | 0 | 4 | 28 |
| 9 | 25 | 0 | 0 | 0 | 0 |
| 20 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,27,0,0,0,0,25,7],[15,26,0,0,0,0,1,25,0,0,0,0,0,0,26,0,0,0,0,0,0,19,0,0,0,0,0,0,1,4,0,0,0,0,14,28],[9,20,0,0,0,0,25,20,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17] >;
C2×C28.3Q8 in GAP, Magma, Sage, TeX
C_2\times C_{28}._3Q_8 % in TeX
G:=Group("C2xC28.3Q8"); // GroupNames label
G:=SmallGroup(448,952);
// by ID
G=gap.SmallGroup(448,952);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,184,1571,297,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^28=1,d^2=c^14,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations