direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: Q16×C26, C52.43D4, C52.46C23, C104.27C22, C8.5(C2×C26), (C2×C8).4C26, C4.8(D4×C13), C2.13(D4×C26), C26.76(C2×D4), (C2×C26).54D4, Q8.1(C2×C26), (C2×Q8).4C26, (Q8×C26).9C2, (C2×C104).14C2, C4.3(C22×C26), C22.16(D4×C13), (C2×C52).131C22, (Q8×C13).12C22, (C2×C4).27(C2×C26), SmallGroup(416,195)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C26
G = < a,b,c | a26=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 76 in 60 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, C2×C4, Q8, Q8, C13, C2×C8, Q16, C2×Q8, C26, C26, C2×Q16, C52, C52, C2×C26, C104, C2×C52, C2×C52, Q8×C13, Q8×C13, C2×C104, C13×Q16, Q8×C26, Q16×C26
Quotients: C1, C2, C22, D4, C23, C13, Q16, C2×D4, C26, C2×Q16, C2×C26, D4×C13, C22×C26, C13×Q16, D4×C26, Q16×C26
►Regular action on 416 points
Generators in S416
(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)
(1 237 108 351 59 296 92 31)(2 238 109 352 60 297 93 32)(3 239 110 353 61 298 94 33)(4 240 111 354 62 299 95 34)(5 241 112 355 63 300 96 35)(6 242 113 356 64 301 97 36)(7 243 114 357 65 302 98 37)(8 244 115 358 66 303 99 38)(9 245 116 359 67 304 100 39)(10 246 117 360 68 305 101 40)(11 247 118 361 69 306 102 41)(12 248 119 362 70 307 103 42)(13 249 120 363 71 308 104 43)(14 250 121 364 72 309 79 44)(15 251 122 339 73 310 80 45)(16 252 123 340 74 311 81 46)(17 253 124 341 75 312 82 47)(18 254 125 342 76 287 83 48)(19 255 126 343 77 288 84 49)(20 256 127 344 78 289 85 50)(21 257 128 345 53 290 86 51)(22 258 129 346 54 291 87 52)(23 259 130 347 55 292 88 27)(24 260 105 348 56 293 89 28)(25 235 106 349 57 294 90 29)(26 236 107 350 58 295 91 30)(131 160 279 386 211 399 195 336)(132 161 280 387 212 400 196 337)(133 162 281 388 213 401 197 338)(134 163 282 389 214 402 198 313)(135 164 283 390 215 403 199 314)(136 165 284 365 216 404 200 315)(137 166 285 366 217 405 201 316)(138 167 286 367 218 406 202 317)(139 168 261 368 219 407 203 318)(140 169 262 369 220 408 204 319)(141 170 263 370 221 409 205 320)(142 171 264 371 222 410 206 321)(143 172 265 372 223 411 207 322)(144 173 266 373 224 412 208 323)(145 174 267 374 225 413 183 324)(146 175 268 375 226 414 184 325)(147 176 269 376 227 415 185 326)(148 177 270 377 228 416 186 327)(149 178 271 378 229 391 187 328)(150 179 272 379 230 392 188 329)(151 180 273 380 231 393 189 330)(152 181 274 381 232 394 190 331)(153 182 275 382 233 395 191 332)(154 157 276 383 234 396 192 333)(155 158 277 384 209 397 193 334)(156 159 278 385 210 398 194 335)
(1 408 59 169)(2 409 60 170)(3 410 61 171)(4 411 62 172)(5 412 63 173)(6 413 64 174)(7 414 65 175)(8 415 66 176)(9 416 67 177)(10 391 68 178)(11 392 69 179)(12 393 70 180)(13 394 71 181)(14 395 72 182)(15 396 73 157)(16 397 74 158)(17 398 75 159)(18 399 76 160)(19 400 77 161)(20 401 78 162)(21 402 53 163)(22 403 54 164)(23 404 55 165)(24 405 56 166)(25 406 57 167)(26 407 58 168)(27 200 347 284)(28 201 348 285)(29 202 349 286)(30 203 350 261)(31 204 351 262)(32 205 352 263)(33 206 353 264)(34 207 354 265)(35 208 355 266)(36 183 356 267)(37 184 357 268)(38 185 358 269)(39 186 359 270)(40 187 360 271)(41 188 361 272)(42 189 362 273)(43 190 363 274)(44 191 364 275)(45 192 339 276)(46 193 340 277)(47 194 341 278)(48 195 342 279)(49 196 343 280)(50 197 344 281)(51 198 345 282)(52 199 346 283)(79 332 121 382)(80 333 122 383)(81 334 123 384)(82 335 124 385)(83 336 125 386)(84 337 126 387)(85 338 127 388)(86 313 128 389)(87 314 129 390)(88 315 130 365)(89 316 105 366)(90 317 106 367)(91 318 107 368)(92 319 108 369)(93 320 109 370)(94 321 110 371)(95 322 111 372)(96 323 112 373)(97 324 113 374)(98 325 114 375)(99 326 115 376)(100 327 116 377)(101 328 117 378)(102 329 118 379)(103 330 119 380)(104 331 120 381)(131 254 211 287)(132 255 212 288)(133 256 213 289)(134 257 214 290)(135 258 215 291)(136 259 216 292)(137 260 217 293)(138 235 218 294)(139 236 219 295)(140 237 220 296)(141 238 221 297)(142 239 222 298)(143 240 223 299)(144 241 224 300)(145 242 225 301)(146 243 226 302)(147 244 227 303)(148 245 228 304)(149 246 229 305)(150 247 230 306)(151 248 231 307)(152 249 232 308)(153 250 233 309)(154 251 234 310)(155 252 209 311)(156 253 210 312)
G:=sub<Sym(416)| (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), (1,237,108,351,59,296,92,31)(2,238,109,352,60,297,93,32)(3,239,110,353,61,298,94,33)(4,240,111,354,62,299,95,34)(5,241,112,355,63,300,96,35)(6,242,113,356,64,301,97,36)(7,243,114,357,65,302,98,37)(8,244,115,358,66,303,99,38)(9,245,116,359,67,304,100,39)(10,246,117,360,68,305,101,40)(11,247,118,361,69,306,102,41)(12,248,119,362,70,307,103,42)(13,249,120,363,71,308,104,43)(14,250,121,364,72,309,79,44)(15,251,122,339,73,310,80,45)(16,252,123,340,74,311,81,46)(17,253,124,341,75,312,82,47)(18,254,125,342,76,287,83,48)(19,255,126,343,77,288,84,49)(20,256,127,344,78,289,85,50)(21,257,128,345,53,290,86,51)(22,258,129,346,54,291,87,52)(23,259,130,347,55,292,88,27)(24,260,105,348,56,293,89,28)(25,235,106,349,57,294,90,29)(26,236,107,350,58,295,91,30)(131,160,279,386,211,399,195,336)(132,161,280,387,212,400,196,337)(133,162,281,388,213,401,197,338)(134,163,282,389,214,402,198,313)(135,164,283,390,215,403,199,314)(136,165,284,365,216,404,200,315)(137,166,285,366,217,405,201,316)(138,167,286,367,218,406,202,317)(139,168,261,368,219,407,203,318)(140,169,262,369,220,408,204,319)(141,170,263,370,221,409,205,320)(142,171,264,371,222,410,206,321)(143,172,265,372,223,411,207,322)(144,173,266,373,224,412,208,323)(145,174,267,374,225,413,183,324)(146,175,268,375,226,414,184,325)(147,176,269,376,227,415,185,326)(148,177,270,377,228,416,186,327)(149,178,271,378,229,391,187,328)(150,179,272,379,230,392,188,329)(151,180,273,380,231,393,189,330)(152,181,274,381,232,394,190,331)(153,182,275,382,233,395,191,332)(154,157,276,383,234,396,192,333)(155,158,277,384,209,397,193,334)(156,159,278,385,210,398,194,335), (1,408,59,169)(2,409,60,170)(3,410,61,171)(4,411,62,172)(5,412,63,173)(6,413,64,174)(7,414,65,175)(8,415,66,176)(9,416,67,177)(10,391,68,178)(11,392,69,179)(12,393,70,180)(13,394,71,181)(14,395,72,182)(15,396,73,157)(16,397,74,158)(17,398,75,159)(18,399,76,160)(19,400,77,161)(20,401,78,162)(21,402,53,163)(22,403,54,164)(23,404,55,165)(24,405,56,166)(25,406,57,167)(26,407,58,168)(27,200,347,284)(28,201,348,285)(29,202,349,286)(30,203,350,261)(31,204,351,262)(32,205,352,263)(33,206,353,264)(34,207,354,265)(35,208,355,266)(36,183,356,267)(37,184,357,268)(38,185,358,269)(39,186,359,270)(40,187,360,271)(41,188,361,272)(42,189,362,273)(43,190,363,274)(44,191,364,275)(45,192,339,276)(46,193,340,277)(47,194,341,278)(48,195,342,279)(49,196,343,280)(50,197,344,281)(51,198,345,282)(52,199,346,283)(79,332,121,382)(80,333,122,383)(81,334,123,384)(82,335,124,385)(83,336,125,386)(84,337,126,387)(85,338,127,388)(86,313,128,389)(87,314,129,390)(88,315,130,365)(89,316,105,366)(90,317,106,367)(91,318,107,368)(92,319,108,369)(93,320,109,370)(94,321,110,371)(95,322,111,372)(96,323,112,373)(97,324,113,374)(98,325,114,375)(99,326,115,376)(100,327,116,377)(101,328,117,378)(102,329,118,379)(103,330,119,380)(104,331,120,381)(131,254,211,287)(132,255,212,288)(133,256,213,289)(134,257,214,290)(135,258,215,291)(136,259,216,292)(137,260,217,293)(138,235,218,294)(139,236,219,295)(140,237,220,296)(141,238,221,297)(142,239,222,298)(143,240,223,299)(144,241,224,300)(145,242,225,301)(146,243,226,302)(147,244,227,303)(148,245,228,304)(149,246,229,305)(150,247,230,306)(151,248,231,307)(152,249,232,308)(153,250,233,309)(154,251,234,310)(155,252,209,311)(156,253,210,312)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286)(287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338)(339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364)(365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390)(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416), (1,237,108,351,59,296,92,31)(2,238,109,352,60,297,93,32)(3,239,110,353,61,298,94,33)(4,240,111,354,62,299,95,34)(5,241,112,355,63,300,96,35)(6,242,113,356,64,301,97,36)(7,243,114,357,65,302,98,37)(8,244,115,358,66,303,99,38)(9,245,116,359,67,304,100,39)(10,246,117,360,68,305,101,40)(11,247,118,361,69,306,102,41)(12,248,119,362,70,307,103,42)(13,249,120,363,71,308,104,43)(14,250,121,364,72,309,79,44)(15,251,122,339,73,310,80,45)(16,252,123,340,74,311,81,46)(17,253,124,341,75,312,82,47)(18,254,125,342,76,287,83,48)(19,255,126,343,77,288,84,49)(20,256,127,344,78,289,85,50)(21,257,128,345,53,290,86,51)(22,258,129,346,54,291,87,52)(23,259,130,347,55,292,88,27)(24,260,105,348,56,293,89,28)(25,235,106,349,57,294,90,29)(26,236,107,350,58,295,91,30)(131,160,279,386,211,399,195,336)(132,161,280,387,212,400,196,337)(133,162,281,388,213,401,197,338)(134,163,282,389,214,402,198,313)(135,164,283,390,215,403,199,314)(136,165,284,365,216,404,200,315)(137,166,285,366,217,405,201,316)(138,167,286,367,218,406,202,317)(139,168,261,368,219,407,203,318)(140,169,262,369,220,408,204,319)(141,170,263,370,221,409,205,320)(142,171,264,371,222,410,206,321)(143,172,265,372,223,411,207,322)(144,173,266,373,224,412,208,323)(145,174,267,374,225,413,183,324)(146,175,268,375,226,414,184,325)(147,176,269,376,227,415,185,326)(148,177,270,377,228,416,186,327)(149,178,271,378,229,391,187,328)(150,179,272,379,230,392,188,329)(151,180,273,380,231,393,189,330)(152,181,274,381,232,394,190,331)(153,182,275,382,233,395,191,332)(154,157,276,383,234,396,192,333)(155,158,277,384,209,397,193,334)(156,159,278,385,210,398,194,335), (1,408,59,169)(2,409,60,170)(3,410,61,171)(4,411,62,172)(5,412,63,173)(6,413,64,174)(7,414,65,175)(8,415,66,176)(9,416,67,177)(10,391,68,178)(11,392,69,179)(12,393,70,180)(13,394,71,181)(14,395,72,182)(15,396,73,157)(16,397,74,158)(17,398,75,159)(18,399,76,160)(19,400,77,161)(20,401,78,162)(21,402,53,163)(22,403,54,164)(23,404,55,165)(24,405,56,166)(25,406,57,167)(26,407,58,168)(27,200,347,284)(28,201,348,285)(29,202,349,286)(30,203,350,261)(31,204,351,262)(32,205,352,263)(33,206,353,264)(34,207,354,265)(35,208,355,266)(36,183,356,267)(37,184,357,268)(38,185,358,269)(39,186,359,270)(40,187,360,271)(41,188,361,272)(42,189,362,273)(43,190,363,274)(44,191,364,275)(45,192,339,276)(46,193,340,277)(47,194,341,278)(48,195,342,279)(49,196,343,280)(50,197,344,281)(51,198,345,282)(52,199,346,283)(79,332,121,382)(80,333,122,383)(81,334,123,384)(82,335,124,385)(83,336,125,386)(84,337,126,387)(85,338,127,388)(86,313,128,389)(87,314,129,390)(88,315,130,365)(89,316,105,366)(90,317,106,367)(91,318,107,368)(92,319,108,369)(93,320,109,370)(94,321,110,371)(95,322,111,372)(96,323,112,373)(97,324,113,374)(98,325,114,375)(99,326,115,376)(100,327,116,377)(101,328,117,378)(102,329,118,379)(103,330,119,380)(104,331,120,381)(131,254,211,287)(132,255,212,288)(133,256,213,289)(134,257,214,290)(135,258,215,291)(136,259,216,292)(137,260,217,293)(138,235,218,294)(139,236,219,295)(140,237,220,296)(141,238,221,297)(142,239,222,298)(143,240,223,299)(144,241,224,300)(145,242,225,301)(146,243,226,302)(147,244,227,303)(148,245,228,304)(149,246,229,305)(150,247,230,306)(151,248,231,307)(152,249,232,308)(153,250,233,309)(154,251,234,310)(155,252,209,311)(156,253,210,312) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26),(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182),(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208),(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234),(235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260),(261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286),(287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312),(313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338),(339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364),(365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390),(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416)], [(1,237,108,351,59,296,92,31),(2,238,109,352,60,297,93,32),(3,239,110,353,61,298,94,33),(4,240,111,354,62,299,95,34),(5,241,112,355,63,300,96,35),(6,242,113,356,64,301,97,36),(7,243,114,357,65,302,98,37),(8,244,115,358,66,303,99,38),(9,245,116,359,67,304,100,39),(10,246,117,360,68,305,101,40),(11,247,118,361,69,306,102,41),(12,248,119,362,70,307,103,42),(13,249,120,363,71,308,104,43),(14,250,121,364,72,309,79,44),(15,251,122,339,73,310,80,45),(16,252,123,340,74,311,81,46),(17,253,124,341,75,312,82,47),(18,254,125,342,76,287,83,48),(19,255,126,343,77,288,84,49),(20,256,127,344,78,289,85,50),(21,257,128,345,53,290,86,51),(22,258,129,346,54,291,87,52),(23,259,130,347,55,292,88,27),(24,260,105,348,56,293,89,28),(25,235,106,349,57,294,90,29),(26,236,107,350,58,295,91,30),(131,160,279,386,211,399,195,336),(132,161,280,387,212,400,196,337),(133,162,281,388,213,401,197,338),(134,163,282,389,214,402,198,313),(135,164,283,390,215,403,199,314),(136,165,284,365,216,404,200,315),(137,166,285,366,217,405,201,316),(138,167,286,367,218,406,202,317),(139,168,261,368,219,407,203,318),(140,169,262,369,220,408,204,319),(141,170,263,370,221,409,205,320),(142,171,264,371,222,410,206,321),(143,172,265,372,223,411,207,322),(144,173,266,373,224,412,208,323),(145,174,267,374,225,413,183,324),(146,175,268,375,226,414,184,325),(147,176,269,376,227,415,185,326),(148,177,270,377,228,416,186,327),(149,178,271,378,229,391,187,328),(150,179,272,379,230,392,188,329),(151,180,273,380,231,393,189,330),(152,181,274,381,232,394,190,331),(153,182,275,382,233,395,191,332),(154,157,276,383,234,396,192,333),(155,158,277,384,209,397,193,334),(156,159,278,385,210,398,194,335)], [(1,408,59,169),(2,409,60,170),(3,410,61,171),(4,411,62,172),(5,412,63,173),(6,413,64,174),(7,414,65,175),(8,415,66,176),(9,416,67,177),(10,391,68,178),(11,392,69,179),(12,393,70,180),(13,394,71,181),(14,395,72,182),(15,396,73,157),(16,397,74,158),(17,398,75,159),(18,399,76,160),(19,400,77,161),(20,401,78,162),(21,402,53,163),(22,403,54,164),(23,404,55,165),(24,405,56,166),(25,406,57,167),(26,407,58,168),(27,200,347,284),(28,201,348,285),(29,202,349,286),(30,203,350,261),(31,204,351,262),(32,205,352,263),(33,206,353,264),(34,207,354,265),(35,208,355,266),(36,183,356,267),(37,184,357,268),(38,185,358,269),(39,186,359,270),(40,187,360,271),(41,188,361,272),(42,189,362,273),(43,190,363,274),(44,191,364,275),(45,192,339,276),(46,193,340,277),(47,194,341,278),(48,195,342,279),(49,196,343,280),(50,197,344,281),(51,198,345,282),(52,199,346,283),(79,332,121,382),(80,333,122,383),(81,334,123,384),(82,335,124,385),(83,336,125,386),(84,337,126,387),(85,338,127,388),(86,313,128,389),(87,314,129,390),(88,315,130,365),(89,316,105,366),(90,317,106,367),(91,318,107,368),(92,319,108,369),(93,320,109,370),(94,321,110,371),(95,322,111,372),(96,323,112,373),(97,324,113,374),(98,325,114,375),(99,326,115,376),(100,327,116,377),(101,328,117,378),(102,329,118,379),(103,330,119,380),(104,331,120,381),(131,254,211,287),(132,255,212,288),(133,256,213,289),(134,257,214,290),(135,258,215,291),(136,259,216,292),(137,260,217,293),(138,235,218,294),(139,236,219,295),(140,237,220,296),(141,238,221,297),(142,239,222,298),(143,240,223,299),(144,241,224,300),(145,242,225,301),(146,243,226,302),(147,244,227,303),(148,245,228,304),(149,246,229,305),(150,247,230,306),(151,248,231,307),(152,249,232,308),(153,250,233,309),(154,251,234,310),(155,252,209,311),(156,253,210,312)]])
182 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 13A | ··· | 13L | 26A | ··· | 26AJ | 52A | ··· | 52X | 52Y | ··· | 52BT | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
182 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C13 | C26 | C26 | C26 | D4 | D4 | Q16 | D4×C13 | D4×C13 | C13×Q16 |
| kernel | Q16×C26 | C2×C104 | C13×Q16 | Q8×C26 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C52 | C2×C26 | C26 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 12 | 12 | 48 | 24 | 1 | 1 | 4 | 12 | 12 | 48 |
Matrix representation of Q16×C26 ►in GL4(𝔽313) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 92 | 238 | 0 | 0 |
| 92 | 221 | 0 | 0 |
| 0 | 0 | 253 | 60 |
| 0 | 0 | 253 | 253 |
| 312 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 105 | 68 |
| 0 | 0 | 68 | 208 |
G:=sub<GL(4,GF(313))| [36,0,0,0,0,36,0,0,0,0,27,0,0,0,0,27],[92,92,0,0,238,221,0,0,0,0,253,253,0,0,60,253],[312,0,0,0,2,1,0,0,0,0,105,68,0,0,68,208] >;
Q16×C26 in GAP, Magma, Sage, TeX
Q_{16}\times C_{26} % in TeX
G:=Group("Q16xC26"); // GroupNames label
G:=SmallGroup(416,195);
// by ID
G=gap.SmallGroup(416,195);
# by ID
G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1248,1273,1255,9364,4690,88]);
// Polycyclic
G:=Group<a,b,c|a^26=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations