metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊Dic7⋊8C4, C14.3(C4×Q8), C14.45(C4×D4), (C2×C28).34Q8, C2.6(C4×Dic14), C22.55(D4×D7), (C2×C4).21Dic14, (C22×C4).10D14, C14.3(C22⋊Q8), C2.4(D28⋊C4), (C2×Dic7).128D4, C2.1(C28.3Q8), C14.9(C42.C2), C2.2(D14.D4), C22.31(C4○D28), (C22×C28).43C22, C22.14(C2×Dic14), C2.C42.10D7, C23.251(C22×D7), C14.19(C42⋊C2), C14.17(C42⋊2C2), C22.33(D4⋊2D7), C14.C42.23C2, (C22×C14).279C23, C7⋊3(C23.63C23), C2.3(C22⋊Dic14), C22.14(Q8⋊2D7), C14.3(C22.D4), (C22×Dic7).5C22, C2.8(C23.11D14), (C2×C4).25(C4×D7), C22.86(C2×C4×D7), (C2×C28).33(C2×C4), (C2×C4⋊Dic7).4C2, (C2×C14).19(C2×Q8), (C2×C4×Dic7).27C2, C2.2(C4⋊C4⋊D7), (C2×C14).192(C2×D4), (C2×Dic7⋊C4).5C2, (C2×C14).45(C22×C4), (C2×Dic7).43(C2×C4), (C2×C14).180(C4○D4), (C7×C2.C42).17C2, SmallGroup(448,188)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊Dic7⋊8C4
G = < a,b,c,d | a4=b14=d4=1, c2=b7, ab=ba, cac-1=a-1, dad-1=a-1b7, cbc-1=b-1, bd=db, cd=dc >
Subgroups: 604 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.63C23, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C22×Dic7, C22×C28, C14.C42, C7×C2.C42, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C4⋊Dic7⋊8C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C22×C4, C2×D4, C2×Q8, C4○D4, D14, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, Dic14, C4×D7, C22×D7, C23.63C23, C2×Dic14, C2×C4×D7, C4○D28, D4×D7, D4⋊2D7, Q8⋊2D7, C4×Dic14, C23.11D14, C22⋊Dic14, D14.D4, C28.3Q8, D28⋊C4, C4⋊C4⋊D7, C4⋊Dic7⋊8C4
►Regular action on 448 points
Generators in S448
(1 178 427 188)(2 179 428 189)(3 180 429 190)(4 181 430 191)(5 182 431 192)(6 169 432 193)(7 170 433 194)(8 171 434 195)(9 172 421 196)(10 173 422 183)(11 174 423 184)(12 175 424 185)(13 176 425 186)(14 177 426 187)(15 440 122 41)(16 441 123 42)(17 442 124 29)(18 443 125 30)(19 444 126 31)(20 445 113 32)(21 446 114 33)(22 447 115 34)(23 448 116 35)(24 435 117 36)(25 436 118 37)(26 437 119 38)(27 438 120 39)(28 439 121 40)(43 417 233 296)(44 418 234 297)(45 419 235 298)(46 420 236 299)(47 407 237 300)(48 408 238 301)(49 409 225 302)(50 410 226 303)(51 411 227 304)(52 412 228 305)(53 413 229 306)(54 414 230 307)(55 415 231 308)(56 416 232 295)(57 203 282 275)(58 204 283 276)(59 205 284 277)(60 206 285 278)(61 207 286 279)(62 208 287 280)(63 209 288 267)(64 210 289 268)(65 197 290 269)(66 198 291 270)(67 199 292 271)(68 200 293 272)(69 201 294 273)(70 202 281 274)(71 399 243 222)(72 400 244 223)(73 401 245 224)(74 402 246 211)(75 403 247 212)(76 404 248 213)(77 405 249 214)(78 406 250 215)(79 393 251 216)(80 394 252 217)(81 395 239 218)(82 396 240 219)(83 397 241 220)(84 398 242 221)(85 107 356 153)(86 108 357 154)(87 109 358 141)(88 110 359 142)(89 111 360 143)(90 112 361 144)(91 99 362 145)(92 100 363 146)(93 101 364 147)(94 102 351 148)(95 103 352 149)(96 104 353 150)(97 105 354 151)(98 106 355 152)(127 347 370 253)(128 348 371 254)(129 349 372 255)(130 350 373 256)(131 337 374 257)(132 338 375 258)(133 339 376 259)(134 340 377 260)(135 341 378 261)(136 342 365 262)(137 343 366 263)(138 344 367 264)(139 345 368 265)(140 346 369 266)(155 330 383 313)(156 331 384 314)(157 332 385 315)(158 333 386 316)(159 334 387 317)(160 335 388 318)(161 336 389 319)(162 323 390 320)(163 324 391 321)(164 325 392 322)(165 326 379 309)(166 327 380 310)(167 328 381 311)(168 329 382 312)
(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)
(1 370 8 377)(2 369 9 376)(3 368 10 375)(4 367 11 374)(5 366 12 373)(6 365 13 372)(7 378 14 371)(15 245 22 252)(16 244 23 251)(17 243 24 250)(18 242 25 249)(19 241 26 248)(20 240 27 247)(21 239 28 246)(29 222 36 215)(30 221 37 214)(31 220 38 213)(32 219 39 212)(33 218 40 211)(34 217 41 224)(35 216 42 223)(43 315 50 322)(44 314 51 321)(45 313 52 320)(46 312 53 319)(47 311 54 318)(48 310 55 317)(49 309 56 316)(57 360 64 353)(58 359 65 352)(59 358 66 351)(60 357 67 364)(61 356 68 363)(62 355 69 362)(63 354 70 361)(71 117 78 124)(72 116 79 123)(73 115 80 122)(74 114 81 121)(75 113 82 120)(76 126 83 119)(77 125 84 118)(85 293 92 286)(86 292 93 285)(87 291 94 284)(88 290 95 283)(89 289 96 282)(90 288 97 281)(91 287 98 294)(99 208 106 201)(100 207 107 200)(101 206 108 199)(102 205 109 198)(103 204 110 197)(104 203 111 210)(105 202 112 209)(127 434 134 427)(128 433 135 426)(129 432 136 425)(130 431 137 424)(131 430 138 423)(132 429 139 422)(133 428 140 421)(141 270 148 277)(142 269 149 276)(143 268 150 275)(144 267 151 274)(145 280 152 273)(146 279 153 272)(147 278 154 271)(155 305 162 298)(156 304 163 297)(157 303 164 296)(158 302 165 295)(159 301 166 308)(160 300 167 307)(161 299 168 306)(169 342 176 349)(170 341 177 348)(171 340 178 347)(172 339 179 346)(173 338 180 345)(174 337 181 344)(175 350 182 343)(183 258 190 265)(184 257 191 264)(185 256 192 263)(186 255 193 262)(187 254 194 261)(188 253 195 260)(189 266 196 259)(225 326 232 333)(226 325 233 332)(227 324 234 331)(228 323 235 330)(229 336 236 329)(230 335 237 328)(231 334 238 327)(379 416 386 409)(380 415 387 408)(381 414 388 407)(382 413 389 420)(383 412 390 419)(384 411 391 418)(385 410 392 417)(393 441 400 448)(394 440 401 447)(395 439 402 446)(396 438 403 445)(397 437 404 444)(398 436 405 443)(399 435 406 442)
(1 352 77 322)(2 353 78 309)(3 354 79 310)(4 355 80 311)(5 356 81 312)(6 357 82 313)(7 358 83 314)(8 359 84 315)(9 360 71 316)(10 361 72 317)(11 362 73 318)(12 363 74 319)(13 364 75 320)(14 351 76 321)(15 230 138 294)(16 231 139 281)(17 232 140 282)(18 233 127 283)(19 234 128 284)(20 235 129 285)(21 236 130 286)(22 237 131 287)(23 238 132 288)(24 225 133 289)(25 226 134 290)(26 227 135 291)(27 228 136 292)(28 229 137 293)(29 302 266 268)(30 303 253 269)(31 304 254 270)(32 305 255 271)(33 306 256 272)(34 307 257 273)(35 308 258 274)(36 295 259 275)(37 296 260 276)(38 297 261 277)(39 298 262 278)(40 299 263 279)(41 300 264 280)(42 301 265 267)(43 370 58 125)(44 371 59 126)(45 372 60 113)(46 373 61 114)(47 374 62 115)(48 375 63 116)(49 376 64 117)(50 377 65 118)(51 378 66 119)(52 365 67 120)(53 366 68 121)(54 367 69 122)(55 368 70 123)(56 369 57 124)(85 239 329 431)(86 240 330 432)(87 241 331 433)(88 242 332 434)(89 243 333 421)(90 244 334 422)(91 245 335 423)(92 246 336 424)(93 247 323 425)(94 248 324 426)(95 249 325 427)(96 250 326 428)(97 251 327 429)(98 252 328 430)(99 394 388 181)(100 