metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.44D4, C26.1Q16, Dic26⋊5C4, C2.1Dic52, C26.1SD16, C22.7D52, (C2×C8).2D13, C4.7(C4×D13), C52.38(C2×C4), (C2×C104).2C2, (C2×C4).67D26, (C2×C26).12D4, C52⋊3C4.1C2, C13⋊3(Q8⋊C4), C2.1(C104⋊C2), C4.19(C13⋊D4), (C2×C52).80C22, (C2×Dic26).1C2, C2.7(D26⋊C4), C26.16(C22⋊C4), SmallGroup(416,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.44D4
G = < a,b,c | a52=b4=1, c2=a26, bab-1=cac-1=a-1, cbc-1=a13b-1 >
Smallest permutation representation of C52.44D4
►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)
(1 248 55 381)(2 247 56 380)(3 246 57 379)(4 245 58 378)(5 244 59 377)(6 243 60 376)(7 242 61 375)(8 241 62 374)(9 240 63 373)(10 239 64 372)(11 238 65 371)(12 237 66 370)(13 236 67 369)(14 235 68 368)(15 234 69 367)(16 233 70 366)(17 232 71 365)(18 231 72 416)(19 230 73 415)(20 229 74 414)(21 228 75 413)(22 227 76 412)(23 226 77 411)(24 225 78 410)(25 224 79 409)(26 223 80 408)(27 222 81 407)(28 221 82 406)(29 220 83 405)(30 219 84 404)(31 218 85 403)(32 217 86 402)(33 216 87 401)(34 215 88 400)(35 214 89 399)(36 213 90 398)(37 212 91 397)(38 211 92 396)(39 210 93 395)(40 209 94 394)(41 260 95 393)(42 259 96 392)(43 258 97 391)(44 257 98 390)(45 256 99 389)(46 255 100 388)(47 254 101 387)(48 253 102 386)(49 252 103 385)(50 251 104 384)(51 250 53 383)(52 249 54 382)(105 340 298 158)(106 339 299 157)(107 338 300 208)(108 337 301 207)(109 336 302 206)(110 335 303 205)(111 334 304 204)(112 333 305 203)(113 332 306 202)(114 331 307 201)(115 330 308 200)(116 329 309 199)(117 328 310 198)(118 327 311 197)(119 326 312 196)(120 325 261 195)(121 324 262 194)(122 323 263 193)(123 322 264 192)(124 321 265 191)(125 320 266 190)(126 319 267 189)(127 318 268 188)(128 317 269 187)(129 316 270 186)(130 315 271 185)(131 314 272 184)(132 313 273 183)(133 364 274 182)(134 363 275 181)(135 362 276 180)(136 361 277 179)(137 360 278 178)(138 359 279 177)(139 358 280 176)(140 357 281 175)(141 356 282 174)(142 355 283 173)(143 354 284 172)(144 353 285 171)(145 352 286 170)(146 351 287 169)(147 350 288 168)(148 349 289 167)(149 348 290 166)(150 347 291 165)(151 346 292 164)(152 345 293 163)(153 344 294 162)(154 343 295 161)(155 342 296 160)(156 341 297 159)
(1 325 27 351)(2 324 28 350)(3 323 29 349)(4 322 30 348)(5 321 31 347)(6 320 32 346)(7 319 33 345)(8 318 34 344)(9 317 35 343)(10 316 36 342)(11 315 37 341)(12 314 38 340)(13 313 39 339)(14 364 40 338)(15 363 41 337)(16 362 42 336)(17 361 43 335)(18 360 44 334)(19 359 45 333)(20 358 46 332)(21 357 47 331)(22 356 48 330)(23 355 49 329)(24 354 50 328)(25 353 51 327)(26 352 52 326)(53 197 79 171)(54 196 80 170)(55 195 81 169)(56 194 82 168)(57 193 83 167)(58 192 84 166)(59 191 85 165)(60 190 86 164)(61 189 87 163)(62 188 88 162)(63 187 89 161)(64 186 90 160)(65 185 91 159)(66 184 92 158)(67 183 93 157)(68 182 94 208)(69 181 95 207)(70 180 96 206)(71 179 97 205)(72 178 98 204)(73 177 99 203)(74 176 100 202)(75 175 101 201)(76 174 102 200)(77 173 103 199)(78 172 104 198)(105 224 131 250)(106 223 132 249)(107 222 133 248)(108 221 134 247)(109 220 135 246)(110 219 136 245)(111 218 137 244)(112 217 138 243)(113 216 139 242)(114 215 140 241)(115 214 141 240)(116 213 142 239)(117 212 143 238)(118 211 144 237)(119 210 145 236)(120 209 146 235)(121 260 147 234)(122 259 148 233)(123 258 149 232)(124 257 150 231)(125 256 151 230)(126 255 152 229)(127 254 153 228)(128 253 154 227)(129 252 155 226)(130 251 156 225)(261 394 287 368)(262 393 288 367)(263 392 289 366)(264 391 290 365)(265 390 291 416)(266 389 292 415)(267 388 293 414)(268 387 294 413)(269 386 295 412)(270 385 296 411)(271 384 297 410)(272 383 298 409)(273 382 299 408)(274 381 300 407)(275 380 301 406)(276 379 302 405)(277 378 303 404)(278 377 304 403)(279 376 305 402)(280 375 306 401)(281 374 307 400)(282 373 308 399)(283 372 309 398)(284 371 310 397)(285 370 311 396)(286 369 312 395)
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), (1,248,55,381)(2,247,56,380)(3,246,57,379)(4,245,58,378)(5,244,59,377)(6,243,60,376)(7,242,61,375)(8,241,62,374)(9,240,63,373)(10,239,64,372)(11,238,65,371)(12,237,66,370)(13,236,67,369)(14,235,68,368)(15,234,69,367)(16,233,70,366)(17,232,71,365)(18,231,72,416)(19,230,73,415)(20,229,74,414)(21,228,75,413)(22,227,76,412)(23,226,77,411)(24,225,78,410)(25,224,79,409)(26,223,80,408)(27,222,81,407)(28,221,82,406)(29,220,83,405)(30,219,84,404)(31,218,85,403)(32,217,86,402)(33,216,87,401)(34,215,88,400)(35,214,89,399)(36,213,90,398)(37,212,91,397)(38,211,92,396)(39,210,93,395)(40,209,94,394)(41,260,95,393)(42,259,96,392)(43,258,97,391)(44,257,98,390)(45,256,99,389)(46,255,100,388)(47,254,101,387)(48,253,102,386)(49,252,103,385)(50,251,104,384)(51,250,53,383)(52,249,54,382)(105,340,298,158)(106,339,299,157)(107,338,300,208)(108,337,301,207)(109,336,302,206)(110,335,303,205)(111,334,304,204)(112,333,305,203)(113,332,306,202)(114,331,307,201)(115,330,308,200)(116,329,309,199)(117,328,310,198)(118,327,311,197)(119,326,312,196)(120,325,261,195)(121,324,262,194)(122,323,263,193)(123,322,264,192)(124,321,265,191)(125,320,266,190)(126,319,267,189)(127,318,268,188)(128,317,269,187)(129,316,270,186)(130,315,271,185)(131,314,272,184)(132,313,273,183)(133,364,274,182)(134,363,275,181)(135,362,276,180)(136,361,277,179)(137,360,278,178)(138,359,279,177)(139,358,280,176)(140,357,281,175)(141,356,282,174)(142,355,283,173)(143,354,284,172)(144,353,285,171)(145,352,286,170)(146,351,287,169)(147,350,288,168)(148,349,289,167)(149,348,290,166)(150,347,291,165)(151,346,292,164)(152,345,293,163)(153,344,294,162)(154,343,295,161)(155,342,296,160)(156,341,297,159), (1,325,27,351)(2,324,28,350)(3,323,29,349)(4,322,30,348)(5,321,31,347)(6,320,32,346)(7,319,33,345)(8,318,34,344)(9,317,35,343)(10,316,36,342)(11,315,37,341)(12,314,38,340)(13,313,39,339)(14,364,40,338)(15,363,41,337)(16,362,42,336)(17,361,43,335)(18,360,44,334)(19,359,45,333)(20,358,46,332)(21,357,47,331)(22,356,48,330)(23,355,49,329)(24,354,50,328)(25,353,51,327)(26,352,52,326)(53,197,79,171)(54,196,80,170)(55,195,81,169)(56,194,82,168)(57,193,83,167)(58,192,84,166)(59,191,85,165)(60,190,86,164)(61,189,87,163)(62,188,88,162)(63,187,89,161)(64,186,90,160)(65,185,91,159)(66,184,92,158)(67,183,93