direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×Q8⋊C8, Q8⋊C56, C28.30Q16, C28.55SD16, C28.20M4(2), (C7×Q8)⋊3C8, C4⋊C4.4C28, C4⋊C8.1C14, (C4×C56).3C2, (C4×C8).1C14, C4.2(C2×C56), C14.21C4≀C2, C4.8(C7×Q16), C28.31(C2×C8), (C2×Q8).4C28, (C4×Q8).1C14, (C2×C28).529D4, (Q8×C14).14C4, (Q8×C28).14C2, C4.14(C7×SD16), C4.2(C7×M4(2)), C42.63(C2×C14), C14.23(C22⋊C8), (C4×C28).347C22, C14.14(Q8⋊C4), C2.2(C7×C4≀C2), (C7×C4⋊C8).7C2, (C7×C4⋊C4).16C4, (C2×C4).94(C7×D4), C2.6(C7×C22⋊C8), (C2×C4).39(C2×C28), C2.1(C7×Q8⋊C4), (C2×C28).259(C2×C4), C22.26(C7×C22⋊C4), (C2×C14).121(C22⋊C4), SmallGroup(448,130)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×Q8⋊C8
G = < a,b,c,d | a7=b4=d8=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >
Subgroups: 106 in 70 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C14, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C28, C28, C2×C14, C4×C8, C4⋊C8, C4×Q8, C56, C2×C28, C2×C28, C7×Q8, C7×Q8, Q8⋊C8, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C56, Q8×C14, C4×C56, C7×C4⋊C8, Q8×C28, C7×Q8⋊C8
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, D4, C14, C22⋊C4, C2×C8, M4(2), SD16, Q16, C28, C2×C14, C22⋊C8, Q8⋊C4, C4≀C2, C56, C2×C28, C7×D4, Q8⋊C8, C7×C22⋊C4, C2×C56, C7×M4(2), C7×SD16, C7×Q16, C7×C22⋊C8, C7×Q8⋊C4, C7×C4≀C2, C7×Q8⋊C8
►Regular action on 448 points
Generators in S448
(1 287 279 271 263 255 247)(2 288 280 272 264 256 248)(3 281 273 265 257 249 241)(4 282 274 266 258 250 242)(5 283 275 267 259 251 243)(6 284 276 268 260 252 244)(7 285 277 269 261 253 245)(8 286 278 270 262 254 246)(9 114 106 98 90 82 74)(10 115 107 99 91 83 75)(11 116 108 100 92 84 76)(12 117 109 101 93 85 77)(13 118 110 102 94 86 78)(14 119 111 103 95 87 79)(15 120 112 104 96 88 80)(16 113 105 97 89 81 73)(17 65 57 49 41 33 25)(18 66 58 50 42 34 26)(19 67 59 51 43 35 27)(20 68 60 52 44 36 28)(21 69 61 53 45 37 29)(22 70 62 54 46 38 30)(23 71 63 55 47 39 31)(24 72 64 56 48 40 32)(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)(177 231 217 209 201 193 185)(178 232 218 210 202 194 186)(179 225 219 211 203 195 187)(180 226 220 212 204 196 188)(181 227 221 213 205 197 189)(182 228 222 214 206 198 190)(183 229 223 215 207 199 191)(184 230 224 216 208 200 192)(233 329 321 313 305 297 289)(234 330 322 314 306 298 290)(235 331 323 315 307 299 291)(236 332 324 316 308 300 292)(237 333 325 317 309 301 293)(238 334 326 318 310 302 294)(239 335 327 319 311 303 295)(240 336 328 320 312 304 296)(337 385 377 369 361 353 345)(338 386 378 370 362 354 346)(339 387 379 371 363 355 347)(340 388 380 372 364 356 348)(341 389 381 373 365 357 349)(342 390 382 374 366 358 350)(343 391 383 375 367 359 351)(344 392 384 376 368 360 352)(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)
(1 127 343 23)(2 24 344 128)(3 121 337 17)(4 18 338 122)(5 123 339 19)(6 20 340 124)(7 125 341 21)(8 22 342 126)(9 234 232 442)(10 443 225 235)(11 236 226 444)(12 445 227 237)(13 238 228 446)(14 447 229 239)(15 240 230 448)(16 441 231 233)(25 241 129 345)(26 346 130 242)(27 243 131 347)(28 348 132 244)(29 245 133 349)(30 350 134 246)(31 247 135 351)(32 352 136 248)(33 249 137 353)(34 354 138 250)(35 251 139 355)(36 356 140 252)(37 253 141 357)(38 358 142 254)(39 255 143 359)(40 360 144 256)(41 257 145 361)(42 362 146 258)(43 259 147 363)(44 364 148 260)(45 261 149 365)(46 366 150 262)(47 263 151 367)(48 368 152 264)(49 265 153 369)(50 370 154 266)(51 267 155 371)(52 372 156 268)(53 269 157 373)(54 374 158 270)(55 271 159 375)(56 376 160 272)(57 273 161 377)(58 378 162 274)(59 275 163 379)(60 380 164 276)(61 277 165 381)(62 382 166 278)(63 279 167 383)(64 384 168 280)(65 281 169 385)(66 386 170 282)(67 283 171 387)(68 388 172 284)(69 285 173 389)(70 390 174 286)(71 287 175 391)(72 392 176 288)(73 393 177 289)(74 290 178 394)(75 395 179 291)(76 292 180 396)(77 397 181 293)(78 294 182 398)(79 399 183 295)(80 296 184 400)(81 401 185 297)(82 298 186 402)(83 403 187 299)(84 300 188 404)(85 405 189 301)(86 302 190 406)(87 407 191 303)(88 304 192 408)(89 409 193 305)(90 306 194 410)(91 411 195 307)(92 308 196 412)(93 413 197 309)(94 310 198 414)(95 415 199 311)(96 312 200 416)(97 417 201 313)(98 314 202 418)(99 419 203 315)(100 316 204 420)(101 421 205 317)(102 318 206 422)(103 423 207 319)(104 320 208 424)(105 425 209 321)(106 322 210 426)(107 427 211 323)(108 324 212 428)(109 429 213 325)(110 326 214 430)(111 431 215 327)(112 328 216 432)(113 433 217 329)(114 330 218 434)(115 435 219 331)(116 332 220 436)(117 437 221 333)(118 334 222 438)(119 439 223 335)(120 336 224 440)
(1 395 343 291)(2 180 344 76)(3 397 337 293)(4 182 338 78)(5 399 339 295)(6 184 340 80)(7 393 341 289)(8 178 342 74)(9 286 232 390)(10 71 225 175)(11 288 226 392)(12 65 227 169)(13 282 228 386)(14 67 229 171)(15 284 230 388)(16 69 231 173)(17 181 121 77)(18 294 122 398)(19 183 123 79)(20 296 124 400)(21 177 125 73)(22 290 126 394)(23 179 127 75)(24 292 128 396)(25 189 129 85)(26 302 130 406)(27 191 131 87)(28 304 132 408)(29 185 133 81)(30 298 134 402)(31 187 135 83)(32 300 136 404)(33 197 137 93)(34 310 138 414)(35 199 139 95)(36 312 140 416)(37 193 141 89)(38 306 142 410)(39 195 143 91)(40 308 144 412)(41 205 145 101)(42 318 146 422)(43 207 147 103)(44 320 148 424)(45 201 149 97)(46 314 150 418)(47 203 151 99)(48 316 152 420)(49 213 153 109)(50 326 154 430)(51 215 155 111)(52 328 156 432)(53 209 157 105)(54 322 158 426)(55 211 159 107)(56 324 160 428)(57 221 161 117)(58 334 162 438)(59 223 163 119)(60 336 164 440)(61 217 165 113)(62 330 166 434)(63 219 167 115)(64 332 168 436)(66 238 170 446)(68 240 172 448)(70 234 174 442)(72 236 176 444)(82 246 186 350)(84 248 188 352)(86 242 190 346)(88 244 192 348)(90 254 194 358)(92 256 196 360)(94 250 198 354)(96 252 200 356)(98 262 202 366)(100 264 204 368)(102 258 206 362)(104 260 208 364)(106 270 210 374)(108 272 212 376)(110 266 214 370)(112 268 216 372)(114 278 218 382)(116 280 220 384)(118 274 222 378)(120 276 224 380)(233 285 441 389)(235 287 443 391)(237 281 445 385)(239 283 447 387)(241 405 345 301)(243 407 347 303)(245 401 349 297)(247 403 351 299)(249 413 353 309)(251 415 355 311)(253 409 357 305)(255 411 359 307)(257 421 361 317)(259 423 363 319)(261 417 365 313)(263 419 367 315)(265 429 369 325)(267 431 371 327)(269 425 373 321)(271 427 375 323)(273 437 377 333)(275 439 379 335)(277 433 381 329)(279 435 383 331)
(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)
G:=sub<Sym(448)| (1,287,279,271,263,255,247)(2,288,280,272,264,256,248)(3,281,273,265,257,249,241)(4,282,274,266,258,250,242)(5,283,275,267,259,251,243)(6,284,276,268,260,252,244)(7,285,277,269,261,253,245)(8,286,278,270,262,254,246)(9,114,106,98,90,82,74)(10,115,107,99,91,83,75)(11,116,108,100,92,84,76)(12,117,109,101,93,85,77)(13,118,110,102,94,86,78)(14,119,111,103,95,87,79)(15,120,112,104,96,88,80)(16,113,105,97,89,81,73)(17,65,57,49,41,33,25)(18,66,58,50,42,34,26)(19,67,59,51,43,35,27)(20,68,60,52,44,36,28)(21,69,61,53,45,37,29)(22,70,62,54,46,38,30)(23,71,63,55,47,39,31)(24,72,64,56,48,40,32)(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)(177,231,217,209,201,193,185)(178,232,218,210,202,194,186)(179,225,219,211,203,195,187)(180,226,220,212,204,196,188)(181,227,221,213,205,197,189)(182,228,222,214,206,198,190)(183,229,223,215,207,199,191)(184,230,224,216,208,200,192)(233,329,321,313,305,297,289)(234,330,322,314,306,298,290)(235,331,323,315,307,299,291)(236,332,324,316,308,300,292)(237,333,325,317,309,301,293)(238,334,326,318,310,302,294)(239,335,327,319,311,303,295)(240,336,328,320,312,304,296)(337,385,377,369,361,353,345)(338,386,378,370,362,354,346)(339,387,379,371,363,355,347)(340,388,380,372,364,356,348)(341,389,381,373,365,357,349)(342,390,382,374,366,358,350)(343,391,383,375,367,359,351)(344,392,384,376,368,360,352)(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), (1,127,343,23)(2,24,344,128)(3,121,337,17)(4,18,338,122)(5,123,339,19)(6,20,340,124)(7,125,341,21)(8,22,342,126)(9,234,232,442)(10,443,225,235)(11,236,226,444)(12,445,227,237)(13,238,228,446)(14,447,229,239)(15,240,230,448)(16,441,231,233)(25,241,129,345)(26,346,130,242)(27,243,131,347)(28,348,132,244)(29,245,133,349)(30,350,134,246)(31,247,135,351)(32,352,136,248)(33,249,137,353)(34,354,138,250)(35,251,139,355)(36,356,140,252)(37,253,141,357)(38,358,142,254)(39,255,143,359)(40,360,144,256)(41,257,145,361)(42,362,146,258)(43,259,147,363)(44,364,148,260)(45,261,149,365)(46,366,150,262)(47,263,151,367)(48,368,152,264)(49,265,153,369)(50,370,154,266)(51,267,155,371)(52,372,156,268)(53,269,157,373)(54,374,158,270)(55,271,159,375)(56,376,160,272)(57,273,161,377)(58,378,162,274)(59,275,163,379)(60,380,164,276)(61,277,165,381)(62,382,166,278)(63,279,167,383)(64,384,168,280)(65,281,169,385)(66,386,170,282)(67,283,171,387)(68,388,172,284)(69,285,173,389)(70,390,174,286)(71,287,175,391)(72,392,176,288)(73,393,177,289)(74,290,178,394)(75,395,179,291)(76,292,180,396)(77,397,181,293)(78,294,182,398)(79,399,183,295)(80,296,184,400)(81,401,185,297)(82,298,186,402)(83,403,187,299)(84,300,188,404)(85,405,189,301)(86,302,190,406)(87,407,191,303)(88,304,192,408)(89,409,193,305)(90,306,194,410)(91,411,195,307)(92,308,196,412)(93,413,197,309)(94,310,198,414)(95,415,199,311)(96,312,200,416)(97,417,201,313)(98,314,202,418)(99,419,203,315)(100,316,204,420)(101,421,205,317)(102,318,206,422)(103,423,207,319)(104,320,208,424)(105,425,209,321)(106,322,210,426)(107,427,211,323)(108,324,212,428)(109,429,213,325)(110,326,214,430)(111,431,215,327)(112,328,216,432)(113,433,217,329)(114,330,218,434)(115,435,219,331)(116,332,220,436)(117,437,221,333)(118,334,222,438)(119,439,223,335)(120,336,224,440), (1,395,343,291)(2,180,344,76)(3,397,337,293)(4,182,338,78)(5,399,339,295)(6,184,340,80)(7,393,341,289)(8,178,342,74)(9,286,232,390)(10,71,225,175)(11,288,226,392)(12,65,227,169)(13,282,228,386)(14,67,229,171)(15,284,230,388)(16,69,231,173)(17,181,121,77)(18,294,122,398)(19,183,123,79)(20,296,124,400)(21,177,125,73)(22,290,126,394)(23,179,127,75)(24,292,128,396)(25,189,129,85)(26,302,130,406)(27,191,131,87)(28,304,132,408)(29,185,133,81)(30,298,134,402)(31,187,135,83)(32,300,136,404)(33,197,137,93)(34,310,138,414)(35,199,139,95)(36,312,140,416)(37,193,141,89)(38,306,142,410)(39,195,143,91)(40,308,144,412)(41,205,145,101)(42,318,146,422)(43,207,147,103)(44,320,148,424)(45,201,149,97)(46,314,150,418)(47,203,151,99)(48,316,152,420)(49,213,153,109)(50,326,154,430)(51,215,155,111)(52,328,156,432)(53,209,157,105)(54,322,158,426)(55,211,159,107)(56,324,160,428)(57,221,161,117)(58,334,162,438)(59,223,163,119)(60,336,164,440)(61,217,165,113)(62,330,166,434)(63,219,167,115)(64,332,168,436)(66,238,170,446)(68,240,172,448)(70,234,174,442)(72,236,176,444)(82,246,186,350)(84,248,188,352)(86,242,190,346)(88,244,192,348)(90,254,194,358)(92,256,196,360)(94,250,198,354)(96,252,200,356)(98,262,202,366)(100,264,204,368)(102,258,206,362)(104,260,208,364)(106,270,210,374)(108,272,212,376)(110,266,214,370)(112,268,216,372)(114,278,218,382)(116,280,220,384)(118,274,222,378)(120,276,224,380)(233,285,441,389)(235,287,443,391)(237,281,445,385)(239,283,447,387)(241,405,345,301)(243,407,347,303)(245,401,349,297)(247,403,351,299)(249,413,353,309)(251,415,355,311)(253,409,357,305)(255,411,359,307)(257,421,361,317)(259,423,363,319)(261,417,365,313)(263,419,367,315)(265,429,369,325)(267,431,371,327)(269,425,373,321)(271,427,375,323)(273,437,377,333)(275,439,379,335)(277,433,381,329)(279,435,383,331), (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)>;
G:=Group( (1,287,279,271,263,255,247)(2,288,280,272,264,256,248)(3,281,273,265,257,249,241)(4,282,274,266,258,250,242)(5,283,275,267,259,251,243)(6,284,276,268,260,252,244)(7,285,277,269,261,253,245)(8,286,278,270,262,254,246)(9,114,106,98,90,82,74)(10,115,107,99,91,83,75)(11,116,108,100,92,84,76)(12,117,109,101,93,85,77)(13,118,110,102,94,86,78)(14,119,111,103,95,87,79)(15,120,112,104,96,88,80)(16,113,105,97,89,81,73)(17,65,57,49,41,33,25)(18,66,58,50,42,34,26)(19,67,59,51,43,35,27)(20,68,60,52,44,36,28)(21,69,61,53,45,37,29)(22,70,62,54,46,38,30)(23,71,63,55,47,39,31)(24,72,64,56,48,40,32)(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)(177,231,217,209,201,193,185)(178,232,218,210,202,194,186)(179,225,219,211,203,195,187)(180,226,220,212,204,196,188)(181,227,221,213,205,197,189)(182,228,222,214,206,198,190)(183,229,223,215,207,199,191)(184,230,224,216,208,200,192)(233,329,321,313,305,297,289)(234,330,322,314,306,298,290)(235,331,323,315,307,299,291)(236,332,324,316,308,300,292)(237,333,325,317,309,301,293)(238,334,326,318,310,302,294)(239,335,327,319,311,303,295)(240,336,328,320,312,304,296)(337,385,377,369,361,353,345)(338,386,378,370,362,354,346)(339,387,379,371,363,355,347)(340,388,380,372,364,356,348)(341,389,381,373,365,357,349)(342,390,382,374,366,358,350)(343,391,383,375,367,359,351)(344,392,384,376,368,360,352)(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), (1,127,343,23)(2,24,344,128)(3,121,337,17)(4,18,338,122)(5,123,339,19)(6,20,340,124)(7,125,341,21)(8,22,342,126)(9,234,232,442)(10,443,225,235)(11,236,226,444)(12,445,227,237)(13,238,228,446)(14,447,229,239)(15,240,230,448)(16,441,231,233)(25,241,129,345)(26,346,130,242)(27,243,131,347)(28,348,132,244)(29,245,133,349)(30,350,134,246)(31,247,135,351)(32,352,136,248)(33,249,137,353)(34,354,138,250)(35,251,139,355)(36,356,140,252)(37,253,141,357)(38,358,142,254)(39,255,143,359)(40,360,144,256)(41,257,145,361)(42,362,146,258)(43,259,147,363)(44,364,148,260)(45,261,149,365)(46,366,150,262)(47,263,151,367)(48,368,152,264)(49,265,153,369)(50,370,154,266)(51,267,155,371)(52,372,156,268)(53,269,157,373)(54,374,158,270)(55,271,159,375)(56,376,160,272)(57,273,161,377)(58,378,162,274)(59,275,163,379)(60,380,164,276)(61,277,165,381)(62,382,166,278)(63,279,167,383)(64,384,168,280)(65,281,169,385)(66,386,170,282)(67,283,171,387)(68,388,172,284)(69,285,173,389)(70,390,174,286)(71,287,175,391)(72,392,176,288)(73,393,177,289)(74,290,178,394)(75,395,179,291)(76,292,180,396)(77,397,181,293)(78,294,182,398)(79,399,183,295)(80,296,184,400)(81,401,185,297)(82,298,186,402)(83,403,187,299)(84,300,188,404)(85,405,189,301)(86,302,190,406)(87,407,191,303)(88,304,192,408)(89,409,193,305)(90,306,194,410)(91,411,195,307)(92,308,196,412)(93,413,197,309)(94,310,198,414)(95,415,199,311)(96,312,200,416)(97,417,201,313)(98,314,202,418)(99,419,203,315)(100,316,204,420)(101,421,205,317)(102,318,206,422)(103,423,207,319)(104,320,208,424)(105,425,209,321)(106,322,210,426)(107,427,211,323)(108,324,212,428)(109,429,213,325)(110,326,214,430)(111,431,215,327)(112,328,216,432)(113,433,217,329)(114,330,218,434)(115,435,219,331)(116,332,220,436)(117,437,221,333)(118,334,222,438)(119,439,223,335)(120,336,224,440), (1,395,343,291)(2,180,344,76)(3,397,337,293)(4,182,338,78)(5,399,339,295)(6,184,340,80)(7,393,341,289)(8,178,342,74)(9,286,232,390)(10,71,225,175)(11,288,226,392)(12,65,227,169)(13,282,228,386)(14,67,229,171)(15,284,230,388)(16,69,231,173)(17,181,121,77)(18,294,122,398)(19,183,123,79)(20,296,124,400)(21,177,125,73)(22,290,126,394)(23,179,127,75)(24,292,128,396)(25,189,129,85)(26,302,130,406)(27,191,131,87)(28,304,132,408)(29,185,133,81)(30,298,134,402)(31,187,135,83)(32,300,136,404)(33,197,137,93)(34,310,138,414)(35,199,139,95)(36,312,140,416)(37,193,141,89)(38,306,142,410)(39,195,143,91)(40,308,144,412)(41,205,145,101)(42,318,146,422)(43,207,147,103)(44,320,148,424)(45,201,149,97)(46,314,150,418)(47,203,151,99)(48,316,152,420)(49,213,153,109)(50,326,154,430)(51,215,155,111)(52,328,156,432)(53,209,157,105)(54,322,158,426)(55,211,159,107)(56,324,160,428)(57,221,161,117)(58,334,162,438)(59,223,163,119)(60,336,164,440)(61,217,165,113)(62,330,166,434)(63,219,167,115)(64,332,168,436)(66,238,170,446)(68,240,172,448)(70,234,174,442)(72,236,176,444)(82,246,186,350)(84,248,188,352)(86,242,190,346)(88,244,192,348)(90,254,194,358)(92,256,196,360)(94,250,198,354)(96,252,200,356)(98,262,202,366)(100,264,204,368)(102,258,206,362)(104,260,208,364)(106,270,210,374)(108,272,212,376)(110,266,214,370)(112,268,216,372)(114,278,218,382)(116,280,220,384)(118,274,222,378)(120,276,224,380)(233,285,441,389)(235,287,443,391)(237,281,445,385)(239,283,447,387)(241,405,345,301)(243,407,347,303)(245,401,349,297)(247,403,351,299)(249,413,353,309)(251,415,355,311)(253,409,357,305)(255,411,359,307)(257,421,361,317)(259,423,363,319)(261,417,365,313)(263,419,367,315)(265,429,369,325)(267,431,371,327)(269,425,373,321)(271,427,375,323)(273,437,377,333)(275,439,379,335)(277,433,381,329)(279,435,383,331), (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) );
G=PermutationGroup([[(1,287,279,271,263,255,247),(2,288,280,272,264,256,248),(3,281,273,265,257,249,241),(4,282,274,266,258,250,242),(5,283,275,267,259,251,243),(6,284,276,268,260,252,244),(7,285,277,269,261,253,245),(8,286,278,270,262,254,246),(9,114,106,98,90,82,74),(10,115,107,99,91,83,75),(11,116,108,100,92,84,76),(12,117,109,101,93,85,77),(13,118,110,102,94,86,78),(14,119,111,103,95,87,79),(15,120,112,104,96,88,80),(16,113,105,97,89,81,73),(17,65,57,49,41,33,25),(18,66,58,50,42,34,26),(19,67,59,51,43,35,27),(20,68,60,52,44,36,28),(21,69,61,53,45,37,29),(22,70,62,54,46,38,30),(23,71,63,55,47,39,31),(24,72,64,56,48,40,32),(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),(177,231,217,209,201,193,185),(178,232,218,210,202,194,186),(179,225,219,211,203,195,187),(180,226,220,212,204,196,188),(181,227,221,213,205,197,189),(182,228,222,214,206,198,190),(183,229,223,215,207,199,191),(184,230,224,216,208,200,192),(233,329,321,313,305,297,289),(234,330,322,314,306,298,290),(235,331,323,315,307,299,291),(236,332,324,316,308,300,292),(237,333,325,317,309,301,293),(238,334,326,318,310,302,294),(239,335,327,319,311,303,295),(240,336,328,320,312,304,296),(337,385,377,369,361,353,345),(338,386,378,370,362,354,346),(339,387,379,371,363,355,347),(340,388,380,372,364,356,348),(341,389,381,373,365,357,349),(342,390,382,374,366,358,350),(343,391,383,375,367,359,351),(344,392,384,376,368,360,352),(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)], [(1,127,343,23),(2,24,344,128),(3,121,337,17),(4,18,338,122),(5,123,339,19),(6,20,340,124),(7,125,341,21),(8,22,342,126),(9,234,232,442),(10,443,225,235),(11,236,226,444),(12,445,227,237),(13,238,228,446),(14,447,229,239),(15,240,230,448),(16,441,231,233),(25,241,129,345),(26,346,130,242),(27,243,131,347),(28,348,132,244),(29,245,133,349),(30,350,134,246),(31,247,135,351),(32,352,136,248),(33,249,137,353),(34,354,138,250),(35,251,139,355),(36,356,140,252),(37,253,141,357),(38,358,142,254),(39,255,143,359),(40,360,144,256),(41,257,145,361),(42,362,146,258),(43,259,147,363),(44,364,148,260),(45,261,149,365),(46,366,150,262),(47,263,151,367),(48,368,152,264),(49,265,153,369),(50,370,154,266),(51,267,155,371),(52,372,156,268),(53,269,157,373),(54,374,158,270),(55,271,159,375),(56,376,160,272),(57,273,161,377),(58,378,162,274),(59,275,163,379),(60,380,164,276),(61,277,165,381),(62,382,166,278),(63,279,167,383),(64,384,168,280),(65,281,169,385),(66,386,170,282),(67,283,171,387),(68,388,172,284),(69,285,173,389),(70,390,174,286),(71,287,175,391),(72,392,176,288),(73,393,177,289),(74,290,178,394),(75,395,179,291),(76,292,180,396),(77,397,181,293),(78,294,182,398),(79,399,183,295),(80,296,184,400),(81,401,185,297),(82,298,186,402),(83,403,187,299),(84,300,188,404),(85,405,189,301),(86,302,190,406),(87,407,191,303),(88,304,192,408),(89,409,193,305),(90,306,194,410),(91,411,195,307),(92,308,196,412),(93,413,197,309),(94,310,198,414),(95,415,199,311),(96,312,200,416),(97,417,201,313),(98,314,202,418),(99,419,203,315),(100,316,204,420),(101,421,205,317),(102,318,206,422),(103,423,207,319),(104,320,208,424),(105,425,209,321),(106,322,210,426),(107,427,211,323),(108,324,212,428),(109,429,213,325),(110,326,214,430),(111,431,215,327),(112,328,216,432),(113,433,217,329),(114,330,218,434),(115,435,219,331),(116,332,220,436),(117,437,221,333),(118,334,222,438),(119,439,223,335),(120,336,224,440)], [(1,395,343,291),(2,180,344,76),(3,397,337,293),(4,182,338,78),(5,399,339,295),(6,184,340,80),(7,393,341,289),(8,178,342,74),(9,286,232,390),(10,71,225,175),(11,288,226,392),(12,65,227,169),(13,282,228,386),(14,67,229,171),(15,284,230,388),(16,69,231,173),(17,181,121,77),(18,294,122,398),(19,183,123,79),(20,296,124,400),(21,177,125,73),(22,290,126,394),(23,179,127,75),(24,292,128,396),(25,189,129,85),(26,302,130,406),(27,191,131,87),(28,304,132,408),(29,185,133,81),(30,298,134,402),(31,187,135,83),(32,300,136,404),(33,197,137,93),(34,310,138,414),(35,199,139,95),(36,312,140,416),(37,193,141,89),(38,306,142,410),(39,195,143,91),(40,308,144,412),(41,205,145,101),(42,318,146,422),(43,207,147,103),(44,320,148,424),(45,201,149,97),(46,314,150,418),(47,203,151,99),(48,316,152,420),(49,213,153,109),(50,326,154,430),(51,215,155,111),(52,328,156,432),(53,209,157,105),(54,322,158,426),(55,211,159,107),(56,324,160,428),(57,221,161,117),(58,334,162,438),(59,223,163,119),(60,336,164,440),(61,217,165,113),(62,330,166,434),(63,219,167,115),(64,332,168,436),(66,238,170,446),(68,240,172,448),(70,234,174,442),(72,236,176,444),(82,246,186,350),(84,248,188,352),(86,242,190,346),(88,244,192,348),(90,254,194,358),(92,256,196,360),(94,250,198,354),(96,252,200,356),(98,262,202,366),(100,264,204,368),(102,258,206,362),(104,260,208,364),(106,270,210,374),(108,272,212,376),(110,266,214,370),(112,268,216,372),(114,278,218,382),(116,280,220,384),(118,274,222,378),(120,276,224,380),(233,285,441,389),(235,287,443,391),(237,281,445,385),(239,283,447,387),(241,405,345,301),(243,407,347,303),(245,401,349,297),(247,403,351,299),(249,413,353,309),(251,415,355,311),(253,409,357,305),(255,411,359,307),(257,421,361,317),(259,423,363,319),(261,417,365,313),(263,419,367,315),(265,429,369,325),(267,431,371,327),(269,425,373,321),(271,427,375,323),(273,437,377,333),(275,439,379,335),(277,433,381,329),(279,435,383,331)], [(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)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | ··· | 7F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14R | 28A | ··· | 28X | 28Y | ··· | 28AV | 28AW | ··· | 28BT | 56A | ··· | 56AV | 56AW | ··· | 56BT |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C8 | C14 | C14 | C14 | C28 | C28 | C56 | D4 | M4(2) | SD16 | Q16 | C4≀C2 | C7×D4 | C7×M4(2) | C7×SD16 | C7×Q16 | C7×C4≀C2 |
| kernel | C7×Q8⋊C8 | C4×C56 | C7×C4⋊C8 | Q8×C28 | C7×C4⋊C4 | Q8×C14 | Q8⋊C8 | C7×Q8 | C4×C8 | C4⋊C8 | C4×Q8 | C4⋊C4 | C2×Q8 | Q8 | C2×C28 | C28 | C28 | C28 | C14 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 8 | 6 | 6 | 6 | 12 | 12 | 48 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 24 |
Matrix representation of C7×Q8⋊C8 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 0 | 28 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 112 | 0 |
| 112 | 0 | 0 |
| 0 | 39 | 88 |
| 0 | 88 | 74 |
| 95 | 0 | 0 |
| 0 | 44 | 41 |
| 0 | 41 | 69 |
G:=sub<GL(3,GF(113))| [1,0,0,0,28,0,0,0,28],[1,0,0,0,0,112,0,1,0],[112,0,0,0,39,88,0,88,74],[95,0,0,0,44,41,0,41,69] >;
C7×Q8⋊C8 in GAP, Magma, Sage, TeX
C_7\times Q_8\rtimes C_8
% in TeX
G:=Group("C7xQ8:C8"); // GroupNames label
G:=SmallGroup(448,130);
// by ID
G=gap.SmallGroup(448,130);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,3923,1970,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=d^8=1,c^2=b^2,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>;
// generators/relations