metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.6Q8, C4.6Dic26, C42.5D13, (C4×C52).3C2, C26.3(C2×Q8), (C2×C4).74D26, C52⋊3C4.5C2, C26.2(C4○D4), C2.5(C2×Dic26), C13⋊1(C42.C2), (C2×C52).72C22, (C2×C26).11C23, C26.D4.1C2, C2.6(D52⋊5C2), (C2×Dic13).2C22, C22.35(C22×D13), SmallGroup(416,91)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.6Q8
G = < a,b,c | a52=b4=1, c2=a26b2, ab=ba, cac-1=a-1, cbc-1=a26b-1 >
Subgroups: 296 in 56 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, C2×C4, C13, C42, C4⋊C4, C26, C26, C42.C2, Dic13, C52, C52, C2×C26, C2×Dic13, C2×C52, C2×C52, C26.D4, C52⋊3C4, C4×C52, C52.6Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, D13, C42.C2, D26, Dic26, C22×D13, C2×Dic26, D52⋊5C2, C52.6Q8
►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)
(1 187 111 291)(2 188 112 292)(3 189 113 293)(4 190 114 294)(5 191 115 295)(6 192 116 296)(7 193 117 297)(8 194 118 298)(9 195 119 299)(10 196 120 300)(11 197 121 301)(12 198 122 302)(13 199 123 303)(14 200 124 304)(15 201 125 305)(16 202 126 306)(17 203 127 307)(18 204 128 308)(19 205 129 309)(20 206 130 310)(21 207 131 311)(22 208 132 312)(23 157 133 261)(24 158 134 262)(25 159 135 263)(26 160 136 264)(27 161 137 265)(28 162 138 266)(29 163 139 267)(30 164 140 268)(31 165 141 269)(32 166 142 270)(33 167 143 271)(34 168 144 272)(35 169 145 273)(36 170 146 274)(37 171 147 275)(38 172 148 276)(39 173 149 277)(40 174 150 278)(41 175 151 279)(42 176 152 280)(43 177 153 281)(44 178 154 282)(45 179 155 283)(46 180 156 284)(47 181 105 285)(48 182 106 286)(49 183 107 287)(50 184 108 288)(51 185 109 289)(52 186 110 290)(53 239 351 408)(54 240 352 409)(55 241 353 410)(56 242 354 411)(57 243 355 412)(58 244 356 413)(59 245 357 414)(60 246 358 415)(61 247 359 416)(62 248 360 365)(63 249 361 366)(64 250 362 367)(65 251 363 368)(66 252 364 369)(67 253 313 370)(68 254 314 371)(69 255 315 372)(70 256 316 373)(71 257 317 374)(72 258 318 375)(73 259 319 376)(74 260 320 377)(75 209 321 378)(76 210 322 379)(77 211 323 380)(78 212 324 381)(79 213 325 382)(80 214 326 383)(81 215 327 384)(82 216 328 385)(83 217 329 386)(84 218 330 387)(85 219 331 388)(86 220 332 389)(87 221 333 390)(88 222 334 391)(89 223 335 392)(90 224 336 393)(91 225 337 394)(92 226 338 395)(93 227 339 396)(94 228 340 397)(95 229 341 398)(96 230 342 399)(97 231 343 400)(98 232 344 401)(99 233 345 402)(100 234 346 403)(101 235 347 404)(102 236 348 405)(103 237 349 406)(104 238 350 407)
(1 402 137 259)(2 401 138 258)(3 400 139 257)(4 399 140 256)(5 398 141 255)(6 397 142 254)(7 396 143 253)(8 395 144 252)(9 394 145 251)(10 393 146 250)(11 392 147 249)(12 391 148 248)(13 390 149 247)(14 389 150 246)(15 388 151 245)(16 387 152 244)(17 386 153 243)(18 385 154 242)(19 384 155 241)(20 383 156 240)(21 382 105 239)(22 381 106 238)(23 380 107 237)(24 379 108 236)(25 378 109 235)(26 377 110 234)(27 376 111 233)(28 375 112 232)(29 374 113 231)(30 373 114 230)(31 372 115 229)(32 371 116 228)(33 370 117 227)(34 369 118 226)(35 368 119 225)(36 367 120 224)(37 366 121 223)(38 365 122 222)(39 416 123 221)(40 415 124 220)(41 414 125 219)(42 413 126 218)(43 412 127 217)(44 411 128 216)(45 410 129 215)(46 409 130 214)(47 408 131 213)(48 407 132 212)(49 406 133 211)(50 405 134 210)(51 404 135 209)(52 403 136 260)(53 181 325 311)(54 180 326 310)(55 179 327 309)(56 178 328 308)(57 177 329 307)(58 176 330 306)(59 175 331 305)(60 174 332 304)(61 173 333 303)(62 172 334 302)(63 171 335 301)(64 170 336 300)(65 169 337 299)(66 168 338 298)(67 167 339 297)(68 166 340 296)(69 165 341 295)(70 164 342 294)(71 163 343 293)(72 162 344 292)(73 161 345 291)(74 160 346 290)(75 159 347 289)(76 158 348 288)(77 157 349 287)(78 208 350 286)(79 207 351 285)(80 206 352 284)(81 205 353 283)(82 204 354 282)(83 203 355 281)(84 202 356 280)(85 201 357 279)(86 200 358 278)(87 199 359 277)(88 198 360 276)(89 197 361 275)(90 196 362 274)(91 195 363 273)(92 194 364 272)(93 193 313 271)(94 192 314 270)(95 191 315 269)(96 190 316 268)(97 189 317 267)(98 188 318 266)(99 187 319 265)(100 186 320 264)(101 185 321 263)(102 184 322 262)(103 183 323 261)(104 182 324 312)
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), (1,187,111,291)(2,188,112,292)(3,189,113,293)(4,190,114,294)(5,191,115,295)(6,192,116,296)(7,193,117,297)(8,194,118,298)(9,195,119,299)(10,196,120,300)(11,197,121,301)(12,198,122,302)(13,199,123,303)(14,200,124,304)(15,201,125,305)(16,202,126,306)(17,203,127,307)(18,204,128,308)(19,205,129,309)(20,206,130,310)(21,207,131,311)(22,208,132,312)(23,157,133,261)(24,158,134,262)(25,159,135,263)(26,160,136,264)(27,161,137,265)(28,162,138,266)(29,163,139,267)(30,164,140,268)(31,165,141,269)(32,166,142,270)(33,167,143,271)(34,168,144,272)(35,169,145,273)(36,170,146,274)(37,171,147,275)(38,172,148,276)(39,173,149,277)(40,174,150,278)(41,175,151,279)(42,176,152,280)(43,177,153,281)(44,178,154,282)(45,179,155,283)(46,180,156,284)(47,181,105,285)(48,182,106,286)(49,183,107,287)(50,184,108,288)(51,185,109,289)(52,186,110,290)(53,239,351,408)(54,240,352,409)(55,241,353,410)(56,242,354,411)(57,243,355,412)(58,244,356,413)(59,245,357,414)(60,246,358,415)(61,247,359,416)(62,248,360,365)(63,249,361,366)(64,250,362,367)(65,251,363,368)(66,252,364,369)(67,253,313,370)(68,254,314,371)(69,255,315,372)(70,256,316,373)(71,257,317,374)(72,258,318,375)(73,259,319,376)(74,260,320,377)(75,209,321,378)(76,210,322,379)(77,211,323,380)(78,212,324,381)(79,213,325,382)(80,214,326,383)(81,215,327,384)(82,216,328,385)(83,217,329,386)(84,218,330,387)(85,219,331,388)(86,220,332,389)(87,221,333,390)(88,222,334,391)(89,223,335,392)(90,224,336,393)(91,225,337,394)(92,226,338,395)(93,227,339,396)(94,228,340,397)(95,229,341,398)(96,230,342,399)(97,231,343,400)(98,232,344,401)(99,233,345,402)(100,234,346,403)(101,235,347,404)(102,236,348,405)(103,237,349,406)(104,238,350,407), (1,402,137,259)(2,401,138,258)(3,400,139,257)(4,399,140,256)(5,398,141,255)(6,397,142,254)(7,396,143,253)(8,395,144,252)(9,394,145,251)(10,393,146,250)(11,392,147,249)(12,391,148,248)(13,390,149,247)(14,389,150,246)(15,388,151,245)(16,387,152,244)(17,386,153,243)(18,385,154,242)(19,384,155,241)(20,383,156,240)(21,382,105,239)(22,381,106,238)(23,380,107,237)(24,379,108,236)(25,378,109,235)(26,377,110,234)(27,376,111,233)(28,375,112,232)(29,374,113,231)(30,373,114,230)(31,372,115,229)(32,371,116,228)(33,370,117,227)(34,369,118,226)(35,368,119,225)(36,367,120,224)(37,366,121,223)(38,365,122,222)(39,416,123,221)(40,415,124,220)(41,414,125,219)(42,413,126,218)(43,412,127,217)(44,411,128,216)(45,410,129,215)(46,409,130,214)(47,408,131,213)(48,407,132,212)(49,406,133,211)(50,405,134,210)(51,404,135,209)(52,403,136,260)(53,181,325,311)(54,180,326,310)(55,179,327,309)(56,178,328,308)(57,177,329,307)(58,176,330,306)(59,175,331,305)(60,174,332,304)(61,173,333,303)(62,172,334,302)(63,171,335,301)(64,170,336,300)(65,169,337,299)(66,168,338,298)(67,167,339,297)(68,166,340,296)(69,165,341,295)(70,164,342,294)(71,163,343,293)(72,162,344,292)(73,161,345,291)(74,160,346,290)(75,159,347,289)(76,158,348,288)(77,157,349,287)(78,208,350,286)(79,207,351,285)(80,206,352,284)(81,205,353,283)(82,204,354,282)(83,203,355,281)(84,202,356,280)(85,201,357,279)(86,200,358,278)(87,199,359,277)(88,198,360,276)(89,197,361,275)(90,196,362,274)(91,195,363,273)(92,194,364,272)(93,193,313,271)(94,192,314,270)(95,191,315,269)(96,190,316,268)(97,189,317,267)(98,188,318,266)(99,187,319,265)(100,186,320,264)(101,185,321,263)(102,184,322,262)(103,183,323,261)(104,182,324,312)>;
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), (1,187,111,291)(2,188,112,292)(3,189,113,293)(4,190,114,294)(5,191,115,295)(6,192,116,296)(7,193,117,297)(8,194,118,298)(9,195,119,299)(10,196,120,300)(11,197,121,301)(12,198,122,302)(13,199,123,303)(14,200,124,304)(15,201,125,305)(16,202,126,306)(17,203,127,307)(18,204,128,308)(19,205,129,309)(20,206,130,310)(21,207,131,311)(22,208,132,312)(23,157,133,261)(24,158,134,262)(25,159,135,263)(26,160,136,264)(27,161,137,265)(28,162,138,266)(29,163,139,267)(30,164,140,268)(31,165,141,269)(32,166,142,270)(33,167,143,271)(34,168,144,272)(35,169,145,273)(36,170,146,274)(37,171,147,275)(38,172,148,276)(39,173,149,277)(40,174,150,278)(41,175,151,279)(42,176,152,280)(43,177,153,281)(44,178,154,282)(45,179,155,283)(46,180,156,284)(47,181,105,285)(48,182,106,286)(49,183,107,287)(50,184,108,288)(51,185,109,289)(52,186,110,290)(53,239,351,408)(54,240,352,409)(55,241,353,410)(56,242,354,411)(57,243,355,412)(58,244,356,413)(59,245,357,414)(60,246,358,415)(61,247,359,416)(62,248,360,365)(63,249,361,366)(64,250,362,367)(65,251,363,368)(66,252,364,369)(67,253,313,370)(68,254,314,371)(69,255,315,372)(70,256,316,373)(71,257,317,374)(72,258,318,375)(73,259,319,376)(74,260,320,377)(75,209,321,378)(76,210,322,379)(77,211,323,380)(78,212,324,381)(79,213,325,382)(80,214,326,383)(81,215,327,384)(82,216,328,385)(83,217,329,386)(84,218,330,387)(85,219,331,388)(86,220,332,389)(87,221,333,390)(88,222,334,391)(89,223,335,392)(90,224,336,393)(91,225,337,394)(92,226,338,395)(93,227,339,396)(94,228,340,397)(95,229,341,398)(96,230,342,399)(97,231,343,400)(98,232,344,401)(99,233,345,402)(100,234,346,403)(101,235,347,404)(102,236,348,405)(103,237,349,406)(104,238,350,407), (1,402,137,259)(2,401,138,258)(3,400,139,257)(4,399,140,256)(5,398,141,255)(6,397,142,254)(7,396,143,253)(8,395,144,252)(9,394,145,251)(10,393,146,250)(11,392,147,249)(12,391,148,248)(13,390,149,247)(14,389,150,246)(15,388,151,245)(16,387,152,244)(17,386,153,243)(18,385,154,242)(19,384,155,241)(20,383,156,240)(21,382,105,239)(22,381,106,238)(23,380,107,237)(24,379,108,236)(25,378,109,235)(26,377,110,234)(27,376,111,233)(28,375,112,232)(29,374,113,231)(30,373,114,230)(31,372,115,229)(32,371,116,228)(33,370,117,227)(34,369,118,226)(35,368,119,225)(36,367,120,224)(37,366,121,223)(38,365,122,222)(39,416,123,221)(40,415,124,220)(41,414,125,219)(42,413,126,218)(43,412,127,217)(44,411,128,216)(45,410,129,215)(46,409,130,214)(47,408,131,213)(48,407,132,212)(49,406,133,211)(50,405,134,210)(51,404,135,209)(52,403,136,260)(53,181,325,311)(54,180,326,310)(55,179,327,309)(56,178,328,308)(57,177,329,307)(58,176,330,306)(59,175,331,305)(60,174,332,304)(61,173,333,303)(62,172,334,302)(63,171,335,301)(64,170,336,300)(65,169,337,299)(66,168,338,298)(67,167,339,297)(68,166,340,296)(69,165,341,295)(70,164,342,294)(71,163,343,293)(72,162,344,292)(73,161,345,291)(74,160,346,290)(75,159,347,289)(76,158,348,288)(77,157,349,287)(78,208,350,286)(79,207,351,285)(80,206,352,284)(81,205,353,283)(82,204,354,282)(83,203,355,281)(84,202,356,280)(85,201,357,279)(86,200,358,278)(87,199,359,277)(88,198,360,276)(89,197,361,275)(90,196,362,274)(91,195,363,273)(92,194,364,272)(93,193,313,271)(94,192,314,270)(95,191,315,269)(96,190,316,268)(97,189,317,267)(98,188,318,266)(99,187,319,265)(100,186,320,264)(101,185,321,263)(102,184,322,262)(103,183,323,261)(104,182,324,312) );
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)], [(1,187,111,291),(2,188,112,292),(3,189,113,293),(4,190,114,294),(5,191,115,295),(6,192,116,296),(7,193,117,297),(8,194,118,298),(9,195,119,299),(10,196,120,300),(11,197,121,301),(12,198,122,302),(13,199,123,303),(14,200,124,304),(15,201,125,305),(16,202,126,306),(17,203,127,307),(18,204,128,308),(19,205,129,309),(20,206,130,310),(21,207,131,311),(22,208,132,312),(23,157,133,261),(24,158,134,262),(25,159,135,263),(26,160,136,264),(27,161,137,265),(28,162,138,266),(29,163,139,267),(30,164,140,268),(31,165,141,269),(32,166,142,270),(33,167,143,271),(34,168,144,272),(35,169,145,273),(36,170,146,274),(37,171,147,275),(38,172,148,276),(39,173,149,277),(40,174,150,278),(41,175,151,279),(42,176,152,280),(43,177,153,281),(44,178,154,282),(45,179,155,283),(46,180,156,284),(47,181,105,285),(48,182,106,286),(49,183,107,287),(50,184,108,288),(51,185,109,289),(52,186,110,290),(53,239,351,408),(54,240,352,409),(55,241,353,410),(56,242,354,411),(57,243,355,412),(58,244,356,413),(59,245,357,414),(60,246,358,415),(61,247,359,416),(62,248,360,365),(63,249,361,366),(64,250,362,367),(65,251,363,368),(66,252,364,369),(67,253,313,370),(68,254,314,371),(69,255,315,372),(70,256,316,373),(71,257,317,374),(72,258,318,375),(73,259,319,376),(74,260,320,377),(75,209,321,378),(76,210,322,379),(77,211,323,380),(78,212,324,381),(79,213,325,382),(80,214,326,383),(81,215,327,384),(82,216,328,385),(83,217,329,386),(84,218,330,387),(85,219,331,388),(86,220,332,389),(87,221,333,390),(88,222,334,391),(89,223,335,392),(90,224,336,393),(91,225,337,394),(92,226,338,395),(93,227,339,396),(94,228,340,397),(95,229,341,398),(96,230,342,399),(97,231,343,400),(98,232,344,401),(99,233,345,402),(100,234,346,403),(101,235,347,404),(102,236,348,405),(103,237,349,406),(104,238,350,407)], [(1,402,137,259),(2,401,138,258),(3,400,139,257),(4,399,140,256),(5,398,141,255),(6,397,142,254),(7,396,143,253),(8,395,144,252),(9,394,145,251),(10,393,146,250),(11,392,147,249),(12,391,148,248),(13,390,149,247),(14,389,150,246),(15,388,151,245),(16,387,152,244),(17,386,153,243),(18,385,154,242),(19,384,155,241),(20,383,156,240),(21,382,105,239),(22,381,106,238),(23,380,107,237),(24,379,108,236),(25,378,109,235),(26,377,110,234),(27,376,111,233),(28,375,112,232),(29,374,113,231),(30,373,114,230),(31,372,115,229),(32,371,116,228),(33,370,117,227),(34,369,118,226),(35,368,119,225),(36,367,120,224),(37,366,121,223),(38,365,122,222),(39,416,123,221),(40,415,124,220),(41,414,125,219),(42,413,126,218),(43,412,127,217),(44,411,128,216),(45,410,129,215),(46,409,130,214),(47,408,131,213),(48,407,132,212),(49,406,133,211),(50,405,134,210),(51,404,135,209),(52,403,136,260),(53,181,325,311),(54,180,326,310),(55,179,327,309),(56,178,328,308),(57,177,329,307),(58,176,330,306),(59,175,331,305),(60,174,332,304),(61,173,333,303),(62,172,334,302),(63,171,335,301),(64,170,336,300),(65,169,337,299),(66,168,338,298),(67,167,339,297),(68,166,340,296),(69,165,341,295),(70,164,342,294),(71,163,343,293),(72,162,344,292),(73,161,345,291),(74,160,346,290),(75,159,347,289),(76,158,348,288),(77,157,349,287),(78,208,350,286),(79,207,351,285),(80,206,352,284),(81,205,353,283),(82,204,354,282),(83,203,355,281),(84,202,356,280),(85,201,357,279),(86,200,358,278),(87,199,359,277),(88,198,360,276),(89,197,361,275),(90,196,362,274),(91,195,363,273),(92,194,364,272),(93,193,313,271),(94,192,314,270),(95,191,315,269),(96,190,316,268),(97,189,317,267),(98,188,318,266),(99,187,319,265),(100,186,320,264),(101,185,321,263),(102,184,322,262),(103,183,323,261),(104,182,324,312)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52BT |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 52 | 52 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | Q8 | C4○D4 | D13 | D26 | Dic26 | D52⋊5C2 |
| kernel | C52.6Q8 | C26.D4 | C52⋊3C4 | C4×C52 | C52 | C26 | C42 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 4 | 6 | 18 | 24 | 48 |
Matrix representation of C52.6Q8 ►in GL4(𝔽53) generated by
| 15 | 25 | 0 | 0 |
| 28 | 22 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 26 | 1 |
| 0 | 0 | 12 | 27 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 |
| 0 | 0 | 35 | 52 |
G:=sub<GL(4,GF(53))| [15,28,0,0,25,22,0,0,0,0,52,0,0,0,0,52],[30,0,0,0,0,30,0,0,0,0,26,12,0,0,1,27],[0,1,0,0,1,0,0,0,0,0,1,35,0,0,6,52] >;
C52.6Q8 in GAP, Magma, Sage, TeX
C_{52}._6Q_8 % in TeX
G:=Group("C52.6Q8"); // GroupNames label
G:=SmallGroup(416,91);
// by ID
G=gap.SmallGroup(416,91);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,217,55,218,86,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^4=1,c^2=a^26*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^26*b^-1>;
// generators/relations