395 389 182)(101 396 390 169)(102 397 391 170)(103 398 392 171)(104 399 379 172)(105 400 380 173)(106 401 381 174)(107 402 382 175)(108 403 383 176)(109 404 384 177)(110 405 385 178)(111 406 386 179)(112 393 387 180)(141 213 156 187)(142 214 157 188)(143 215 158 189)(144 216 159 190)(145 217 160 191)(146 218 161 192)(147 219 162 193)(148 220 163 194)(149 221 164 195)(150 222 165 196)(151 223 166 183)(152 224 167 184)(153 211 168 185)(154 212 155 186)(197 443 410 347)(198 444 411 348)(199 445 412 349)(200 446 413 350)(201 447 414 337)(202 448 415 338)(203 435 416 339)(204 436 417 340)(205 437 418 341)(206 438 419 342)(207 439 420 343)(208 440 407 344)(209 441 408 345)(210 442 409 346)
G:=sub<Sym(448)| (1,178,427,188)(2,179,428,189)(3,180,429,190)(4,181,430,191)(5,182,431,192)(6,169,432,193)(7,170,433,194)(8,171,434,195)(9,172,421,196)(10,173,422,183)(11,174,423,184)(12,175,424,185)(13,176,425,186)(14,177,426,187)(15,440,122,41)(16,441,123,42)(17,442,124,29)(18,443,125,30)(19,444,126,31)(20,445,113,32)(21,446,114,33)(22,447,115,34)(23,448,116,35)(24,435,117,36)(25,436,118,37)(26,437,119,38)(27,438,120,39)(28,439,121,40)(43,417,233,296)(44,418,234,297)(45,419,235,298)(46,420,236,299)(47,407,237,300)(48,408,238,301)(49,409,225,302)(50,410,226,303)(51,411,227,304)(52,412,228,305)(53,413,229,306)(54,414,230,307)(55,415,231,308)(56,416,232,295)(57,203,282,275)(58,204,283,276)(59,205,284,277)(60,206,285,278)(61,207,286,279)(62,208,287,280)(63,209,288,267)(64,210,289,268)(65,197,290,269)(66,198,291,270)(67,199,292,271)(68,200,293,272)(69,201,294,273)(70,202,281,274)(71,399,243,222)(72,400,244,223)(73,401,245,224)(74,402,246,211)(75,403,247,212)(76,404,248,213)(77,405,249,214)(78,406,250,215)(79,393,251,216)(80,394,252,217)(81,395,239,218)(82,396,240,219)(83,397,241,220)(84,398,242,221)(85,107,356,153)(86,108,357,154)(87,109,358,141)(88,110,359,142)(89,111,360,143)(90,112,361,144)(91,99,362,145)(92,100,363,146)(93,101,364,147)(94,102,351,148)(95,103,352,149)(96,104,353,150)(97,105,354,151)(98,106,355,152)(127,347,370,253)(128,348,371,254)(129,349,372,255)(130,350,373,256)(131,337,374,257)(132,338,375,258)(133,339,376,259)(134,340,377,260)(135,341,378,261)(136,342,365,262)(137,343,366,263)(138,344,367,264)(139,345,368,265)(140,346,369,266)(155,330,383,313)(156,331,384,314)(157,332,385,315)(158,333,386,316)(159,334,387,317)(160,335,388,318)(161,336,389,319)(162,323,390,320)(163,324,391,321)(164,325,392,322)(165,326,379,309)(166,327,380,310)(167,328,381,311)(168,329,382,312), (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), (1,370,8,377)(2,369,9,376)(3,368,10,375)(4,367,11,374)(5,366,12,373)(6,365,13,372)(7,378,14,371)(15,245,22,252)(16,244,23,251)(17,243,24,250)(18,242,25,249)(19,241,26,248)(20,240,27,247)(21,239,28,246)(29,222,36,215)(30,221,37,214)(31,220,38,213)(32,219,39,212)(33,218,40,211)(34,217,41,224)(35,216,42,223)(43,315,50,322)(44,314,51,321)(45,313,52,320)(46,312,53,319)(47,311,54,318)(48,310,55,317)(49,309,56,316)(57,360,64,353)(58,359,65,352)(59,358,66,351)(60,357,67,364)(61,356,68,363)(62,355,69,362)(63,354,70,361)(71,117,78,124)(72,116,79,123)(73,115,80,122)(74,114,81,121)(75