,157)(68,182,94,208)(69,181,95,207)(70,180,96,206)(71,179,97,205)(72,178,98,204)(73,177,99,203)(74,176,100,202)(75,175,101,201)(76,174,102,200)(77,173,103,199)(78,172,104,198)(105,224,131,250)(106,223,132,249)(107,222,133,248)(108,221,134,247)(109,220,135,246)(110,219,136,245)(111,218,137,244)(112,217,138,243)(113,216,139,242)(114,215,140,241)(115,214,141,240)(116,213,142,239)(117,212,143,238)(118,211,144,237)(119,210,145,236)(120,209,146,235)(121,260,147,234)(122,259,148,233)(123,258,149,232)(124,257,150,231)(125,256,151,230)(126,255,152,229)(127,254,153,228)(128,253,154,227)(129,252,155,226)(130,251,156,225)(261,394,287,368)(262,393,288,367)(263,392,289,366)(264,391,290,365)(265,390,291,416)(266,389,292,415)(267,388,293,414)(268,387,294,413)(269,386,295,412)(270,385,296,411)(271,384,297,410)(272,383,298,409)(273,382,299,408)(274,381,300,407)(275,380,301,406)(276,379,302,405)(277,378,303,404)(278,377,304,403)(279,376,305,402)(280,375,306,401)(281,374,307,400)(282,373,308,399)(283,372,309,398)(284,371,310,397)(285,370,311,396)(286,369,312,395)>;
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), (1,248,55,381)(2,247,56,380)(3,246,57,379)(4,245,58,378)(5,244,59,377)(6,243,60,376)(7,242,61,375)(8,241,62,374)(9,240,63,373)(10,239,64,372)(11,238,65,371)(12,237,66,370)(13,236,67,369)(14,235,68,368)(15,234,69,367)(16,233,70,366)(17,232,71,365)(18,231,72,416)(19,230,73,415)(20,229,74,414)(21,228,75,413)(22,227,76,412)(23,226,77,411)(24,225,78,410)(25,224,79,409)(26,223,80,408)(27,222,81,407)(28,221,82,406)(29,220,83,405)(30,219,84,404)(31,218,85,403)(32,217,86,402)(33,216,87,401)(34,215,88,400)(35,214,89,399)(36,213,90,398)(37,212,91,397)(38,211,92,396)(39,210,93,395)(40,209,94,394)(41,260,95,393)(42,259,96,392)(43,258,97,391)(44,257,98,390)(45,256,99,389)(46,255,100,388)(47,254,101,387)(48,253,102,386)(49,252,103,385)(50,251,104,384)(51,250,53,383)(52,249,54,382)(105,340,298,158)(106,339,299,157)(107,338,300,208)(108,337,301,207)(109,336,302,206)(110,335,303,205)(111,334,304,204)(112,333,305,203)(113,332,306,202)(114,331,307,201)(115,330,308,200)(116,329,309,199)(117,328,310,198)(118,327,311,197)(119,326,312,196)(120,325,261,195)(121,324,262,194)(122,323,263,193)(123,322,264,192)(124,321,265,191)(125,320,266,190)(126,319,267,189)(127,318,268,188)(128,317,269,187)(129,316,270,186)(130,315,271,185)(131,314,272,184)(132,313,273,183)(133,364,274,182)(134,363,275,181)(135,362,276,180)(136,361,277,179)(137,360,278,178)(138,359,279,177)(139,358,280,176)(140,357,281,175)(141,356,282,174)(142,355,283,173)(143,354,284,172)(144,353,285,171)(145,352,286,170)(146,351,287,169)(147,350,288,168)(148,349,289,167)(149,348,290,166)(150,347,291,165)(151,346,292,164)(152,345,293,163)(153,344,294,162)(154,343,295,161)(155,342,296,160)(156,341,297,159), (1,325,27,351)(2,324,28,350)(3,323,29,349)(4,322,30,348)(5,321,31,347)(6,320,32,346)(7,319,33,345)(8,318,34,344)(9,317,35,343)(10,316,36,342)(11,315,37,341)(12,314,38,340)(13,313,39,339)(14,364,40,338)(15,363,41,337)(16,362,42,336)(17,361,43,335)(18,360,44,334)(19,359,45,333)(20,358,46,332)(21,357,47,331)(22,356,48,330)(23,355,49,329)(24,354,50,328)(25,353,51,327)(26,352,52,326)(53,197,79,171)(54,196,80,170)(55,195,81,169)(56,194,82,168)(57,193,83,167)(58,192,84,166)(59,191,85,165)(60,190,86,164)(61,189,87,163)(62,188,88,162)(63,187,89,161)(64,186,90,160)(65,185,91,159)(66,184,92,158)(67,183,93,157)(68,182,94,208)(69,181,95,207)(70,180,96,206)(71,179,97,205)(72,178,98,204)(73,177,99,203)(74,176,100,202)(75,175,101,201)(76,174,102,200)(77,173,103,199)(78,172,104,198)(105,224,131,250)(106,223,132,249)(107,222,133,248)(108,221,134,247)(109,220,135,246)(110,219,136,245)(111,218,137,244)(112,217,138,243)(113,216,139,242)(114,215,140,241)(115,214,141,240)(116,213,142,239)(117,212,143,238)(118,211,144,237)(119,210,145,236)(120,209,146,235)(121,260,147,234)(122,259,148,233)(123,258,149,232)(124,257,150,231)(125,256,151,230)(126,255,152,229)(127,254,153,228)(128,253,154,227)(129,252,155,226)(130,251,156,225)(261,394,287,368)(262,393,288,367)(263,392,289,366)(264,391,290,365)(265,390,291,416)(266,389,292,415)(267,388,293,414)(268,387,294,413)(269,386,295,412)(270,385,296,411)(271,384,297,410)(272,383,298,409)(273,382,299,408)(274,381,300,407)(275,380,301,406)(276,379,302,405)(277,378,303,404)(278,377,304,403)(279,376,305,402)(280,375,306,401)(281,374,307,400)(282,373,308,399)(283,372,309,398)(284,371,310,397)(285,370,311,396)(286,369,312,395) );
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)], [(1,248,55,381),(2,247,56,380),(3,246,57,379),(4,245,58,378),(5,244,59,377),(6,243,60,376),(7,242,61,375),(8,241,62,374),(9,240,63,373),(10,239,64,372),(11,238,65,371),(12,237,66,370),(13,236,67,369),(14,235,68,368),(15,234,69,367),(16,233,70,366),(17,232,71,365),(18,231,72,416),(19,230,73,415),(20,229,74,414),(21,228,75,413),(22,227,76,412),(23,226,77,411),(24,225,78,410),(25,224,79,409),(26,223,80,408),(27,222,81,407),(28,221,82,406),(29,220,83,405),(30,219,84,404),(31,218,85,403),(32,217,86,402),(33,216,87,401),(34,215,88,400),(35,214,89,399),(36,213,90,398),(37,212,91,397),(38,211,92,396),(39,210,93,395),(40,209,94,394),(41,260,95,393),(42,259,96,392),(43,258,97,391),(44,257,98,390),(45,256,99,389),(46,255,100,388),(47,254,101,387),(48,253,102,386),(49,252,103,385),(50,251,104,384),(51,250,53,383),(52,249,54,382),(105,340,298,158),(106,339,299,157),(107,338,300,208),(108,337,301,207),(109,336,302,206),(110,335,303,205),(111,334,304,204),(112,333,305,203),(113,332,306,202),(114,331,307,201),(115,330,308,200),(116,329,309,199),(117,328,310,198),(118,327,311,197),(119,326,312,196),(120,325,261,195),(121,324,262,194),(122,323,263,193),(123,322,264,192),(124,321,265,191),(125,320,266,190),(126,319,267,189),(127,318,268,188),(128,317,269,187),(129,316,270,186),(130,315,271,185),(131,314,272,184),(132,313,273,183),(133,364,274,182),(134,363,275,181),(135,362,276,180),(136,361,277,179),(137,360,278,178),(138,359,279,177),(139,358,280,176),(140,357,281,175),(141,356,282,174),(142,355,283,173),(143,354,284,172),(144,353,285,171),(145,352,286,170),(146,351,287,169),(147,350,288,168),(148,349,289,167),(149,348,290,166),(150,347,291,165),(151,346,292,164),(152,345,293,163),(153,344,294,162),(154,343,295,161),(155,342,296,160),(156,341,297,159)], [(1,325,27,351),(2,324,28,350),(3,323,29,349),(4,322,30,348),(5,321,31,347),(6,320,32,346),(7,319,33,345),(8,318,34,344),(9,317,35,343),(10,316,36,342),(11,315,37,341),(12,314,38,340),(13,313,39,339),(14,364,40,338),(15,363,41,337),(16,362,42,336),(17,361,43,335),(18,360,44,334),(19,359,45,333),(20,358,46,332),(21,357,47,331),(22,356,48,330),(23,355,49,329),(24,354,50,328),(25,353,51,327),(26,352,52,326),(53,197,79,171),(54,196,80,170),(55,195,81,169),(56,194,82,168),(57,193,83,167),(58,192,84,166),(59,191,85,165),(60,190,86,164),(61,189,87,163),(62,188,88,162),(63,187,89,161),(64,186,90,160),(65,185,91,159),(66,184,92,158),(67,183,93,157),(68,182,94,208),(69,181,95,207),(70,180,96,206),(71,179,97,205),(72,178,98,204),(73,177,99,203),(74,176,100,202),(75,175,101,201),(76,174,102,200),(77,173,103,199),(78,172,104,198),(105,224,131,250),(106,223,132,249),(107,222,133,248),(108,221,134,247),(109,220,135,246),(110,219,136,245),(111,218,137,244),(112,217,138,243),(113,216,139,242),(114,215,140,241),(115,214,141,240),(116,213,142,239),(117,212,143,238),(118,211,144,237),(119,210,145,236),(120,209,146,235),(121,260,147,234),(122,259,148,233),(123,258,149,232),(124,257,150,231),(125,256,151,230),(126,255,152,229),(127,254,153,228),(128,253,154,227),(129,252,155,226),(130,251,156,225),(261,394,287,368),(262,393,288,367),(263,392,289,366),(264,391,290,365),(265,390,291,416),(266,389,292,415),(267,388,293,414),(268,387,294,413),(269,386,295,412),(270,385,296,411),(271,384,297,410),(272,383,298,409),(273,382,299,408),(274,381,300,407),(275,380,301,406),(276,379,302,405),(277,378,303,404),(278,377,304,403),(279,376,305,402),(280,375,306,401),(281,374,307,400),(282,373,308,399),(283,372,309,398),(284,371,310,397),(285,370,311,396),(286,369,312,395)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 52 | 52 | 52 | 52 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | SD16 | Q16 | D13 | D26 | C4×D13 | C13⋊D4 | D52 | C104⋊C2 | Dic52 |
| kernel | C52.44D4 | C52⋊3C4 | C2×C104 | C2×Dic26 | Dic26 | C52 | C2×C26 | C26 | C26 | C2×C8 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 24 | 24 |
Matrix representation of C52.44D4 ►in GL4(𝔽313) generated by
| 304 | 227 | 0 | 0 |
| 291 | 207 | 0 | 0 |
| 0 | 0 | 26 | 261 |
| 0 | 0 | 179 | 256 |
| 132 | 241 | 0 | 0 |
| 29 | 181 | 0 | 0 |
| 0 | 0 | 168 | 16 |
| 0 | 0 | 251 | 145 |
| 175 | 30 | 0 | 0 |
| 106 | 138 | 0 | 0 |
| 0 | 0 | 95 | 111 |
| 0 | 0 | 305 | 218 |
G:=sub<GL(4,GF(313))| [304,291,0,0,227,207,0,0,0,0,26,179,0,0,261,256],[132,29,0,0,241,181,0,0,0,0,168,251,0,0,16,145],[175,106,0,0,30,138,0,0,0,0,95,305,0,0,111,218] >;
C52.44D4 in GAP, Magma, Sage, TeX
C_{52}._{44}D_4 % in TeX
G:=Group("C52.44D4"); // GroupNames label
G:=SmallGroup(416,23);
// by ID
G=gap.SmallGroup(416,23);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,73,79,362,86,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^4=1,c^2=a^26,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^13*b^-1>;
// generators/relations
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