metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.8Q8, C52.51D4, Dic13⋊1C8, C4.8Dic26, C26.6M4(2), C13⋊4(C4⋊C8), (C2×C8).1D13, C2.4(C8×D13), (C2×C104).1C2, C26.13(C2×C8), (C2×C4).91D26, C26.11(C4⋊C4), C22.9(C4×D13), C2.1(C8⋊D13), C4.26(C13⋊D4), (C2×Dic13).3C4, (C4×Dic13).5C2, (C2×C52).105C22, C2.1(C26.D4), (C2×C13⋊2C8).9C2, (C2×C26).30(C2×C4), SmallGroup(416,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.8Q8
G = < a,b,c | a52=1, b4=a26, c2=a39b2, ab=ba, cac-1=a25, cbc-1=a39b3 >
Smallest permutation representation of C52.8Q8
►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)
(1 411 167 234 27 385 193 260)(2 412 168 235 28 386 194 209)(3 413 169 236 29 387 195 210)(4 414 170 237 30 388 196 211)(5 415 171 238 31 389 197 212)(6 416 172 239 32 390 198 213)(7 365 173 240 33 391 199 214)(8 366 174 241 34 392 200 215)(9 367 175 242 35 393 201 216)(10 368 176 243 36 394 202 217)(11 369 177 244 37 395 203 218)(12 370 178 245 38 396 204 219)(13 371 179 246 39 397 205 220)(14 372 180 247 40 398 206 221)(15 373 181 248 41 399 207 222)(16 374 182 249 42 400 208 223)(17 375 183 250 43 401 157 224)(18 376 184 251 44 402 158 225)(19 377 185 252 45 403 159 226)(20 378 186 253 46 404 160 227)(21 379 187 254 47 405 161 228)(22 380 188 255 48 406 162 229)(23 381 189 256 49 407 163 230)(24 382 190 257 50 408 164 231)(25 383 191 258 51 409 165 232)(26 384 192 259 52 410 166 233)(53 330 285 135 79 356 311 109)(54 331 286 136 80 357 312 110)(55 332 287 137 81 358 261 111)(56 333 288 138 82 359 262 112)(57 334 289 139 83 360 263 113)(58 335 290 140 84 361 264 114)(59 336 291 141 85 362 265 115)(60 337 292 142 86 363 266 116)(61 338 293 143 87 364 267 117)(62 339 294 144 88 313 268 118)(63 340 295 145 89 314 269 119)(64 341 296 146 90 315 270 120)(65 342 297 147 91 316 271 121)(66 343 298 148 92 317 272 122)(67 344 299 149 93 318 273 123)(68 345 300 150 94 319 274 124)(69 346 301 151 95 320 275 125)(70 347 302 152 96 321 276 126)(71 348 303 153 97 322 277 127)(72 349 304 154 98 323 278 128)(73 350 305 155 99 324 279 129)(74 351 306 156 100 325 280 130)(75 352 307 105 101 326 281 131)(76 353 308 106 102 327 282 132)(77 354 309 107 103 328 283 133)(78 355 310 108 104 329 284 134)
(1 140 206 348)(2 113 207 321)(3 138 208 346)(4 111 157 319)(5 136 158 344)(6 109 159 317)(7 134 160 342)(8 107 161 315)(9 132 162 340)(10 105 163 313)(11 130 164 338)(12 155 165 363)(13 128 166 336)(14 153 167 361)(15 126 168 334)(16 151 169 359)(17 124 170 332)(18 149 171 357)(19 122 172 330)(20 147 173 355)(21 120 174 328)(22 145 175 353)(23 118 176 326)(24 143 177 351)(25 116 178 324)(26 141 179 349)(27 114 180 322)(28 139 181 347)(29 112 182 320)(30 137 183 345)(31 110 184 318)(32 135 185 343)(33 108 186 316)(34 133 187 341)(35 106 188 314)(36 131 189 339)(37 156 190 364)(38 129 191 337)(39 154 192 362)(40 127 193 335)(41 152 194 360)(42 125 195 333)(43 150 196 358)(44 123 197 331)(45 148 198 356)(46 121 199 329)(47 146 200 354)(48 119 201 327)(49 144 202 352)(50 117 203 325)(51 142 204 350)(52 115 205 323)(53 416 272 226)(54 389 273 251)(55 414 274 224)(56 387 275 249)(57 412 276 222)(58 385 277 247)(59 410 278 220)(60 383 279 245)(61 408 280 218)(62 381 281 243)(63 406 282 216)(64 379 283 241)(65 404 284 214)(66 377 285 239)(67 402 286 212)(68 375 287 237)(69 400 288 210)(70 373 289 235)(71 398 290 260)(72 371 291 233)(73 396 292 258)(74 369 293 231)(75 394 294 256)(76 367 295 229)(77 392 296 254)(78 365 297 227)(79 390 298 252)(80 415 299 225)(81 388 300 250)(82 413 301 223)(83 386 302 248)(84 411 303 221)(85 384 304 246)(86 409 305 219)(87 382 306 244)(88 407 307 217)(89 380 308 242)(90 405 309 215)(91 378 310 240)(92 403 311 213)(93 376 312 238)(94 401 261 211)(95 374 262 236)(96 399 263 209)(97 372 264 234)(98 397 265 259)(99 370 266 232)(100 395 267 257)(101 368 268 230)(102 393 269 255)(103 366 270 228)(104 391 271 253)
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), (1,411,167,234,27,385,193,260)(2,412,168,235,28,386,194,209)(3,413,169,236,29,387,195,210)(4,414,170,237,30,388,196,211)(5,415,171,238,31,389,197,212)(6,416,172,239,32,390,198,213)(7,365,173,240,33,391,199,214)(8,366,174,241,34,392,200,215)(9,367,175,242,35,393,201,216)(10,368,176,243,36,394,202,217)(11,369,177,244,37,395,203,218)(12,370,178,245,38,396,204,219)(13,371,179,246,39,397,205,220)(14,372,180,247,40,398,206,221)(15,373,181,248,41,399,207,222)(16,374,182,249,42,400,208,223)(17,375,183,250,43,401,157,224)(18,376,184,251,44,402,158,225)(19,377,185,252,45,403,159,226)(20,378,186,253,46,404,160,227)(21,379,187,254,47,405,161,228)(22,380,188,255,48,406,162,229)(23,381,189,256,49,407,163,230)(24,382,190,257,50,408,164,231)(25,383,191,258,51,409,165,232)(26,384,192,259,52,410,166,233)(53,330,285,135,79,356,311,109)(54,331,286,136,80,357,312,110)(55,332,287,137,81,358,261,111)(56,333,288,138,82,359,262,112)(57,334,289,139,83,360,263,113)(58,335,290,140,84,361,264,114)(59,336,291,141,85,362,265,115)(60,337,292,142,86,363,266,116)(61,338,293,143,87,364,267,117)(62,339,294,144,88,313,268,118)(63,340,295,145,89,314,269,119)(64,341,296,146,90,315,270,120)(65,342,297,147,91,316,271,121)(66,343,298,148,92,317,272,122)(67,344,299,149,93,318,273,123)(68,345,300,150,94,319,274,124)(69,346,301,151,95,320,275,125)(70,347,302,152,96,321,276,126)(71,348,303,153,97,322,277,127)(72,349,304,154,98,323,278,128)(73,350,305,155,99,324,279,129)(74,351,306,156,100,325,280,130)(75,352,307,105,101,326,281,131)(76,353,308,106,102,327,282,132)(77,354,309,107,103,328,283,133)(78,355,310,108,104,329,284,134), (1,140,206,348)(2,113,207,321)(3,138,208,346)(4,111,157,319)(5,136,158,344)(6,109,159,317)(7,134,160,342)(8,107,161,315)(9,132,162,340)(10,105,163,313)(11,130,164,338)(12,155,165,363)(13,128,166,336)(14,153,167,361)(15,126,168,334)(16,151,169,359)(17,124,170,332)(18,149,171,357)(19,122,172,330)(20,147,173,355)(21,120,174,328)(22,145,175,353)(23,118,176,326)(24,143,177,351)(25,116,178,324)(26,141,179,349)(27,114,180,322)(28,139,181,347)(29,112,182,320)(30,137,183,345)(31,110,184,318)(32,135,185,343)(33,108,186,316)(34,133,187,341)(35,106,188,314)(36,131,189,339)(37,156,190,364)(38,129,191,337)(39,154,192,362)(40,127,193,335)(41,152,194,360)(42,125,195,333)(43,150,196,358)(44,123,197,331)(45,148,198,356)(46,121,199,329)(47,146,200,354)(48,119,201,327)(49,144,202,352)(50,117,203,325)(51,142,204,350)(52,115,205,323)(53,416,272,226)(54,389,273,251)(55,414,274,224)(56,387,275,249)(57,412,276,222)(58,385,277,247)(59,410,278,220)(60,383,279,245)(61,408,280,218)(62,381,281,243)(63,406,282,216)(64,379,283,241)(65,404,284,214)(66,377,285,239)(67,402,286,212)(68,375,287,237)(69,400,288,210)(70,373,289,235)(71,398,290,260)(72,371,291,233)(73,396,292,258)(74,369,293,231)(75,394,294,256)(76,367,295,229)(77,392,296,254)(78,365,297,227)(79,390,298,252)(80,415,299,225)(81,388,300,250)(82,413,301,223)(83,386,302,248)(84,411,303,221)(85,384,304,246)(86,409,305,219)(87,382,306,244)(88,407,307,217)(89,380,308,242)(90,405,309,215)(91,378,310,240)(92,403,311,213)(93,376,312,238)(94,401,261,211)(95,374,262,236)(96,399,263,209)(97,372,264,234)(98,397,265,259)(99,370,266,232)(100,395,267,257)(101,368,268,230)(102,393,269,255)(103,366,270,228)(104,391,271,253)>;
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), (1,411,167,234,27,385,193,260)(2,412,168,235,28,386,194,209)(3,413,169,236,29,387,195,210)(4,414,170,237,30,388,196,211)(5,415,171,238,31,389,197,212)(6,416,172,239,32,390,198,213)(7,365,173,240,33,391,199,214)(8,366,174,241,34,392,200,215)(9,367,175,242,35,393,201,216)(10,368,176,243,36,394,202,217)(11,369,177,244,37,395,203,218)(12,370,178,245,38,396,204,219)(13,371,179,246,39,397,205,220)(14,372,180,247,40,398,206,221)(15,373,181,248,41,399,207,222)(16,374,182,249,42,400,208,223)(17,375,183,250,43,401,157,224)(18,376,184,251,44,402,158,225)(19,377,185,252,45,403,159,226)(20,378,186,253,46,404,160,227)(21,379,187,254,47,405,161,228)(22,380,188,255,48,406,162,229)(23,381,189,256,49,407,163,230)(24,382,190,257,50,408,164,231)(25,383,191,258,51,409,165,232)(26,384,192,259,52,410,166,233)(53,330,285,135,79,356,311,109)(54,331,286,136,80,357,312,110)(55,332,287,137,81,358,261,111)(56,333,288,138,82,359,262,112)(57,334,289,139,83,360,263,113)(58,335,290,140,84,361,264,114)(59,336,291,141,85,362,265,115)(60,337,292,142,86,363,266,116)(61,338,293,143,87,364,267,117)(62,339,294,144,88,313,268,118)(63,340,295,145,89,314,269,119)(64,341,296,146,90,315,270,120)(65,342,297,147,91,316,271,121)(66,343,298,148,92,317,272,122)(67,344,299,149,93,318,273,123)(68,345,300,150,94,319,274,124)(69,346,301,151,95,320,275,125)(70,347,302,152,96,321,276,126)(71,348,303,153,97,322,277,127)(72,349,304,154,98,323,278,128)(73,350,305,155,99,324,279,129)(74,351,306,156,100,325,280,130)(75,352,307,105,101,326,281,131)(76,353,308,106,102,327,282,132)(77,354,309,107,103,328,283,133)(78,355,310,108,104,329,284,134), (1,140,206,348)(2,113,207,321)(3,138,208,346)(4,111,157,319)(5,136,158,344)(6,109,159,317)(7,134,160,342)(8,107,161,315)(9,132,162,340)(10,105,163,313)(11,130,164,338)(12,155,165,363)(13,128,166,336)(14,153,167,361)(15,126,168,334)(16,151,169,359)(17,124,170,332)(18,149,171,357)(19,122,172,330)(20,147,173,355)(21,120,174,328)(22,145,175,353)(23,118,176,326)(24,143,177,351)(25,116,178,324)(26,141,179,349)(27,114,180,322)(28,139,181,347)(29,112,182,320)(30,137,183,345)(31,110,184,318)(32,135,185,343)(33,108,186,316)(34,133,187,341)(35,106,188,314)(36,131,189,339)(37,156,190,364)(38,129,191,337)(39,154,192,362)(40,127,193,335)(41,152,194,360)(42,125,195,333)(43,150,196,358)(44,123,197,331)(45,148,198,356)(46,121,199,329)(47,146,200,354)(48,119,201,327)(49,144,202,352)(50,117,203,325)(51,142,204,350)(52,115,205,323)(53,416,272,226)(54,389,273,251)(55,414,274,224)(56,387,275,249)(57,412,276,222)(58,385,277,247)(59,410,278,220)(60,383,279,245)(61,408,280,218)(62,381,281,243)(63,406,282,216)(64,379,283,241)(65,404,284,214)(66,377,285,239)(67,402,286,212)(68,375,287,237)(69,400,288,210)(70,373,289,235)(71,398,290,260)(72,371,291,233)(73,396,292,258)(74,369,293,231)(75,394,294,256)(76,367,295,229)(77,392,296,254)(78,365,297,227)(79,390,298,252)(80,415,299,225)(81,388,300,250)(82,413,301,223)(83,386,302,248)(84,411,303,221)(85,384,304,246)(86,409,305,219)(87,382,306,244)(88,407,307,217)(89,380,308,242)(90,405,309,215)(91,378,310,240)(92,403,311,213)(93,376,312,238)(94,401,261,211)(95,374,262,236)(96,399,263,209)(97,372,264,234)(98,397,265,259)(99,370,266,232)(100,395,267,257)(101,368,268,230)(102,393,269,255)(103,366,270,228)(104,391,271,253) );
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)], [(1,411,167,234,27,385,193,260),(2,412,168,235,28,386,194,209),(3,413,169,236,29,387,195,210),(4,414,170,237,30,388,196,211),(5,415,171,238,31,389,197,212),(6,416,172,239,32,390,198,213),(7,365,173,240,33,391,199,214),(8,366,174,241,34,392,200,215),(9,367,175,242,35,393,201,216),(10,368,176,243,36,394,202,217),(11,369,177,244,37,395,203,218),(12,370,178,245,38,396,204,219),(13,371,179,246,39,397,205,220),(14,372,180,247,40,398,206,221),(15,373,181,248,41,399,207,222),(16,374,182,249,42,400,208,223),(17,375,183,250,43,401,157,224),(18,376,184,251,44,402,158,225),(19,377,185,252,45,403,159,226),(20,378,186,253,46,404,160,227),(21,379,187,254,47,405,161,228),(22,380,188,255,48,406,162,229),(23,381,189,256,49,407,163,230),(24,382,190,257,50,408,164,231),(25,383,191,258,51,409,165,232),(26,384,192,259,52,410,166,233),(53,330,285,135,79,356,311,109),(54,331,286,136,80,357,312,110),(55,332,287,137,81,358,261,111),(56,333,288,138,82,359,262,112),(57,334,289,139,83,360,263,113),(58,335,290,140,84,361,264,114),(59,336,291,141,85,362,265,115),(60,337,292,142,86,363,266,116),(61,338,293,143,87,364,267,117),(62,339,294,144,88,313,268,118),(63,340,295,145,89,314,269,119),(64,341,296,146,90,315,270,120),(65,342,297,147,91,316,271,121),(66,343,298,148,92,317,272,122),(67,344,299,149,93,318,273,123),(68,345,300,150,94,319,274,124),(69,346,301,151,95,320,275,125),(70,347,302,152,96,321,276,126),(71,348,303,153,97,322,277,127),(72,349,304,154,98,323,278,128),(73,350,305,155,99,324,279,129),(74,351,306,156,100,325,280,130),(75,352,307,105,101,326,281,131),(76,353,308,106,102,327,282,132),(77,354,309,107,103,328,283,133),(78,355,310,108,104,329,284,134)], [(1,140,206,348),(2,113,207,321),(3,138,208,346),(4,111,157,319),(5,136,158,344),(6,109,159,317),(7,134,160,342),(8,107,161,315),(9,132,162,340),(10,105,163,313),(11,130,164,338),(12,155,165,363),(13,128,166,336),(14,153,167,361),(15,126,168,334),(16,151,169,359),(17,124,170,332),(18,149,171,357),(19,122,172,330),(20,147,173,355),(21,120,174,328),(22,145,175,353),(23,118,176,326),(24,143,177,351),(25,116,178,324),(26,141,179,349),(27,114,180,322),(28,139,181,347),(29,112,182,320),(30,137,183,345),(31,110,184,318),(32,135,185,343),(33,108,186,316),(34,133,187,341),(35,106,188,314),(36,131,189,339),(37,156,190,364),(38,129,191,337),(39,154,192,362),(40,127,193,335),(41,152,194,360),(42,125,195,333),(43,150,196,358),(44,123,197,331),(45,148,198,356),(46,121,199,329),(47,146,200,354),(48,119,201,327),(49,144,202,352),(50,117,203,325),(51,142,204,350),(52,115,205,323),(53,416,272,226),(54,389,273,251),(55,414,274,224),(56,387,275,249),(57,412,276,222),(58,385,277,247),(59,410,278,220),(60,383,279,245),(61,408,280,218),(62,381,281,243),(63,406,282,216),(64,379,283,241),(65,404,284,214),(66,377,285,239),(67,402,286,212),(68,375,287,237),(69,400,288,210),(70,373,289,235),(71,398,290,260),(72,371,291,233),(73,396,292,258),(74,369,293,231),(75,394,294,256),(76,367,295,229),(77,392,296,254),(78,365,297,227),(79,390,298,252),(80,415,299,225),(81,388,300,250),(82,413,301,223),(83,386,302,248),(84,411,303,221),(85,384,304,246),(86,409,305,219),(87,382,306,244),(88,407,307,217),(89,380,308,242),(90,405,309,215),(91,378,310,240),(92,403,311,213),(93,376,312,238),(94,401,261,211),(95,374,262,236),(96,399,263,209),(97,372,264,234),(98,397,265,259),(99,370,266,232),(100,395,267,257),(101,368,268,230),(102,393,269,255),(103,366,270,228),(104,391,271,253)]])
116 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 26 | 26 | 26 | 26 | 2 | 2 | 2 | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | D4 | Q8 | M4(2) | D13 | D26 | Dic26 | C13⋊D4 | C4×D13 | C8×D13 | C8⋊D13 |
| kernel | C52.8Q8 | C2×C13⋊2C8 | C4×Dic13 | C2×C104 | C2×Dic13 | Dic13 | C52 | C52 | C26 | C2×C8 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 2 | 6 | 6 | 12 | 12 | 12 | 24 | 24 |
Matrix representation of C52.8Q8 ►in GL3(𝔽313) generated by
| 25 | 0 | 0 |
| 0 | 25 | 19 |
| 0 | 294 | 236 |
| 308 | 0 | 0 |
| 0 | 265 | 111 |
| 0 | 202 | 48 |
| 1 | 0 | 0 |
| 0 | 242 | 7 |
| 0 | 174 | 71 |
G:=sub<GL(3,GF(313))| [25,0,0,0,25,294,0,19,236],[308,0,0,0,265,202,0,111,48],[1,0,0,0,242,174,0,7,71] >;
C52.8Q8 in GAP, Magma, Sage, TeX
C_{52}._8Q_8 % in TeX
G:=Group("C52.8Q8"); // GroupNames label
G:=SmallGroup(416,21);
// by ID
G=gap.SmallGroup(416,21);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,121,31,86,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=1,b^4=a^26,c^2=a^39*b^2,a*b=b*a,c*a*c^-1=a^25,c*b*c^-1=a^39*b^3>;
// generators/relations
Export