metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.2Q8, C26.4SD16, C4.2Dic26, C13⋊2C8⋊2C4, C4⋊C4.2D13, C13⋊2(C4.Q8), C52.23(C2×C4), (C2×C26).29D4, C4.12(C4×D13), (C2×C4).34D26, C26.10(C4⋊C4), C52⋊3C4.9C2, C2.1(Q8⋊D13), (C2×C52).9C22, C2.1(D4.D13), C2.4(C26.D4), C22.13(C13⋊D4), (C13×C4⋊C4).2C2, (C2×C13⋊2C8).2C2, SmallGroup(416,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.Q8
G = < a,b,c | a52=b4=1, c2=a13b2, bab-1=a27, cac-1=a25, cbc-1=a39b-1 >
Smallest permutation representation of C52.Q8
►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)
(1 290 251 395)(2 265 252 370)(3 292 253 397)(4 267 254 372)(5 294 255 399)(6 269 256 374)(7 296 257 401)(8 271 258 376)(9 298 259 403)(10 273 260 378)(11 300 209 405)(12 275 210 380)(13 302 211 407)(14 277 212 382)(15 304 213 409)(16 279 214 384)(17 306 215 411)(18 281 216 386)(19 308 217 413)(20 283 218 388)(21 310 219 415)(22 285 220 390)(23 312 221 365)(24 287 222 392)(25 262 223 367)(26 289 224 394)(27 264 225 369)(28 291 226 396)(29 266 227 371)(30 293 228 398)(31 268 229 373)(32 295 230 400)(33 270 231 375)(34 297 232 402)(35 272 233 377)(36 299 234 404)(37 274 235 379)(38 301 236 406)(39 276 237 381)(40 303 238 408)(41 278 239 383)(42 305 240 410)(43 280 241 385)(44 307 242 412)(45 282 243 387)(46 309 244 414)(47 284 245 389)(48 311 246 416)(49 286 247 391)(50 261 248 366)(51 288 249 393)(52 263 250 368)(53 193 108 339)(54 168 109 314)(55 195 110 341)(56 170 111 316)(57 197 112 343)(58 172 113 318)(59 199 114 345)(60 174 115 320)(61 201 116 347)(62 176 117 322)(63 203 118 349)(64 178 119 324)(65 205 120 351)(66 180 121 326)(67 207 122 353)(68 182 123 328)(69 157 124 355)(70 184 125 330)(71 159 126 357)(72 186 127 332)(73 161 128 359)(74 188 129 334)(75 163 130 361)(76 190 131 336)(77 165 132 363)(78 192 133 338)(79 167 134 313)(80 194 135 340)(81 169 136 315)(82 196 137 342)(83 171 138 317)(84 198 139 344)(85 173 140 319)(86 200 141 346)(87 175 142 321)(88 202 143 348)(89 177 144 323)(90 204 145 350)(91 179 146 325)(92 206 147 352)(93 181 148 327)(94 208 149 354)(95 183 150 329)(96 158 151 356)(97 185 152 331)(98 160 153 358)(99 187 154 333)(100 162 155 360)(101 189 156 335)(102 164 105 362)(103 191 106 337)(104 166 107 364)
(1 92 212 108 27 66 238 134)(2 65 213 133 28 91 239 107)(3 90 214 106 29 64 240 132)(4 63 215 131 30 89 241 105)(5 88 216 156 31 62 242 130)(6 61 217 129 32 87 243 155)(7 86 218 154 33 60 244 128)(8 59 219 127 34 85 245 153)(9 84 220 152 35 58 246 126)(10 57 221 125 36 83 247 151)(11 82 222 150 37 56 248 124)(12 55 223 123 38 81 249 149)(13 80 224 148 39 54 250 122)(14 53 225 121 40 79 251 147)(15 78 226 146 41 104 252 120)(16 103 227 119 42 77 253 145)(17 76 228 144 43 102 254 118)(18 101 229 117 44 75 255 143)(19 74 230 142 45 100 256 116)(20 99 231 115 46 73 257 141)(21 72 232 140 47 98 258 114)(22 97 233 113 48 71 259 139)(23 70 234 138 49 96 260 112)(24 95 235 111 50 69 209 137)(25 68 236 136 51 94 210 110)(26 93 237 109 52 67 211 135)(157 287 316 405 183 261 342 379)(158 312 317 378 184 286 343 404)(159 285 318 403 185 311 344 377)(160 310 319 376 186 284 345 402)(161 283 320 401 187 309 346 375)(162 308 321 374 188 282 347 400)(163 281 322 399 189 307 348 373)(164 306 323 372 190 280 349 398)(165 279 324 397 191 305 350 371)(166 304 325 370 192 278 351 396)(167 277 326 395 193 303 352 369)(168 302 327 368 194 276 353 394)(169 275 328 393 195 301 354 367)(170 300 329 366 196 274 355 392)(171 273 330 391 197 299 356 365)(172 298 331 416 198 272 357 390)(173 271 332 389 199 297 358 415)(174 296 333 414 200 270 359 388)(175 269 334 387 201 295 360 413)(176 294 335 412 202 268 361 386)(177 267 336 385 203 293 362 411)(178 292 337 410 204 266 363 384)(179 265 338 383 205 291 364 409)(180 290 339 408 206 264 313 382)(181 263 340 381 207 289 314 407)(182 288 341 406 208 262 315 380)
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), (1,290,251,395)(2,265,252,370)(3,292,253,397)(4,267,254,372)(5,294,255,399)(6,269,256,374)(7,296,257,401)(8,271,258,376)(9,298,259,403)(10,273,260,378)(11,300,209,405)(12,275,210,380)(13,302,211,407)(14,277,212,382)(15,304,213,409)(16,279,214,384)(17,306,215,411)(18,281,216,386)(19,308,217,413)(20,283,218,388)(21,310,219,415)(22,285,220,390)(23,312,221,365)(24,287,222,392)(25,262,223,367)(26,289,224,394)(27,264,225,369)(28,291,226,396)(29,266,227,371)(30,293,228,398)(31,268,229,373)(32,295,230,400)(33,270,231,375)(34,297,232,402)(35,272,233,377)(36,299,234,404)(37,274,235,379)(38,301,236,406)(39,276,237,381)(40,303,238,408)(41,278,239,383)(42,305,240,410)(43,280,241,385)(44,307,242,412)(45,282,243,387)(46,309,244,414)(47,284,245,389)(48,311,246,416)(49,286,247,391)(50,261,248,366)(51,288,249,393)(52,263,250,368)(53,193,108,339)(54,168,109,314)(55,195,110,341)(56,170,111,316)(57,197,112,343)(58,172,113,318)(59,199,114,345)(60,174,115,320)(61,201,116,347)(62,176,117,322)(63,203,118,349)(64,178,119,324)(65,205,120,351)(66,180,121,326)(67,207,122,353)(68,182,123,328)(69,157,124,355)(70,184,125,330)(71,159,126,357)(72,186,127,332)(73,161,128,359)(74,188,129,334)(75,163,130,361)(76,190,131,336)(77,165,132,363)(78,192,133,338)(79,167,134,313)(80,194,135,340)(81,169,136,315)(82,196,137,342)(83,171,138,317)(84,198,139,344)(85,173,140,319)(86,200,141,346)(87,175,142,321)(88,202,143,348)(89,177,144,323)(90,204,145,350)(91,179,146,325)(92,206,147,352)(93,181,148,327)(94,208,149,354)(95,183,150,329)(96,158,151,356)(97,185,152,331)(98,160,153,358)(99,187,154,333)(100,162,155,360)(101,189,156,335)(102,164,105,362)(103,191,106,337)(104,166,107,364), (1,92,212,108,27,66,238,134)(2,65,213,133,28,91,239,107)(3,90,214,106,29,64,240,132)(4,63,215,131,30,89,241,105)(5,88,216,156,31,62,242,130)(6,61,217,129,32,87,243,155)(7,86,218,154,33,60,244,128)(8,59,219,127,34,85,245,153)(9,84,220,152,35,58,246,126)(10,57,221,125,36,83,247,151)(11,82,222,150,37,56,248,124)(12,55,223,123,38,81,249,149)(13,80,224,148,39,54,250,122)(14,53,225,121,40,79,251,147)(15,78,226,146,41,104,252,120)(16,103,227,119,42,77,253,145)(17,76,228,144,43,102,254,118)(18,101,229,117,44,75,255,143)(19,74,230,142,45,100,256,116)(20,99,231,115,46,73,257,141)(21,72,232,140,47,98,258,114)(22,97,233,113,48,71,259,139)(23,70,234,138,49,96,260,112)(24,95,235,111,50,69,209,137)(25,68,236,136,51,94,210,110)(26,93,237,109,52,67,211,135)(157,287,316,405,183,261,342,379)(158,312,317,378,184,286,343,404)(159,285,318,403,185,311,344,377)(160,310,319,376,186,284,345,402)(161,283,320,401,187,309,346,375)(162,308,321,374,188,282,347,400)(163,281,322,399,189,307,348,373)(164,306,323,372,190,280,349,398)(165,279,324,397,191,305,350,371)(166,304,325,370,192,278,351,396)(167,277,326,395,193,303,352,369)(168,302,327,368,194,276,353,394)(169,275,328,393,195,301,354,367)(170,300,329,366,196,274,355,392)(171,273,330,391,197,299,356,365)(172,298,331,416,198,272,357,390)(173,271,332,389,199,297,358,415)(174,296,333,414,200,270,359,388)(175,269,334,387,201,295,360,413)(176,294,335,412,202,268,361,386)(177,267,336,385,203,293,362,411)(178,292,337,410,204,266,363,384)(179,265,338,383,205,291,364,409)(180,290,339,408,206,264,313,382)(181,263,340,381,207,289,314,407)(182,288,341,406,208,262,315,380)>;
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), (1,290,251,395)(2,265,252,370)(3,292,253,397)(4,267,254,372)(5,294,255,399)(6,269,256,374)(7,296,257,401)(8,271,258,376)(9,298,259,403)(10,273,260,378)(11,300,209,405)(12,275,210,380)(13,302,211,407)(14,277,212,382)(15,304,213,409)(16,279,214,384)(17,306,215,411)(18,281,216,386)(19,308,217,413)(20,283,218,388)(21,310,219,415)(22,285,220,390)(23,312,221,365)(24,287,222,392)(25,262,223,367)(26,289,224,394)(27,264,225,369)(28,291,226,396)(29,266,227,371)(30,293,228,398)(31,268,229,373)(32,295,230,400)(33,270,231,375)(34,297,232,402)(35,272,233,377)(36,299,234,404)(37,274,235,379)(38,301,236,406)(39,276,237,381)(40,303,238,408)(41,278,239,383)(42,305,240,410)(43,280,241,385)(44,307,242,412)(45,282,243,387)(46,309,244,414)(47,284,245,389)(48,311,246,416)(49,286,247,391)(50,261,248,366)(51,288,249,393)(52,263,250,368)(53,193,108,339)(54,168,109,314)(55,195,110,341)(56,170,111,316)(57,197,112,343)(58,172,113,318)(59,199,114,345)(60,174,115,320)(61,201,116,347)(62,176,117,322)(63,203,118,349)(64,178,119,324)(65,205,120,351)(66,180,121,326)(67,207,122,353)(68,182,123,328)(69,157,124,355)(70,184,125,330)(71,159,126,357)(72,186,127,332)(73,161,128,359)(74,188,129,334)(75,163,130,361)(76,190,131,336)(77,165,132,363)(78,192,133,338)(79,167,134,313)(80,194,135,340)(81,169,136,315)(82,196,137,342)(83,171,138,317)(84,198,139,344)(85,173,140,319)(86,200,141,346)(87,175,142,321)(88,202,143,348)(89,177,144,323)(90,204,145,350)(91,179,146,325)(92,206,147,352)(93,181,148,327)(94,208,149,354)(95,183,150,329)(96,158,151,356)(97,185,152,331)(98,160,153,358)(99,187,154,333)(100,162,155,360)(101,189,156,335)(102,164,105,362)(103,191,106,337)(104,166,107,364), (1,92,212,108,27,66,238,134)(2,65,213,133,28,91,239,107)(3,90,214,106,29,64,240,132)(4,63,215,131,30,89,241,105)(5,88,216,156,31,62,242,130)(6,61,217,129,32,87,243,155)(7,86,218,154,33,60,244,128)(8,59,219,127,34,85,245,153)(9,84,220,152,35,58,246,126)(10,57,221,125,36,83,247,151)(11,82,222,150,37,56,248,124)(12,55,223,123,38,81,249,149)(13,80,224,148,39,54,250,122)(14,53,225,121,40,79,251,147)(15,78,226,146,41,104,252,120)(16,103,227,119,42,77,253,145)(17,76,228,144,43,102,254,118)(18,101,229,117,44,75,255,143)(19,74,230,142,45,100,256,116)(20,99,231,115,46,73,257,141)(21,72,232,140,47,98,258,114)(22,97,233,113,48,71,259,139)(23,70,234,138,49,96,260,112)(24,95,235,111,50,69,209,137)(25,68,236,136,51,94,210,110)(26,93,237,109,52,67,211,135)(157,287,316,405,183,261,342,379)(158,312,317,378,184,286,343,404)(159,285,318,403,185,311,344,377)(160,310,319,376,186,284,345,402)(161,283,320,401,187,309,346,375)(162,308,321,374,188,282,347,400)(163,281,322,399,189,307,348,373)(164,306,323,372,190,280,349,398)(165,279,324,397,191,305,350,371)(166,304,325,370,192,278,351,396)(167,277,326,395,193,303,352,369)(168,302,327,368,194,276,353,394)(169,275,328,393,195,301,354,367)(170,300,329,366,196,274,355,392)(171,273,330,391,197,299,356,365)(172,298,331,416,198,272,357,390)(173,271,332,389,199,297,358,415)(174,296,333,414,200,270,359,388)(175,269,334,387,201,295,360,413)(176,294,335,412,202,268,361,386)(177,267,336,385,203,293,362,411)(178,292,337,410,204,266,363,384)(179,265,338,383,205,291,364,409)(180,290,339,408,206,264,313,382)(181,263,340,381,207,289,314,407)(182,288,341,406,208,262,315,380) );
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)], [(1,290,251,395),(2,265,252,370),(3,292,253,397),(4,267,254,372),(5,294,255,399),(6,269,256,374),(7,296,257,401),(8,271,258,376),(9,298,259,403),(10,273,260,378),(11,300,209,405),(12,275,210,380),(13,302,211,407),(14,277,212,382),(15,304,213,409),(16,279,214,384),(17,306,215,411),(18,281,216,386),(19,308,217,413),(20,283,218,388),(21,310,219,415),(22,285,220,390),(23,312,221,365),(24,287,222,392),(25,262,223,367),(26,289,224,394),(27,264,225,369),(28,291,226,396),(29,266,227,371),(30,293,228,398),(31,268,229,373),(32,295,230,400),(33,270,231,375),(34,297,232,402),(35,272,233,377),(36,299,234,404),(37,274,235,379),(38,301,236,406),(39,276,237,381),(40,303,238,408),(41,278,239,383),(42,305,240,410),(43,280,241,385),(44,307,242,412),(45,282,243,387),(46,309,244,414),(47,284,245,389),(48,311,246,416),(49,286,247,391),(50,261,248,366),(51,288,249,393),(52,263,250,368),(53,193,108,339),(54,168,109,314),(55,195,110,341),(56,170,111,316),(57,197,112,343),(58,172,113,318),(59,199,114,345),(60,174,115,320),(61,201,116,347),(62,176,117,322),(63,203,118,349),(64,178,119,324),(65,205,120,351),(66,180,121,326),(67,207,122,353),(68,182,123,328),(69,157,124,355),(70,184,125,330),(71,159,126,357),(72,186,127,332),(73,161,128,359),(74,188,129,334),(75,163,130,361),(76,190,131,336),(77,165,132,363),(78,192,133,338),(79,167,134,313),(80,194,135,340),(81,169,136,315),(82,196,137,342),(83,171,138,317),(84,198,139,344),(85,173,140,319),(86,200,141,346),(87,175,142,321),(88,202,143,348),(89,177,144,323),(90,204,145,350),(91,179,146,325),(92,206,147,352),(93,181,148,327),(94,208,149,354),(95,183,150,329),(96,158,151,356),(97,185,152,331),(98,160,153,358),(99,187,154,333),(100,162,155,360),(101,189,156,335),(102,164,105,362),(103,191,106,337),(104,166,107,364)], [(1,92,212,108,27,66,238,134),(2,65,213,133,28,91,239,107),(3,90,214,106,29,64,240,132),(4,63,215,131,30,89,241,105),(5,88,216,156,31,62,242,130),(6,61,217,129,32,87,243,155),(7,86,218,154,33,60,244,128),(8,59,219,127,34,85,245,153),(9,84,220,152,35,58,246,126),(10,57,221,125,36,83,247,151),(11,82,222,150,37,56,248,124),(12,55,223,123,38,81,249,149),(13,80,224,148,39,54,250,122),(14,53,225,121,40,79,251,147),(15,78,226,146,41,104,252,120),(16,103,227,119,42,77,253,145),(17,76,228,144,43,102,254,118),(18,101,229,117,44,75,255,143),(19,74,230,142,45,100,256,116),(20,99,231,115,46,73,257,141),(21,72,232,140,47,98,258,114),(22,97,233,113,48,71,259,139),(23,70,234,138,49,96,260,112),(24,95,235,111,50,69,209,137),(25,68,236,136,51,94,210,110),(26,93,237,109,52,67,211,135),(157,287,316,405,183,261,342,379),(158,312,317,378,184,286,343,404),(159,285,318,403,185,311,344,377),(160,310,319,376,186,284,345,402),(161,283,320,401,187,309,346,375),(162,308,321,374,188,282,347,400),(163,281,322,399,189,307,348,373),(164,306,323,372,190,280,349,398),(165,279,324,397,191,305,350,371),(166,304,325,370,192,278,351,396),(167,277,326,395,193,303,352,369),(168,302,327,368,194,276,353,394),(169,275,328,393,195,301,354,367),(170,300,329,366,196,274,355,392),(171,273,330,391,197,299,356,365),(172,298,331,416,198,272,357,390),(173,271,332,389,199,297,358,415),(174,296,333,414,200,270,359,388),(175,269,334,387,201,295,360,413),(176,294,335,412,202,268,361,386),(177,267,336,385,203,293,362,411),(178,292,337,410,204,266,363,384),(179,265,338,383,205,291,364,409),(180,290,339,408,206,264,313,382),(181,263,340,381,207,289,314,407),(182,288,341,406,208,262,315,380)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 52 | 52 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | SD16 | D13 | D26 | Dic26 | C4×D13 | C13⋊D4 | D4.D13 | Q8⋊D13 |
| kernel | C52.Q8 | C2×C13⋊2C8 | C52⋊3C4 | C13×C4⋊C4 | C13⋊2C8 | C52 | C2×C26 | C26 | C4⋊C4 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 6 | 6 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of C52.Q8 ►in GL4(𝔽313) generated by
| 146 | 51 | 0 | 0 |
| 206 | 147 | 0 | 0 |
| 0 | 0 | 1 | 311 |
| 0 | 0 | 1 | 312 |
| 107 | 41 | 0 | 0 |
| 49 | 206 | 0 | 0 |
| 0 | 0 | 240 | 221 |
| 0 | 0 | 194 | 73 |
| 115 | 187 | 0 | 0 |
| 100 | 198 | 0 | 0 |
| 0 | 0 | 0 | 183 |
| 0 | 0 | 65 | 183 |
G:=sub<GL(4,GF(313))| [146,206,0,0,51,147,0,0,0,0,1,1,0,0,311,312],[107,49,0,0,41,206,0,0,0,0,240,194,0,0,221,73],[115,100,0,0,187,198,0,0,0,0,0,65,0,0,183,183] >;
C52.Q8 in GAP, Magma, Sage, TeX
C_{52}.Q_8 % in TeX
G:=Group("C52.Q8"); // GroupNames label
G:=SmallGroup(416,15);
// by ID
G=gap.SmallGroup(416,15);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,313,31,297,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^4=1,c^2=a^13*b^2,b*a*b^-1=a^27,c*a*c^-1=a^25,c*b*c^-1=a^39*b^-1>;
// generators/relations
Export