direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4.6Q16, C28.28Q16, C28.42SD16, C4⋊C8.3C14, C4⋊Q8.2C14, C4.6(C7×Q16), (Q8×C14).5C4, (C2×Q8).2C28, C4.7(C7×SD16), (C2×C28).505D4, C42.5(C2×C14), (C4×C28).245C22, C14.17(Q8⋊C4), C14.14(C4.D4), (C7×C4⋊C8).9C2, (C7×C4⋊Q8).17C2, (C2×C4).13(C2×C28), (C2×C4).111(C7×D4), C2.5(C7×Q8⋊C4), C2.5(C7×C4.D4), (C2×C28).180(C2×C4), C22.41(C7×C22⋊C4), (C2×C14).128(C22⋊C4), SmallGroup(448,137)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.6Q16
G = < a,b,c,d | a7=b4=c8=1, d2=b2c4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c-1 >
Subgroups: 114 in 64 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×C8, C2×Q8, C28, C28, C2×C14, C4⋊C8, C4⋊Q8, C56, C2×C28, C2×C28, C2×C28, C7×Q8, C4.6Q16, C4×C28, C7×C4⋊C4, C2×C56, Q8×C14, C7×C4⋊C8, C7×C4⋊Q8, C7×C4.6Q16
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, SD16, Q16, C28, C2×C14, C4.D4, Q8⋊C4, C2×C28, C7×D4, C4.6Q16, C7×C22⋊C4, C7×SD16, C7×Q16, C7×C4.D4, C7×Q8⋊C4, C7×C4.6Q16
►Regular action on 448 points
Generators in S448
(1 391 383 375 367 359 351)(2 392 384 376 368 360 352)(3 385 377 369 361 353 345)(4 386 378 370 362 354 346)(5 387 379 371 363 355 347)(6 388 380 372 364 356 348)(7 389 381 373 365 357 349)(8 390 382 374 366 358 350)(9 57 49 41 33 25 17)(10 58 50 42 34 26 18)(11 59 51 43 35 27 19)(12 60 52 44 36 28 20)(13 61 53 45 37 29 21)(14 62 54 46 38 30 22)(15 63 55 47 39 31 23)(16 64 56 48 40 32 24)(65 113 105 97 89 81 73)(66 114 106 98 90 82 74)(67 115 107 99 91 83 75)(68 116 108 100 92 84 76)(69 117 109 101 93 85 77)(70 118 110 102 94 86 78)(71 119 111 103 95 87 79)(72 120 112 104 96 88 80)(121 169 161 153 145 137 129)(122 170 162 154 146 138 130)(123 171 163 155 147 139 131)(124 172 164 156 148 140 132)(125 173 165 157 149 141 133)(126 174 166 158 150 142 134)(127 175 167 159 151 143 135)(128 176 168 160 152 144 136)(177 225 217 209 201 193 185)(178 226 218 210 202 194 186)(179 227 219 211 203 195 187)(180 228 220 212 204 196 188)(181 229 221 213 205 197 189)(182 230 222 214 206 198 190)(183 231 223 215 207 199 191)(184 232 224 216 208 200 192)(233 281 273 265 257 249 241)(234 282 274 266 258 250 242)(235 283 275 267 259 251 243)(236 284 276 268 260 252 244)(237 285 277 269 261 253 245)(238 286 278 270 262 254 246)(239 287 279 271 263 255 247)(240 288 280 272 264 256 248)(289 337 329 321 313 305 297)(290 338 330 322 314 306 298)(291 339 331 323 315 307 299)(292 340 332 324 316 308 300)(293 341 333 325 317 309 301)(294 342 334 326 318 310 302)(295 343 335 327 319 311 303)(296 344 336 328 320 312 304)(393 441 433 425 417 409 401)(394 442 434 426 418 410 402)(395 443 435 427 419 411 403)(396 444 436 428 420 412 404)(397 445 437 429 421 413 405)(398 446 438 430 422 414 406)(399 447 439 431 423 415 407)(400 448 440 432 424 416 408)
(1 121 11 237)(2 238 12 122)(3 123 13 239)(4 240 14 124)(5 125 15 233)(6 234 16 126)(7 127 9 235)(8 236 10 128)(17 243 349 135)(18 136 350 244)(19 245 351 129)(20 130 352 246)(21 247 345 131)(22 132 346 248)(23 241 347 133)(24 134 348 242)(25 251 357 143)(26 144 358 252)(27 253 359 137)(28 138 360 254)(29 255 353 139)(30 140 354 256)(31 249 355 141)(32 142 356 250)(33 259 365 151)(34 152 366 260)(35 261 367 145)(36 146 368 262)(37 263 361 147)(38 148 362 264)(39 257 363 149)(40 150 364 258)(41 267 373 159)(42 160 374 268)(43 269 375 153)(44 154 376 270)(45 271 369 155)(46 156 370 272)(47 265 371 157)(48 158 372 266)(49 275 381 167)(50 168 382 276)(51 277 383 161)(52 162 384 278)(53 279 377 163)(54 164 378 280)(55 273 379 165)(56 166 380 274)(57 283 389 175)(58 176 390 284)(59 285 391 169)(60 170 392 286)(61 287 385 171)(62 172 386 288)(63 281 387 173)(64 174 388 282)(65 179 397 295)(66 296 398 180)(67 181 399 289)(68 290 400 182)(69 183 393 291)(70 292 394 184)(71 177 395 293)(72 294 396 178)(73 187 405 303)(74 304 406 188)(75 189 407 297)(76 298 408 190)(77 191 401 299)(78 300 402 192)(79 185 403 301)(80 302 404 186)(81 195 413 311)(82 312 414 196)(83 197 415 305)(84 306 416 198)(85 199 409 307)(86 308 410 200)(87 193 411 309)(88 310 412 194)(89 203 421 319)(90 320 422 204)(91 205 423 313)(92 314 424 206)(93 207 417 315)(94 316 418 208)(95 201 419 317)(96 318 420 202)(97 211 429 327)(98 328 430 212)(99 213 431 321)(100 322 432 214)(101 215 425 323)(102 324 426 216)(103 209 427 325)(104 326 428 210)(105 219 437 335)(106 336 438 220)(107 221 439 329)(108 330 440 222)(109 223 433 331)(110 332 434 224)(111 217 435 333)(112 334 436 218)(113 227 445 343)(114 344 446 228)(115 229 447 337)(116 338 448 230)(117 231 441 339)(118 340 442 232)(119 225 443 341)(120 342 444 226)
(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 67 15 395)(2 296 16 184)(3 65 9 393)(4 294 10 182)(5 71 11 399)(6 292 12 180)(7 69 13 397)(8 290 14 178)(17 401 345 73)(18 190 346 302)(19 407 347 79)(20 188 348 300)(21 405 349 77)(22 186 350 298)(23 403 351 75)(24 192 352 304)(25 409 353 81)(26 198 354 310)(27 415 355 87)(28 196 356 308)(29 413 357 85)(30 194 358 306)(31 411 359 83)(32 200 360 312)(33 417 361 89)(34 206 362 318)(35 423 363 95)(36 204 364 316)(37 421 365 93)(38 202 366 314)(39 419 367 91)(40 208 368 320)(41 425 369 97)(42 214 370 326)(43 431 371 103)(44 212 372 324)(45 429 373 101)(46 210 374 322)(47 427 375 99)(48 216 376 328)(49 433 377 105)(50 222 378 334)(51 439 379 111)(52 220 380 332)(53 437 381 109)(54 218 382 330)(55 435 383 107)(56 224 384 336)(57 441 385 113)(58 230 386 342)(59 447 387 119)(60 228 388 340)(61 445 389 117)(62 226 390 338)(63 443 391 115)(64 232 392 344)(66 126 394 238)(68 124 396 236)(70 122 398 234)(72 128 400 240)(74 134 402 246)(76 132 404 244)(78 130 406 242)(80 136 408 248)(82 142 410 254)(84 140 412 252)(86 138 414 250)(88 144 416 256)(90 150 418 262)(92 148 420 260)(94 146 422 258)(96 152 424 264)(98 158 426 270)(100 156 428 268)(102 154 430 266)(104 160 432 272)(106 166 434 278)(108 164 436 276)(110 162 438 274)(112 168 440 280)(114 174 442 286)(116 172 444 284)(118 170 446 282)(120 176 448 288)(121 289 233 177)(123 295 235 183)(125 293 237 181)(127 291 239 179)(129 297 241 185)(131 303 243 191)(133 301 245 189)(135 299 247 187)(137 305 249 193)(139 311 251 199)(141 309 253 197)(143 307 255 195)(145 313 257 201)(147 319 259 207)(149 317 261 205)(151 315 263 203)(153 321 265 209)(155 327 267 215)(157 325 269 213)(159 323 271 211)(161 329 273 217)(163 335 275 223)(165 333 277 221)(167 331 279 219)(169 337 281 225)(171 343 283 231)(173 341 285 229)(175 339 287 227)
G:=sub<Sym(448)| (1,391,383,375,367,359,351)(2,392,384,376,368,360,352)(3,385,377,369,361,353,345)(4,386,378,370,362,354,346)(5,387,379,371,363,355,347)(6,388,380,372,364,356,348)(7,389,381,373,365,357,349)(8,390,382,374,366,358,350)(9,57,49,41,33,25,17)(10,58,50,42,34,26,18)(11,59,51,43,35,27,19)(12,60,52,44,36,28,20)(13,61,53,45,37,29,21)(14,62,54,46,38,30,22)(15,63,55,47,39,31,23)(16,64,56,48,40,32,24)(65,113,105,97,89,81,73)(66,114,106,98,90,82,74)(67,115,107,99,91,83,75)(68,116,108,100,92,84,76)(69,117,109,101,93,85,77)(70,118,110,102,94,86,78)(71,119,111,103,95,87,79)(72,120,112,104,96,88,80)(121,169,161,153,145,137,129)(122,170,162,154,146,138,130)(123,171,163,155,147,139,131)(124,172,164,156,148,140,132)(125,173,165,157,149,141,133)(126,174,166,158,150,142,134)(127,175,167,159,151,143,135)(128,176,168,160,152,144,136)(177,225,217,209,201,193,185)(178,226,218,210,202,194,186)(179,227,219,211,203,195,187)(180,228,220,212,204,196,188)(181,229,221,213,205,197,189)(182,230,222,214,206,198,190)(183,231,223,215,207,199,191)(184,232,224,216,208,200,192)(233,281,273,265,257,249,241)(234,282,274,266,258,250,242)(235,283,275,267,259,251,243)(236,284,276,268,260,252,244)(237,285,277,269,261,253,245)(238,286,278,270,262,254,246)(239,287,279,271,263,255,247)(240,288,280,272,264,256,248)(289,337,329,321,313,305,297)(290,338,330,322,314,306,298)(291,339,331,323,315,307,299)(292,340,332,324,316,308,300)(293,341,333,325,317,309,301)(294,342,334,326,318,310,302)(295,343,335,327,319,311,303)(296,344,336,328,320,312,304)(393,441,433,425,417,409,401)(394,442,434,426,418,410,402)(395,443,435,427,419,411,403)(396,444,436,428,420,412,404)(397,445,437,429,421,413,405)(398,446,438,430,422,414,406)(399,447,439,431,423,415,407)(400,448,440,432,424,416,408), (1,121,11,237)(2,238,12,122)(3,123,13,239)(4,240,14,124)(5,125,15,233)(6,234,16,126)(7,127,9,235)(8,236,10,128)(17,243,349,135)(18,136,350,244)(19,245,351,129)(20,130,352,246)(21,247,345,131)(22,132,346,248)(23,241,347,133)(24,134,348,242)(25,251,357,143)(26,144,358,252)(27,253,359,137)(28,138,360,254)(29,255,353,139)(30,140,354,256)(31,249,355,141)(32,142,356,250)(33,259,365,151)(34,152,366,260)(35,261,367,145)(36,146,368,262)(37,263,361,147)(38,148,362,264)(39,257,363,149)(40,150,364,258)(41,267,373,159)(42,160,374,268)(43,269,375,153)(44,154,376,270)(45,271,369,155)(46,156,370,272)(47,265,371,157)(48,158,372,266)(49,275,381,167)(50,168,382,276)(51,277,383,161)(52,162,384,278)(53,279,377,163)(54,164,378,280)(55,273,379,165)(56,166,380,274)(57,283,389,175)(58,176,390,284)(59,285,391,169)(60,170,392,286)(61,287,385,171)(62,172,386,288)(63,281,387,173)(64,174,388,282)(65,179,397,295)(66,296,398,180)(67,181,399,289)(68,290,400,182)(69,183,393,291)(70,292,394,184)(71,177,395,293)(72,294,396,178)(73,187,405,303)(74,304,406,188)(75,189,407,297)(76,298,408,190)(77,191,401,299)(78,300,402,192)(79,185,403,301)(80,302,404,186)(81,195,413,311)(82,312,414,196)(83,197,415,305)(84,306,416,198)(85,199,409,307)(86,308,410,200)(87,193,411,309)(88,310,412,194)(89,203,421,319)(90,320,422,204)(91,205,423,313)(92,314,424,206)(93,207,417,315)(94,316,418,208)(95,201,419,317)(96,318,420,202)(97,211,429,327)(98,328,430,212)(99,213,431,321)(100,322,432,214)(101,215,425,323)(102,324,426,216)(103,209,427,325)(104,326,428,210)(105,219,437,335)(106,336,438,220)(107,221,439,329)(108,330,440,222)(109,223,433,331)(110,332,434,224)(111,217,435,333)(112,334,436,218)(113,227,445,343)(114,344,446,228)(115,229,447,337)(116,338,448,230)(117,231,441,339)(118,340,442,232)(119,225,443,341)(120,342,444,226), (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), (1,67,15,395)(2,296,16,184)(3,65,9,393)(4,294,10,182)(5,71,11,399)(6,292,12,180)(7,69,13,397)(8,290,14,178)(17,401,345,73)(18,190,346,302)(19,407,347,79)(20,188,348,300)(21,405,349,77)(22,186,350,298)(23,403,351,75)(24,192,352,304)(25,409,353,81)(26,198,354,310)(27,415,355,87)(28,196,356,308)(29,413,357,85)(30,194,358,306)(31,411,359,83)(32,200,360,312)(33,417,361,89)(34,206,362,318)(35,423,363,95)(36,204,364,316)(37,421,365,93)(38,202,366,314)(39,419,367,91)(40,208,368,320)(41,425,369,97)(42,214,370,326)(43,431,371,103)(44,212,372,324)(45,429,373,101)(46,210,374,322)(47,427,375,99)(48,216,376,328)(49,433,377,105)(50,222,378,334)(51,439,379,111)(52,220,380,332)(53,437,381,109)(54,218,382,330)(55,435,383,107)(56,224,384,336)(57,441,385,113)(58,230,386,342)(59,447,387,119)(60,228,388,340)(61,445,389,117)(62,226,390,338)(63,443,391,115)(64,232,392,344)(66,126,394,238)(68,124,396,236)(70,122,398,234)(72,128,400,240)(74,134,402,246)(76,132,404,244)(78,130,406,242)(80,136,408,248)(82,142,410,254)(84,140,412,252)(86,138,414,250)(88,144,416,256)(90,150,418,262)(92,148,420,260)(94,146,422,258)(96,152,424,264)(98,158,426,270)(100,156,428,268)(102,154,430,266)(104,160,432,272)(106,166,434,278)(108,164,436,276)(110,162,438,274)(112,168,440,280)(114,174,442,286)(116,172,444,284)(118,170,446,282)(120,176,448,288)(121,289,233,177)(123,295,235,183)(125,293,237,181)(127,291,239,179)(129,297,241,185)(131,303,243,191)(133,301,245,189)(135,299,247,187)(137,305,249,193)(139,311,251,199)(141,309,253,197)(143,307,255,195)(145,313,257,201)(147,319,259,207)(149,317,261,205)(151,315,263,203)(153,321,265,209)(155,327,267,215)(157,325,269,213)(159,323,271,211)(161,329,273,217)(163,335,275,223)(165,333,277,221)(167,331,279,219)(169,337,281,225)(171,343,283,231)(173,341,285,229)(175,339,287,227)>;
G:=Group( (1,391,383,375,367,359,351)(2,392,384,376,368,360,352)(3,385,377,369,361,353,345)(4,386,378,370,362,354,346)(5,387,379,371,363,355,347)(6,388,380,372,364,356,348)(7,389,381,373,365,357,349)(8,390,382,374,366,358,350)(9,57,49,41,33,25,17)(10,58,50,42,34,26,18)(11,59,51,43,35,27,19)(12,60,52,44,36,28,20)(13,61,53,45,37,29,21)(14,62,54,46,38,30,22)(15,63,55,47,39,31,23)(16,64,56,48,40,32,24)(65,113,105,97,89,81,73)(66,114,106,98,90,82,74)(67,115,107,99,91,83,75)(68,116,108,100,92,84,76)(69,117,109,101,93,85,77)(70,118,110,102,94,86,78)(71,119,111,103,95,87,79)(72,120,112,104,96,88,80)(121,169,161,153,145,137,129)(122,170,162,154,146,138,130)(123,171,163,155,147,139,131)(124,172,164,156,148,140,132)(125,173,165,157,149,141,133)(126,174,166,158,150,142,134)(127,175,167,159,151,143,135)(128,176,168,160,152,144,136)(177,225,217,209,201,193,185)(178,226,218,210,202,194,186)(179,227,219,211,203,195,187)(180,228,220,212,204,196,188)(181,229,221,213,205,197,189)(182,230,222,214,206,198,190)(183,231,223,215,207,199,191)(184,232,224,216,208,200,192)(233,281,273,265,257,249,241)(234,282,274,266,258,250,242)(235,283,275,267,259,251,243)(236,284,276,268,260,252,244)(237,285,277,269,261,253,245)(238,286,278,270,262,254,246)(239,287,279,271,263,255,247)(240,288,280,272,264,256,248)(289,337,329,321,313,305,297)(290,338,330,322,314,306,298)(291,339,331,323,315,307,299)(292,340,332,324,316,308,300)(293,341,333,325,317,309,301)(294,342,334,326,318,310,302)(295,343,335,327,319,311,303)(296,344,336,328,320,312,304)(393,441,433,425,417,409,401)(394,442,434,426,418,410,402)(395,443,435,427,419,411,403)(396,444,436,428,420,412,404)(397,445,437,429,421,413,405)(398,446,438,430,422,414,406)(399,447,439,431,423,415,407)(400,448,440,432,424,416,408), (1,121,11,237)(2,238,12,122)(3,123,13,239)(4,240,14,124)(5,125,15,233)(6,234,16,126)(7,127,9,235)(8,236,10,128)(17,243,349,135)(18,136,350,244)(19,245,351,129)(20,130,352,246)(21,247,345,131)(22,132,346,248)(23,241,347,133)(24,134,348,242)(25,251,357,143)(26,144,358,252)(27,253,359,137)(28,138,360,254)(29,255,353,139)(30,140,354,256)(31,249,355,141)(32,142,356,250)(33,259,365,151)(34,152,366,260)(35,261,367,145)(36,146,368,262)(37,263,361,147)(38,148,362,264)(39,257,363,149)(40,150,364,258)(41,267,373,159)(42,160,374,268)(43,269,375,153)(44,154,376,270)(45,271,369,155)(46,156,370,272)(47,265,371,157)(48,158,372,266)(49,275,381,167)(50,168,382,276)(51,277,383,161)(52,162,384,278)(53,279,377,163)(54,164,378,280)(55,273,379,165)(56,166,380,274)(57,283,389,175)(58,176,390,284)(59,285,391,169)(60,170,392,286)(61,287,385,171)(62,172,386,288)(63,281,387,173)(64,174,388,282)(65,179,397,295)(66,296,398,180)(67,181,399,289)(68,290,400,182)(69,183,393,291)(70,292,394,184)(71,177,395,293)(72,294,396,178)(73,187,405,303)(74,304,406,188)(75,189,407,297)(76,298,408,190)(77,191,401,299)(78,300,402,192)(79,185,403,301)(80,302,404,186)(81,195,413,311)(82,312,414,196)(83,197,415,305)(84,306,416,198)(85,199,409,307)(86,308,410,200)(87,193,411,309)(88,310,412,194)(89,203,421,319)(90,320,422,204)(91,205,423,313)(92,314,424,206)(93,207,417,315)(94,316,418,208)(95,201,419,317)(96,318,420,202)(97,211,429,327)(98,328,430,212)(99,213,431,321)(100,322,432,214)(101,215,425,323)(102,324,426,216)(103,209,427,325)(104,326,428,210)(105,219,437,335)(106,336,438,220)(107,221,439,329)(108,330,440,222)(109,223,433,331)(110,332,434,224)(111,217,435,333)(112,334,436,218)(113,227,445,343)(114,344,446,228)(115,229,447,337)(116,338,448,230)(117,231,441,339)(118,340,442,232)(119,225,443,341)(120,342,444,226), (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), (1,67,15,395)(2,296,16,184)(3,65,9,393)(4,294,10,182)(5,71,11,399)(6,292,12,180)(7,69,13,397)(8,290,14,178)(17,401,345,73)(18,190,346,302)(19,407,347,79)(20,188,348,300)(21,405,349,77)(22,186,350,298)(23,403,351,75)(24,192,352,304)(25,409,353,81)(26,198,354,310)(27,415,355,87)(28,196,356,308)(29,413,357,85)(30,194,358,306)(31,411,359,83)(32,200,360,312)(33,417,361,89)(34,206,362,318)(35,423,363,95)(36,204,364,316)(37,421,365,93)(38,202,366,314)(39,419,367,91)(40,208,368,320)(41,425,369,97)(42,214,370,326)(43,431,371,103)(44,212,372,324)(45,429,373,101)(46,210,374,322)(47,427,375,99)(48,216,376,328)(49,433,377,105)(50,222,378,334)(51,439,379,111)(52,220,380,332)(53,437,381,109)(54,218,382,330)(55,435,383,107)(56,224,384,336)(57,441,385,113)(58,230,386,342)(59,447,387,119)(60,228,388,340)(61,445,389,117)(62,226,390,338)(63,443,391,115)(64,232,392,344)(66,126,394,238)(68,124,396,236)(70,122,398,234)(72,128,400,240)(74,134,402,246)(76,132,404,244)(78,130,406,242)(80,136,408,248)(82,142,410,254)(84,140,412,252)(86,138,414,250)(88,144,416,256)(90,150,418,262)(92,148,420,260)(94,146,422,258)(96,152,424,264)(98,158,426,270)(100,156,428,268)(102,154,430,266)(104,160,432,272)(106,166,434,278)(108,164,436,276)(110,162,438,274)(112,168,440,280)(114,174,442,286)(116,172,444,284)(118,170,446,282)(120,176,448,288)(121,289,233,177)(123,295,235,183)(125,293,237,181)(127,291,239,179)(129,297,241,185)(131,303,243,191)(133,301,245,189)(135,299,247,187)(137,305,249,193)(139,311,251,199)(141,309,253,197)(143,307,255,195)(145,313,257,201)(147,319,259,207)(149,317,261,205)(151,315,263,203)(153,321,265,209)(155,327,267,215)(157,325,269,213)(159,323,271,211)(161,329,273,217)(163,335,275,223)(165,333,277,221)(167,331,279,219)(169,337,281,225)(171,343,283,231)(173,341,285,229)(175,339,287,227) );
G=PermutationGroup([[(1,391,383,375,367,359,351),(2,392,384,376,368,360,352),(3,385,377,369,361,353,345),(4,386,378,370,362,354,346),(5,387,379,371,363,355,347),(6,388,380,372,364,356,348),(7,389,381,373,365,357,349),(8,390,382,374,366,358,350),(9,57,49,41,33,25,17),(10,58,50,42,34,26,18),(11,59,51,43,35,27,19),(12,60,52,44,36,28,20),(13,61,53,45,37,29,21),(14,62,54,46,38,30,22),(15,63,55,47,39,31,23),(16,64,56,48,40,32,24),(65,113,105,97,89,81,73),(66,114,106,98,90,82,74),(67,115,107,99,91,83,75),(68,116,108,100,92,84,76),(69,117,109,101,93,85,77),(70,118,110,102,94,86,78),(71,119,111,103,95,87,79),(72,120,112,104,96,88,80),(121,169,161,153,145,137,129),(122,170,162,154,146,138,130),(123,171,163,155,147,139,131),(124,172,164,156,148,140,132),(125,173,165,157,149,141,133),(126,174,166,158,150,142,134),(127,175,167,159,151,143,135),(128,176,168,160,152,144,136),(177,225,217,209,201,193,185),(178,226,218,210,202,194,186),(179,227,219,211,203,195,187),(180,228,220,212,204,196,188),(181,229,221,213,205,197,189),(182,230,222,214,206,198,190),(183,231,223,215,207,199,191),(184,232,224,216,208,200,192),(233,281,273,265,257,249,241),(234,282,274,266,258,250,242),(235,283,275,267,259,251,243),(236,284,276,268,260,252,244),(237,285,277,269,261,253,245),(238,286,278,270,262,254,246),(239,287,279,271,263,255,247),(240,288,280,272,264,256,248),(289,337,329,321,313,305,297),(290,338,330,322,314,306,298),(291,339,331,323,315,307,299),(292,340,332,324,316,308,300),(293,341,333,325,317,309,301),(294,342,334,326,318,310,302),(295,343,335,327,319,311,303),(296,344,336,328,320,312,304),(393,441,433,425,417,409,401),(394,442,434,426,418,410,402),(395,443,435,427,419,411,403),(396,444,436,428,420,412,404),(397,445,437,429,421,413,405),(398,446,438,430,422,414,406),(399,447,439,431,423,415,407),(400,448,440,432,424,416,408)], [(1,121,11,237),(2,238,12,122),(3,123,13,239),(4,240,14,124),(5,125,15,233),(6,234,16,126),(7,127,9,235),(8,236,10,128),(17,243,349,135),(18,136,350,244),(19,245,351,129),(20,130,352,246),(21,247,345,131),(22,132,346,248),(23,241,347,133),(24,134,348,242),(25,251,357,143),(26,144,358,252),(27,253,359,137),(28,138,360,254),(29,255,353,139),(30,140,354,256),(31,249,355,141),(32,142,356,250),(33,259,365,151),(34,152,366,260),(35,261,367,145),(36,146,368,262),(37,263,361,147),(38,148,362,264),(39,257,363,149),(40,150,364,258),(41,267,373,159),(42,160,374,268),(43,269,375,153),(44,154,376,270),(45,271,369,155),(46,156,370,272),(47,265,371,157),(48,158,372,266),(49,275,381,167),(50,168,382,276),(51,277,383,161),(52,162,384,278),(53,279,377,163),(54,164,378,280),(55,273,379,165),(56,166,380,274),(57,283,389,175),(58,176,390,284),(59,285,391,169),(60,170,392,286),(61,287,385,171),(62,172,386,288),(63,281,387,173),(64,174,388,282),(65,179,397,295),(66,296,398,180),(67,181,399,289),(68,290,400,182),(69,183,393,291),(70,292,394,184),(71,177,395,293),(72,294,396,178),(73,187,405,303),(74,304,406,188),(75,189,407,297),(76,298,408,190),(77,191,401,299),(78,300,402,192),(79,185,403,301),(80,302,404,186),(81,195,413,311),(82,312,414,196),(83,197,415,305),(84,306,416,198),(85,199,409,307),(86,308,410,200),(87,193,411,309),(88,310,412,194),(89,203,421,319),(90,320,422,204),(91,205,423,313),(92,314,424,206),(93,207,417,315),(94,316,418,208),(95,201,419,317),(96,318,420,202),(97,211,429,327),(98,328,430,212),(99,213,431,321),(100,322,432,214),(101,215,425,323),(102,324,426,216),(103,209,427,325),(104,326,428,210),(105,219,437,335),(106,336,438,220),(107,221,439,329),(108,330,440,222),(109,223,433,331),(110,332,434,224),(111,217,435,333),(112,334,436,218),(113,227,445,343),(114,344,446,228),(115,229,447,337),(116,338,448,230),(117,231,441,339),(118,340,442,232),(119,225,443,341),(120,342,444,226)], [(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)], [(1,67,15,395),(2,296,16,184),(3,65,9,393),(4,294,10,182),(5,71,11,399),(6,292,12,180),(7,69,13,397),(8,290,14,178),(17,401,345,73),(18,190,346,302),(19,407,347,79),(20,188,348,300),(21,405,349,77),(22,186,350,298),(23,403,351,75),(24,192,352,304),(25,409,353,81),(26,198,354,310),(27,415,355,87),(28,196,356,308),(29,413,357,85),(30,194,358,306),(31,411,359,83),(32,200,360,312),(33,417,361,89),(34,206,362,318),(35,423,363,95),(36,204,364,316),(37,421,365,93),(38,202,366,314),(39,419,367,91),(40,208,368,320),(41,425,369,97),(42,214,370,326),(43,431,371,103),(44,212,372,324),(45,429,373,101),(46,210,374,322),(47,427,375,99),(48,216,376,328),(49,433,377,105),(50,222,378,334),(51,439,379,111),(52,220,380,332),(53,437,381,109),(54,218,382,330),(55,435,383,107),(56,224,384,336),(57,441,385,113),(58,230,386,342),(59,447,387,119),(60,228,388,340),(61,445,389,117),(62,226,390,338),(63,443,391,115),(64,232,392,344),(66,126,394,238),(68,124,396,236),(70,122,398,234),(72,128,400,240),(74,134,402,246),(76,132,404,244),(78,130,406,242),(80,136,408,248),(82,142,410,254),(84,140,412,252),(86,138,414,250),(88,144,416,256),(90,150,418,262),(92,148,420,260),(94,146,422,258),(96,152,424,264),(98,158,426,270),(100,156,428,268),(102,154,430,266),(104,160,432,272),(106,166,434,278),(108,164,436,276),(110,162,438,274),(112,168,440,280),(114,174,442,286),(116,172,444,284),(118,170,446,282),(120,176,448,288),(121,289,233,177),(123,295,235,183),(125,293,237,181),(127,291,239,179),(129,297,241,185),(131,303,243,191),(133,301,245,189),(135,299,247,187),(137,305,249,193),(139,311,251,199),(141,309,253,197),(143,307,255,195),(145,313,257,201),(147,319,259,207),(149,317,261,205),(151,315,263,203),(153,321,265,209),(155,327,267,215),(157,325,269,213),(159,323,271,211),(161,329,273,217),(163,335,275,223),(165,333,277,221),(167,331,279,219),(169,337,281,225),(171,343,283,231),(173,341,285,229),(175,339,287,227)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 28A | ··· | 28X | 28Y | ··· | 28AD | 28AE | ··· | 28AP | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D4 | SD16 | Q16 | C7×D4 | C7×SD16 | C7×Q16 | C4.D4 | C7×C4.D4 |
| kernel | C7×C4.6Q16 | C7×C4⋊C8 | C7×C4⋊Q8 | Q8×C14 | C4.6Q16 | C4⋊C8 | C4⋊Q8 | C2×Q8 | C2×C28 | C28 | C28 | C2×C4 | C4 | C4 | C14 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 2 | 4 | 4 | 12 | 24 | 24 | 1 | 6 |
Matrix representation of C7×C4.6Q16 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 111 | 0 | 0 |
| 1 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 50 | 3 | 0 | 0 |
| 108 | 63 | 0 | 0 |
| 0 | 0 | 0 | 82 |
| 0 | 0 | 62 | 62 |
| 84 | 49 | 0 | 0 |
| 52 | 29 | 0 | 0 |
| 0 | 0 | 47 | 20 |
| 0 | 0 | 59 | 66 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[1,1,0,0,111,112,0,0,0,0,1,0,0,0,0,1],[50,108,0,0,3,63,0,0,0,0,0,62,0,0,82,62],[84,52,0,0,49,29,0,0,0,0,47,59,0,0,20,66] >;
C7×C4.6Q16 in GAP, Magma, Sage, TeX
C_7\times C_4._6Q_{16} % in TeX
G:=Group("C7xC4.6Q16"); // GroupNames label
G:=SmallGroup(448,137);
// by ID
G=gap.SmallGroup(448,137);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,1576,3923,3538,248,6871,242]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^8=1,d^2=b^2*c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^-1>;
// generators/relations