,113,82,120)(76,126,83,119)(77,125,84,118)(85,293,92,286)(86,292,93,285)(87,291,94,284)(88,290,95,283)(89,289,96,282)(90,288,97,281)(91,287,98,294)(99,208,106,201)(100,207,107,200)(101,206,108,199)(102,205,109,198)(103,204,110,197)(104,203,111,210)(105,202,112,209)(127,434,134,427)(128,433,135,426)(129,432,136,425)(130,431,137,424)(131,430,138,423)(132,429,139,422)(133,428,140,421)(141,270,148,277)(142,269,149,276)(143,268,150,275)(144,267,151,274)(145,280,152,273)(146,279,153,272)(147,278,154,271)(155,305,162,298)(156,304,163,297)(157,303,164,296)(158,302,165,295)(159,301,166,308)(160,300,167,307)(161,299,168,306)(169,342,176,349)(170,341,177,348)(171,340,178,347)(172,339,179,346)(173,338,180,345)(174,337,181,344)(175,350,182,343)(183,258,190,265)(184,257,191,264)(185,256,192,263)(186,255,193,262)(187,254,194,261)(188,253,195,260)(189,266,196,259)(225,326,232,333)(226,325,233,332)(227,324,234,331)(228,323,235,330)(229,336,236,329)(230,335,237,328)(231,334,238,327)(379,416,386,409)(380,415,387,408)(381,414,388,407)(382,413,389,420)(383,412,390,419)(384,411,391,418)(385,410,392,417)(393,441,400,448)(394,440,401,447)(395,439,402,446)(396,438,403,445)(397,437,404,444)(398,436,405,443)(399,435,406,442), (1,352,77,322)(2,353,78,309)(3,354,79,310)(4,355,80,311)(5,356,81,312)(6,357,82,313)(7,358,83,314)(8,359,84,315)(9,360,71,316)(10,361,72,317)(11,362,73,318)(12,363,74,319)(13,364,75,320)(14,351,76,321)(15,230,138,294)(16,231,139,281)(17,232,140,282)(18,233,127,283)(19,234,128,284)(20,235,129,285)(21,236,130,286)(22,237,131,287)(23,238,132,288)(24,225,133,289)(25,226,134,290)(26,227,135,291)(27,228,136,292)(28,229,137,293)(29,302,266,268)(30,303,253,269)(31,304,254,270)(32,305,255,271)(33,306,256,272)(34,307,257,273)(35,308,258,274)(36,295,259,275)(37,296,260,276)(38,297,261,277)(39,298,262,278)(40,299,263,279)(41,300,264,280)(42,301,265,267)(43,370,58,125)(44,371,59,126)(45,372,60,113)(46,373,61,114)(47,374,62,115)(48,375,63,116)(49,376,64,117)(50,377,65,118)(51,378,66,119)(52,365,67,120)(53,366,68,121)(54,367,69,122)(55,368,70,123)(56,369,57,124)(85,239,329,431)(86,240,330,432)(87,241,331,433)(88,242,332,434)(89,243,333,421)(90,244,334,422)(91,245,335,423)(92,246,336,424)(93,247,323,425)(94,248,324,426)(95,249,325,427)(96,250,326,428)(97,251,327,429)(98,252,328,430)(99,394,388,181)(100,395,389,182)(101,396,390,169)(102,397,391,170)(103,398,392,171)(104,399,379,172)(105,400,380,173)(106,401,381,174)(107,402,382,175)(108,403,383,176)(109,404,384,177)(110,405,385,178)(111,406,386,179)(112,393,387,180)(141,213,156,187)(142,214,157,188)(143,215,158,189)(144,216,159,190)(145,217,160,191)(146,218,161,192)(147,219,162,193)(148,220,163,194)(149,221,164,195)(150,222,165,196)(151,223,166,183)(152,224,167,184)(153,211,168,185)(154,212,155,186)(197,443,410,347)(198,444,411,348)(199,445,412,349)(200,446,413,350)(201,447,414,337)(202,448,415,338)(203,435,416,339)(204,436,417,340)(205,437,418,341)(206,438,419,342)(207,439,420,343)(208,440,407,344)(209,441,408,345)(210,442,409,346)>;
G:=Group( (1,178,427,188)(2,179,428,189)(3,180,429,190)(4,181,430,191)(5,182,431,192)(6,169,432,193)(7,170,433,194)(8,171,434,195)(9,172,421,196)(10,173,422,183)(11,174,423,184)(12,175,424,185)(13,176,425,186)(14,177,426,187)(15,440,122,41)(16,441,123,42)(17,442,124,29)(18,443,125,30)(19,444,126,31)(20,445,113,32)(21,446,114,33)(22,447,115,34)(23,448,116,35)(24,435,117,36)(25,436,118,37)(26,437,119,38)(27,438,120,39)(28,439,121,40)(43,417,233,296)(44,418,234,297)(45,419,235,298)(46,420,236,299)(47,407,237,300)(48,408,238,301)(49,409,225,302)(50,410,226,303)(51,411,227,304)(52,412,228,305)(53,413,229,306)(54,414,230,307)(55,415,231,308)(56,416,232,295)(57,203,282,275)(58,204,283,276)(59,205,284,277)(60,206,285,278)(61,207,286,279)(62,208,287,280)(63,209,288,267)(64,210,289,268)(65,197,290,269)(66,198,291,270)(67,199,292,271)(68,200,293,272)(69,201,294,273)(70,202,281,274)(71,399,243,222)(72,400,244,223)(73,401,245,224)(74,402,246,211)(75,403,247,212)(76,404,248,213)(77,405,249,214)(78,406,250,215)(79,393,251,216)(80,394,252,217)(81,395,239,218)(82,396,240,219)(83,397,241,220)(84,398,242,221)(85,107,356,153)(86,108,357,154)(87,109,358,141)(88,110,359,142)(89,111,360,143)(90,112,361,144)(91,99,362,145)(92,100,363,146)(93,101,364,147)(94,102,351,148)(95,103,352,149)(96,104,353,150)(97,105,354,151)(98,106,355,152)(127,347,370,253)(128,348,371,254)(129,349,372,255)(130,350,373,256)(131,337,374,257)(132,338,375,258)(133,339,376,259)(134,340,377,260)(135,341,378,261)(136,342,365,262)(137,343,366,263)(138,344,367,264)(139,345,368,265)(140,346,369,266)(155,330,383,313)(156,331,384,314)(157,332,385,315)(158,333,386,316)(159,334,387,317)(160,335,388,318)(161,336,389,319)(162,323,390,320)(163,324,391,321)(164,325,392,322)(165,326,379,309)(166,327,380,310)(167,328,381,311)(168,329,382,312), (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), (1,370,8,377)(2,369,9,376)(3,368,10,375)(4,367,11,374)(5,366,12,373)(6,365,13,372)(7,378,14,371)(15,245,22,252)(16,244,23,251)(17,243,24,250)(18,242,25,249)(19,241,26,248)(20,240,27,247)(21,239,28,246)(29,222,36,215)(30,221,37,214)(31,220,38,213)(32,219,39,212)(33,218,40,211)(34,217,41,224)(35,216,42,223)(43,315,50,322)(44,314,51,321)(45,313,52,320)(46,312,53,319)(47,311,54,318)(48,310,55,317)(49,309,56,316)(57,360,64,353)(58,359,65,352)(59,358,66,351)(60,357,67,364)(61,356,68,363)(62,355,69,362)(63,354,70,361)(71,117,78,124)(72,116,79,123)(73,115,80,122)(74,114,81,121)(75,113,82,120)(76,126,83,119)(77,125,84,118)(85,293,92,286)(86,292,93,285)(87,291,94,284)(88,290,95,283)(89,289,96,282)(90,288,97,281)(91,287,98,294)(99,208,106,201)(100,207,107,200)(101,206,108,199)(102,205,109,198)(103,204,110,197)(104,203,111,210)(105,202,112,209)(127,434,134,427)(128,433,135,426)(129,432,136,425)(130,431,137,424)(131,430,138,423)(132,429,139,422)(133,428,140,421)(141,270,148,277)(142,269,149,276)(143,268,150,275)(144,267,151,274)(145,280,152,273)(146,279,153,272)(147,278,154,271)(155,305,162,298)(156,304,163,297)(157,303,164,296)(158,302,165,295)(159,301,166,308)(160,300,167,307)(161,299,168,306)(169,342,176,349)(170,341,177,348)(171,340,178,347)(172,339,179,346)(173,338,180,345)(174,337,181,344)(175,350,182,343)(183,258,190,265)(184,257,191,264)(185,256,192,263)(186,255,193,262)(187,254,194,261)(188,253,195,260)(189,266,196,259)(225,326,232,333)(226,325,233,332)(227,324,234,331)(228,323,235,330)(229,336,236,329)(230,335,237,328)(231,334,238,327)(379,416,386,409)(380,415,387,408)(381,414,388,407)(382,413,389,420)(383,412,390,419)(384,411,391,418)(385,410,392,417)(393,441,400,448)(394,440,401,447)(395,439,402,446)(396,438,403,445)(397,437,404,444)(398,436,405,443)(399,435,406,442), (1,352,77,322)(2,353,78,309)(3,354,79,310)(4,355,80,311)(5,356,81,312)(6,357,82,313)(7,358,83,314)(8,359,84,315)(9,360,71,316)(10,361,72,317)(11,362,73,318)(12,363,74,319)(13,364,75,320)(14,351,76,321)(15,230,138,294)(16,231,139,281)(17,232,140,282)(18,233,127,283)(19,234,128,284)(20,235,129,285)(21,236,130,286)(22,237,131,287)(23,238,132,288)(24,225,133,289)(25,226,134,290)(26,227,135,291)(27,228,136,292)(28,229,137,293)(29,302,266,268)(30,303,253,269)(31,304,254,270)(32,305,255,271)(33,306,256,272)(34,307,257,273)(35,308,258,274)(36,295,259,275)(37,296,260,276)(38,297,261,277)(39,298,262,278)(40,299,263,279)(41,300,264,280)(42,301,265,267)(43,370,58,125)(44,371,59,126)(45,372,60,113)(46,373,61,114)(47,374,62,115)(48,375,63,116)(49,376,64,117)(50,377,65,118)(51,378,66,119)(52,365,67,120)(53,366,68,121)(54,367,69,122)(55,368,70,123)(56,369,57,124)(85,239,329,431)(86,240,330,432)(87,241,331,433)(88,242,332,434)(89,243,333,421)(90,244,334,422)(91,245,335,423)(92,246,336,424)(93,247,323,425)(94,248,324,426)(95,249,325,427)(96,250,326,428)(97,251,327,429)(98,252,328,430)(99,394,388,181)(100,395,389,182)(101,396,390,169)(102,397,391,170)(103,398,392,171)(104,399,379,172)(105,400,380,173)(106,401,381,174)(107,402,382,175)(108,403,383,176)(109,404,384,177)(110,405,385,178)(111,406,386,179)(112,393,387,180)(141,213,156,187)(142,214,157,188)(143,215,158,189)(144,216,159,190)(145,217,160,191)(146,218,161,192)(147,219,162,193)(148,220,163,194)(149,221,164,195)(150,222,165,196)(151,223,166,183)(152,224,167,184)(153,211,168,185)(154,212,155,186)(197,443,410,347)(198,444,411,348)(199,445,412,349)(200,446,413,350)(201,447,414,337)(202,448,415,338)(203,435,416,339)(204,436,417,340)(205,437,418,341)(206,438,419,342)(207,439,420,343)(208,440,407,344)(209,441,408,345)(210,442,409,346) );
G=PermutationGroup([[(1,178,427,188),(2,179,428,189),(3,180,429,190),(4,181,430,191),(5,182,431,192),(6,169,432,193),(7,170,433,194),(8,171,434,195),(9,172,421,196),(10,173,422,183),(11,174,423,184),(12,175,424,185),(13,176,425,186),(14,177,426,187),(15,440,122,41),(16,441,123,42),(17,442,124,29),(18,443,125,30),(19,444,126,31),(20,445,113,32),(21,446,114,33),(22,447,115,34),(23,448,116,35),(24,435,117,36),(25,436,118,37),(26,437,119,38),(27,438,120,39),(28,439,121,40),(43,417,233,296),(44,418,234,297),(45,419,235,298),(46,420,236,299),(47,407,237,300),(48,408,238,301),(49,409,225,302),(50,410,226,303),(51,411,227,304),(52,412,228,305),(53,413,229,306),(54,414,230,307),(55,415,231,308),(56,416,232,295),(57,203,282,275),(58,204,283,276),(59,205,284,277),(60,206,285,278),(61,207,286,279),(62,208,287,280),(63,209,288,267),(64,210,289,268),(65,197,290,269),(66,198,291,270),(67,199,292,271),(68,200,293,272),(69,201,294,273),(70,202,281,274),(71,399,243,222),(72,400,244,223),(73,401,245,224),(74,402,246,211),(75,403,247,212),(76,404,248,213),(77,405,249,214),(78,406,250,215),(79,393,251,216),(80,394,252,217),(81,395,239,218),(82,396,240,219),(83,397,241,220),(84,398,242,221),(85,107,356,153),(86,108,357,154),(87,109,358,141),(88,110,359,142),(89,111,360,143),(90,112,361,144),(91,99,362,145),(92,100,363,146),(93,101,364,147),(94,102,351,148),(95,103,352,149),(96,104,353,150),(97,105,354,151),(98,106,355,152),(127,347,370,253),(128,348,371,254),(129,349,372,255),(130,350,373,256),(131,337,374,257),(132,338,375,258),(133,339,376,259),(134,340,377,260),(135,341,378,261),(136,342,365,262),(137,343,366,263),(138,344,367,264),(139,345,368,265),(140,346,369,266),(155,330,383,313),(156,331,384,314),(157,332,385,315),(158,333,386,316),(159,334,387,317),(160,335,388,318),(161,336,389,319),(162,323,390,320),(163,324,391,321),(164,325,392,322),(165,326,379,309),(166,327,380,310),(167,328,381,311),(168,329,382,312)], [(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)], [(1,370,8,377),(2,369,9,376),(3,368,10,375),(4,367,11,374),(5,366,12,373),(6,365,13,372),(7,378,14,371),(15,245,22,252),(16,244,23,251),(17,243,24,250),(18,242,25,249),(19,241,26,248),(20,240,27,247),(21,239,28,246),(29,222,36,215),(30,221,37,214),(31,220,38,213),(32,219,39,212),(33,218,40,211),(34,217,41,224),(35,216,42,223),(43,315,50,322),(44,314,51,321),(45,313,52,320),(46,312,53,319),(47,311,54,318),(48,310,55,317),(49,309,56,316),(57,360,64,353),(58,359,65,352),(59,358,66,351),(60,357,67,364),(61,356,68,363),(62,355,69,362),(63,354,70,361),(71,117,78,124),(72,116,79,123),(73,115,80,122),(74,114,81,121),(75,113,82,120),(76,126,83,119),(77,125,84,118),(85,293,92,286),(86,292,93,285),(87,291,94,284),(88,290,95,283),(89,289,96,282),(90,288,97,281),(91,287,98,294),(99,208,106,201),(100,207,107,200),(101,206,108,199),(102,205,109,198),(103,204,110,197),(104,203,111,210),(105,202,112,209),(127,434,134,427),(128,433,135,426),(129,432,136,425),(130,431,137,424),(131,430,138,423),(132,429,139,422),(133,428,140,421),(141,270,148,277),(142,269,149,276),(143,268,150,275),(144,267,151,274),(145,280,152,273),(146,279,153,272),(147,278,154,271),(155,305,162,298),(156,304,163,297),(157,303,164,296),(158,302,165,295),(159,301,166,308),(160,300,167,307),(161,299,168,306),(169,342,176,349),(170,341,177,348),(171,340,178,347),(172,339,179,346),(173,338,180,345),(174,337,181,344),(175,350,182,343),(183,258,190,265),(184,257,191,264),(185,256,192,263),(186,255,193,262),(187,254,194,261),(188,253,195,260),(189,266,196,259),(225,326,232,333),(226,325,233,332),(227,324,234,331),(228,323,235,330),(229,336,236,329),(230,335,237,328),(231,334,238,327),(379,416,386,409),(380,415,387,408),(381,414,388,407),(382,413,389,420),(383,412,390,419),(384,411,391,418),(385,410,392,417),(393,441,400,448),(394,440,401,447),(395,439,402,446),(396,438,403,445),(397,437,404,444),(398,436,405,443),(399,435,406,442)], [(1,352,77,322),(2,353,78,309),(3,354,79,310),(4,355,80,311),(5,356,81,312),(6,357,82,313),(7,358,83,314),(8,359,84,315),(9,360,71,316),(10,361,72,317),(11,362,73,318),(12,363,74,319),(13,364,75,320),(14,351,76,321),(15,230,138,294),(16,231,139,281),(17,232,140,282),(18,233,127,283),(19,234,128,284),(20,235,129,285),(21,236,130,286),(22,237,131,287),(23,238,132,288),(24,225,133,289),(25,226,134,290),(26,227,135,291),(27,228,136,292),(28,229,137,293),(29,302,266,268),(30,303,253,269),(31,304,254,270),(32,305,255,271),(33,306,256,272),(34,307,257,273),(35,308,258,274),(36,295,259,275),(37,296,260,276),(38,297,261,277),(39,298,262,278),(40,299,263,279),(41,300,264,280),(42,301,265,267),(43,370,58,125),(44,371,59,126),(45,372,60,113),(46,373,61,114),(47,374,62,115),(48,375,63,116),(49,376,64,117),(50,377,65,118),(51,378,66,119),(52,365,67,120),(53,366,68,121),(54,367,69,122),(55,368,70,123),(56,369,57,124),(85,239,329,431),(86,240,330,432),(87,241,331,433),(88,242,332,434),(89,243,333,421),(90,244,334,422),(91,245,335,423),(92,246,336,424),(93,247,323,425),(94,248,324,426),(95,249,325,427),(96,250,326,428),(97,251,327,429),(98,252,328,430),(99,394,388,181),(100,395,389,182),(101,396,390,169),(102,397,391,170),(103,398,392,171),(104,399,379,172),(105,400,380,173),(106,401,381,174),(107,402,382,175),(108,403,383,176),(109,404,384,177),(110,405,385,178),(111,406,386,179),(112,393,387,180),(141,213,156,187),(142,214,157,188),(143,215,158,189),(144,216,159,190),(145,217,160,191),(146,218,161,192),(147,219,162,193),(148,220,163,194),(149,221,164,195),(150,222,165,196),(151,223,166,183),(152,224,167,184),(153,211,168,185),(154,212,155,186),(197,443,410,347),(198,444,411,348),(199,445,412,349),(200,446,413,350),(201,447,414,337),(202,448,415,338),(203,435,416,339),(204,436,417,340),(205,437,418,341),(206,438,419,342),(207,439,420,343),(208,440,407,344),(209,441,408,345),(210,442,409,346)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D7 | C4○D4 | D14 | Dic14 | C4×D7 | C4○D28 | D4×D7 | D4⋊2D7 | Q8⋊2D7 |
| kernel | C4⋊Dic7⋊8C4 | C14.C42 | C7×C2.C42 | C2×C4×Dic7 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C4⋊Dic7 | C2×Dic7 | C2×C28 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 3 | 8 | 9 | 12 | 12 | 12 | 3 | 6 | 3 |
Matrix representation of C4⋊Dic7⋊8C4 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 20 | 18 | 0 | 0 |
| 0 | 18 | 9 | 0 | 0 |
| 0 | 0 | 0 | 16 | 24 |
| 0 | 0 | 0 | 5 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 | 1 |
| 0 | 0 | 0 | 28 | 0 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 16 | 19 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,20,18,0,0,0,18,9,0,0,0,0,0,16,5,0,0,0,24,13],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,28,0,0,0,1,0],[28,0,0,0,0,0,0,17,0,0,0,12,0,0,0,0,0,0,10,16,0,0,0,10,19],[17,0,0,0,0,0,0,28,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,28] >;
C4⋊Dic7⋊8C4 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_7\rtimes_8C_4 % in TeX
G:=Group("C4:Dic7:8C4"); // GroupNames label
G:=SmallGroup(448,188);
// by ID
G=gap.SmallGroup(448,188);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,701,344,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^14=d^4=1,c^2=b^7,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^7,c*b*c^-1=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations