metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52⋊Q8, C4⋊1Dic26, Dic13⋊1Q8, Dic13.6D4, C13⋊2(C4⋊Q8), C4⋊C4.4D13, C2.4(Q8×D13), C26.5(C2×Q8), C26.22(C2×D4), C2.11(D4×D13), (C2×C4).42D26, (C2×C52).4C22, C52⋊3C4.11C2, C2.7(C2×Dic26), (C2×C26).29C23, (C2×Dic26).4C2, (C4×Dic13).1C2, C26.D4.2C2, (C2×Dic13).8C22, C22.46(C22×D13), (C13×C4⋊C4).5C2, SmallGroup(416,109)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊Q8
G = < a,b,c | a52=b4=1, c2=b2, bab-1=a27, cac-1=a25, cbc-1=b-1 >
Subgroups: 408 in 68 conjugacy classes, 37 normal (19 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, C2×C4, Q8, C13, C42, C4⋊C4, C4⋊C4, C2×Q8, C26, C4⋊Q8, Dic13, Dic13, C52, C52, C2×C26, Dic26, C2×Dic13, C2×Dic13, C2×C52, C2×C52, C4×Dic13, C26.D4, C52⋊3C4, C13×C4⋊C4, C2×Dic26, C52⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, D13, C4⋊Q8, D26, Dic26, C22×D13, C2×Dic26, D4×D13, Q8×D13, 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)
(1 361 258 116)(2 336 259 143)(3 363 260 118)(4 338 209 145)(5 313 210 120)(6 340 211 147)(7 315 212 122)(8 342 213 149)(9 317 214 124)(10 344 215 151)(11 319 216 126)(12 346 217 153)(13 321 218 128)(14 348 219 155)(15 323 220 130)(16 350 221 105)(17 325 222 132)(18 352 223 107)(19 327 224 134)(20 354 225 109)(21 329 226 136)(22 356 227 111)(23 331 228 138)(24 358 229 113)(25 333 230 140)(26 360 231 115)(27 335 232 142)(28 362 233 117)(29 337 234 144)(30 364 235 119)(31 339 236 146)(32 314 237 121)(33 341 238 148)(34 316 239 123)(35 343 240 150)(36 318 241 125)(37 345 242 152)(38 320 243 127)(39 347 244 154)(40 322 245 129)(41 349 246 156)(42 324 247 131)(43 351 248 106)(44 326 249 133)(45 353 250 108)(46 328 251 135)(47 355 252 110)(48 330 253 137)(49 357 254 112)(50 332 255 139)(51 359 256 114)(52 334 257 141)(53 386 192 302)(54 413 193 277)(55 388 194 304)(56 415 195 279)(57 390 196 306)(58 365 197 281)(59 392 198 308)(60 367 199 283)(61 394 200 310)(62 369 201 285)(63 396 202 312)(64 371 203 287)(65 398 204 262)(66 373 205 289)(67 400 206 264)(68 375 207 291)(69 402 208 266)(70 377 157 293)(71 404 158 268)(72 379 159 295)(73 406 160 270)(74 381 161 297)(75 408 162 272)(76 383 163 299)(77 410 164 274)(78 385 165 301)(79 412 166 276)(80 387 167 303)(81 414 168 278)(82 389 169 305)(83 416 170 280)(84 391 171 307)(85 366 172 282)(86 393 173 309)(87 368 174 284)(88 395 175 311)(89 370 176 286)(90 397 177 261)(91 372 178 288)(92 399 179 263)(93 374 180 290)(94 401 181 265)(95 376 182 292)(96 403 183 267)(97 378 184 294)(98 405 185 269)(99 380 186 296)(100 407 187 271)(101 382 188 298)(102 409 189 273)(103 384 190 300)(104 411 191 275)
(1 64 258 203)(2 89 259 176)(3 62 260 201)(4 87 209 174)(5 60 210 199)(6 85 211 172)(7 58 212 197)(8 83 213 170)(9 56 214 195)(10 81 215 168)(11 54 216 193)(12 79 217 166)(13 104 218 191)(14 77 219 164)(15 102 220 189)(16 75 221 162)(17 100 222 187)(18 73 223 160)(19 98 224 185)(20 71 225 158)(21 96 226 183)(22 69 227 208)(23 94 228 181)(24 67 229 206)(25 92 230 179)(26 65 231 204)(27 90 232 177)(28 63 233 202)(29 88 234 175)(30 61 235 200)(31 86 236 173)(32 59 237 198)(33 84 238 171)(34 57 239 196)(35 82 240 169)(36 55 241 194)(37 80 242 167)(38 53 243 192)(39 78 244 165)(40 103 245 190)(41 76 246 163)(42 101 247 188)(43 74 248 161)(44 99 249 186)(45 72 250 159)(46 97 251 184)(47 70 252 157)(48 95 253 182)(49 68 254 207)(50 93 255 180)(51 66 256 205)(52 91 257 178)(105 408 350 272)(106 381 351 297)(107 406 352 270)(108 379 353 295)(109 404 354 268)(110 377 355 293)(111 402 356 266)(112 375 357 291)(113 400 358 264)(114 373 359 289)(115 398 360 262)(116 371 361 287)(117 396 362 312)(118 369 363 285)(119 394 364 310)(120 367 313 283)(121 392 314 308)(122 365 315 281)(123 390 316 306)(124 415 317 279)(125 388 318 304)(126 413 319 277)(127 386 320 302)(128 411 321 275)(129 384 322 300)(130 409 323 273)(131 382 324 298)(132 407 325 271)(133 380 326 296)(134 405 327 269)(135 378 328 294)(136 403 329 267)(137 376 330 292)(138 401 331 265)(139 374 332 290)(140 399 333 263)(141 372 334 288)(142 397 335 261)(143 370 336 286)(144 395 337 311)(145 368 338 284)(146 393 339 309)(147 366 340 282)(148 391 341 307)(149 416 342 280)(150 389 343 305)(151 414 344 278)(152 387 345 303)(153 412 346 276)(154 385 347 301)(155 410 348 274)(156 383 349 299)
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), (1,361,258,116)(2,336,259,143)(3,363,260,118)(4,338,209,145)(5,313,210,120)(6,340,211,147)(7,315,212,122)(8,342,213,149)(9,317,214,124)(10,344,215,151)(11,319,216,126)(12,346,217,153)(13,321,218,128)(14,348,219,155)(15,323,220,130)(16,350,221,105)(17,325,222,132)(18,352,223,107)(19,327,224,134)(20,354,225,109)(21,329,226,136)(22,356,227,111)(23,331,228,138)(24,358,229,113)(25,333,230,140)(26,360,231,115)(27,335,232,142)(28,362,233,117)(29,337,234,144)(30,364,235,119)(31,339,236,146)(32,314,237,121)(33,341,238,148)(34,316,239,123)(35,343,240,150)(36,318,241,125)(37,345,242,152)(38,320,243,127)(39,347,244,154)(40,322,245,129)(41,349,246,156)(42,324,247,131)(43,351,248,106)(44,326,249,133)(45,353,250,108)(46,328,251,135)(47,355,252,110)(48,330,253,137)(49,357,254,112)(50,332,255,139)(51,359,256,114)(52,334,257,141)(53,386,192,302)(54,413,193,277)(55,388,194,304)(56,415,195,279)(57,390,196,306)(58,365,197,281)(59,392,198,308)(60,367,199,283)(61,394,200,310)(62,369,201,285)(63,396,202,312)(64,371,203,287)(65,398,204,262)(66,373,205,289)(67,400,206,264)(68,375,207,291)(69,402,208,266)(70,377,157,293)(71,404,158,268)(72,379,159,295)(73,406,160,270)(74,381,161,297)(75,408,162,272)(76,383,163,299)(77,410,164,274)(78,385,165,301)(79,412,166,276)(80,387,167,303)(81,414,168,278)(82,389,169,305)(83,416,170,280)(84,391,171,307)(85,366,172,282)(86,393,173,309)(87,368,174,284)(88,395,175,311)(89,370,176,286)(90,397,177,261)(91,372,178,288)(92,399,179,263)(93,374,180,290)(94,401,181,265)(95,376,182,292)(96,403,183,267)(97,378,184,294)(98,405,185,269)(99,380,186,296)(100,407,187,271)(101,382,188,298)(102,409,189,273)(103,384,190,300)(104,411,191,275), (1,64,258,203)(2,89,259,176)(3,62,260,201)(4,87,209,174)(5,60,210,199)(6,85,211,172)(7,58,212,197)(8,83,213,170)(9,56,214,195)(10,81,215,168)(11,54,216,193)(12,79,217,166)(13,104,218,191)(14,77,219,164)(15,102,220,189)(16,75,221,162)(17,100,222,187)(18,73,223,160)(19,98,224,185)(20,71,225,158)(21,96,226,183)(22,69,227,208)(23,94,228,181)(24,67,229,206)(25,92,230,179)(26,65,231,204)(27,90,232,177)(28,63,233,202)(29,88,234,175)(30,61,235,200)(31,86,236,173)(32,59,237,198)(33,84,238,171)(34,57,239,196)(35,82,240,169)(36,55,241,194)(37,80,242,167)(38,53,243,192)(39,78,244,165)(40,103,245,190)(41,76,246,163)(42,101,247,188)(43,74,248,161)(44,99,249,186)(45,72,250,159)(46,97,251,184)(47,70,252,157)(48,95,253,182)(49,68,254,207)(50,93,255,180)(51,66,256,205)(52,91,257,178)(105,408,350,272)(106,381,351,297)(107,406,352,270)(108,379,353,295)(109,404,354,268)(110,377,355,293)(111,402,356,266)(112,375,357,291)(113,400,358,264)(114,373,359,289)(115,398,360,262)(116,371,361,287)(117,396,362,312)(118,369,363,285)(119,394,364,310)(120,367,313,283)(121,392,314,308)(122,365,315,281)(123,390,316,306)(124,415,317,279)(125,388,318,304)(126,413,319,277)(127,386,320,302)(128,411,321,275)(129,384,322,300)(130,409,323,273)(131,382,324,298)(132,407,325,271)(133,380,326,296)(134,405,327,269)(135,378,328,294)(136,403,329,267)(137,376,330,292)(138,401,331,265)(139,374,332,290)(140,399,333,263)(141,372,334,288)(142,397,335,261)(143,370,336,286)(144,395,337,311)(145,368,338,284)(146,393,339,309)(147,366,340,282)(148,391,341,307)(149,416,342,280)(150,389,343,305)(151,414,344,278)(152,387,345,303)(153,412,346,276)(154,385,347,301)(155,410,348,274)(156,383,349,299)>;
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), (1,361,258,116)(2,336,259,143)(3,363,260,118)(4,338,209,145)(5,313,210,120)(6,340,211,147)(7,315,212,122)(8,342,213,149)(9,317,214,124)(10,344,215,151)(11,319,216,126)(12,346,217,153)(13,321,218,128)(14,348,219,155)(15,323,220,130)(16,350,221,105)(17,325,222,132)(18,352,223,107)(19,327,224,134)(20,354,225,109)(21,329,226,136)(22,356,227,111)(23,331,228,138)(24,358,229,113)(25,333,230,140)(26,360,231,115)(27,335,232,142)(28,362,233,117)(29,337,234,144)(30,364,235,119)(31,339,236,146)(32,314,237,121)(33,341,238,148)(34,316,239,123)(35,343,240,150)(36,318,241,125)(37,345,242,152)(38,320,243,127)(39,347,244,154)(40,322,245,129)(41,349,246,156)(42,324,247,131)(43,351,248,106)(44,326,249,133)(45,353,250,108)(46,328,251,135)(47,355,252,110)(48,330,253,137)(49,357,254,112)(50,332,255,139)(51,359,256,114)(52,334,257,141)(53,386,192,302)(54,413,193,277)(55,388,194,304)(56,415,195,279)(57,390,196,306)(58,365,197,281)(59,392,198,308)(60,367,199,283)(61,394,200,310)(62,369,201,285)(63,396,202,312)(64,371,203,287)(65,398,204,262)(66,373,205,289)(67,400,206,264)(68,375,207,291)(69,402,208,266)(70,377,157,293)(71,404,158,268)(72,379,159,295)(73,406,160,270)(74,381,161,297)(75,408,162,272)(76,383,163,299)(77,410,164,274)(78,385,165,301)(79,412,166,276)(80,387,167,303)(81,414,168,278)(82,389,169,305)(83,416,170,280)(84,391,171,307)(85,366,172,282)(86,393,173,309)(87,368,174,284)(88,395,175,311)(89,370,176,286)(90,397,177,261)(91,372,178,288)(92,399,179,263)(93,374,180,290)(94,401,181,265)(95,376,182,292)(96,403,183,267)(97,378,184,294)(98,405,185,269)(99,380,186,296)(100,407,187,271)(101,382,188,298)(102,409,189,273)(103,384,190,300)(104,411,191,275), (1,64,258,203)(2,89,259,176)(3,62,260,201)(4,87,209,174)(5,60,210,199)(6,85,211,172)(7,58,212,197)(8,83,213,170)(9,56,214,195)(10,81,215,168)(11,54,216,193)(12,79,217,166)(13,104,218,191)(14,77,219,164)(15,102,220,189)(16,75,221,162)(17,100,222,187)(18,73,223,160)(19,98,224,185)(20,71,225,158)(21,96,226,183)(22,69,227,208)(23,94,228,181)(24,67,229,206)(25,92,230,179)(26,65,231,204)(27,90,232,177)(28,63,233,202)(29,88,234,175)(30,61,235,200)(31,86,236,173)(32,59,237,198)(33,84,238,171)(34,57,239,196)(35,82,240,169)(36,55,241,194)(37,80,242,167)(38,53,243,192)(39,78,244,165)(40,103,245,190)(41,76,246,163)(42,101,247,188)(43,74,248,161)(44,99,249,186)(45,72,250,159)(46,97,251,184)(47,70,252,157)(48,95,253,182)(49,68,254,207)(50,93,255,180)(51,66,256,205)(52,91,257,178)(105,408,350,272)(106,381,351,297)(107,406,352,270)(108,379,353,295)(109,404,354,268)(110,377,355,293)(111,402,356,266)(112,375,357,291)(113,400,358,264)(114,373,359,289)(115,398,360,262)(116,371,361,287)(117,396,362,312)(118,369,363,285)(119,394,364,310)(120,367,313,283)(121,392,314,308)(122,365,315,281)(123,390,316,306)(124,415,317,279)(125,388,318,304)(126,413,319,277)(127,386,320,302)(128,411,321,275)(129,384,322,300)(130,409,323,273)(131,382,324,298)(132,407,325,271)(133,380,326,296)(134,405,327,269)(135,378,328,294)(136,403,329,267)(137,376,330,292)(138,401,331,265)(139,374,332,290)(140,399,333,263)(141,372,334,288)(142,397,335,261)(143,370,336,286)(144,395,337,311)(145,368,338,284)(146,393,339,309)(147,366,340,282)(148,391,341,307)(149,416,342,280)(150,389,343,305)(151,414,344,278)(152,387,345,303)(153,412,346,276)(154,385,347,301)(155,410,348,274)(156,383,349,299) );
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)], [(1,361,258,116),(2,336,259,143),(3,363,260,118),(4,338,209,145),(5,313,210,120),(6,340,211,147),(7,315,212,122),(8,342,213,149),(9,317,214,124),(10,344,215,151),(11,319,216,126),(12,346,217,153),(13,321,218,128),(14,348,219,155),(15,323,220,130),(16,350,221,105),(17,325,222,132),(18,352,223,107),(19,327,224,134),(20,354,225,109),(21,329,226,136),(22,356,227,111),(23,331,228,138),(24,358,229,113),(25,333,230,140),(26,360,231,115),(27,335,232,142),(28,362,233,117),(29,337,234,144),(30,364,235,119),(31,339,236,146),(32,314,237,121),(33,341,238,148),(34,316,239,123),(35,343,240,150),(36,318,241,125),(37,345,242,152),(38,320,243,127),(39,347,244,154),(40,322,245,129),(41,349,246,156),(42,324,247,131),(43,351,248,106),(44,326,249,133),(45,353,250,108),(46,328,251,135),(47,355,252,110),(48,330,253,137),(49,357,254,112),(50,332,255,139),(51,359,256,114),(52,334,257,141),(53,386,192,302),(54,413,193,277),(55,388,194,304),(56,415,195,279),(57,390,196,306),(58,365,197,281),(59,392,198,308),(60,367,199,283),(61,394,200,310),(62,369,201,285),(63,396,202,312),(64,371,203,287),(65,398,204,262),(66,373,205,289),(67,400,206,264),(68,375,207,291),(69,402,208,266),(70,377,157,293),(71,404,158,268),(72,379,159,295),(73,406,160,270),(74,381,161,297),(75,408,162,272),(76,383,163,299),(77,410,164,274),(78,385,165,301),(79,412,166,276),(80,387,167,303),(81,414,168,278),(82,389,169,305),(83,416,170,280),(84,391,171,307),(85,366,172,282),(86,393,173,309),(87,368,174,284),(88,395,175,311),(89,370,176,286),(90,397,177,261),(91,372,178,288),(92,399,179,263),(93,374,180,290),(94,401,181,265),(95,376,182,292),(96,403,183,267),(97,378,184,294),(98,405,185,269),(99,380,186,296),(100,407,187,271),(101,382,188,298),(102,409,189,273),(103,384,190,300),(104,411,191,275)], [(1,64,258,203),(2,89,259,176),(3,62,260,201),(4,87,209,174),(5,60,210,199),(6,85,211,172),(7,58,212,197),(8,83,213,170),(9,56,214,195),(10,81,215,168),(11,54,216,193),(12,79,217,166),(13,104,218,191),(14,77,219,164),(15,102,220,189),(16,75,221,162),(17,100,222,187),(18,73,223,160),(19,98,224,185),(20,71,225,158),(21,96,226,183),(22,69,227,208),(23,94,228,181),(24,67,229,206),(25,92,230,179),(26,65,231,204),(27,90,232,177),(28,63,233,202),(29,88,234,175),(30,61,235,200),(31,86,236,173),(32,59,237,198),(33,84,238,171),(34,57,239,196),(35,82,240,169),(36,55,241,194),(37,80,242,167),(38,53,243,192),(39,78,244,165),(40,103,245,190),(41,76,246,163),(42,101,247,188),(43,74,248,161),(44,99,249,186),(45,72,250,159),(46,97,251,184),(47,70,252,157),(48,95,253,182),(49,68,254,207),(50,93,255,180),(51,66,256,205),(52,91,257,178),(105,408,350,272),(106,381,351,297),(107,406,352,270),(108,379,353,295),(109,404,354,268),(110,377,355,293),(111,402,356,266),(112,375,357,291),(113,400,358,264),(114,373,359,289),(115,398,360,262),(116,371,361,287),(117,396,362,312),(118,369,363,285),(119,394,364,310),(120,367,313,283),(121,392,314,308),(122,365,315,281),(123,390,316,306),(124,415,317,279),(125,388,318,304),(126,413,319,277),(127,386,320,302),(128,411,321,275),(129,384,322,300),(130,409,323,273),(131,382,324,298),(132,407,325,271),(133,380,326,296),(134,405,327,269),(135,378,328,294),(136,403,329,267),(137,376,330,292),(138,401,331,265),(139,374,332,290),(140,399,333,263),(141,372,334,288),(142,397,335,261),(143,370,336,286),(144,395,337,311),(145,368,338,284),(146,393,339,309),(147,366,340,282),(148,391,341,307),(149,416,342,280),(150,389,343,305),(151,414,344,278),(152,387,345,303),(153,412,346,276),(154,385,347,301),(155,410,348,274),(156,383,349,299)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 26 | 26 | 26 | 26 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | Q8 | D13 | D26 | Dic26 | D4×D13 | Q8×D13 |
| kernel | C52⋊Q8 | C4×Dic13 | C26.D4 | C52⋊3C4 | C13×C4⋊C4 | C2×Dic26 | Dic13 | Dic13 | C52 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 18 | 24 | 6 | 6 |
Matrix representation of C52⋊Q8 ►in GL6(𝔽53)
| 6 | 39 | 0 | 0 | 0 | 0 |
| 14 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 27 | 0 | 0 |
| 0 | 0 | 21 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 49 | 6 |
| 0 | 0 | 0 | 0 | 6 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 0 | 0 | 0 |
| 0 | 0 | 0 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 52 | 0 |
| 52 | 0 | 0 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 52 | 0 | 0 |
| 0 | 0 | 0 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 49 | 6 |
| 0 | 0 | 0 | 0 | 6 | 4 |
G:=sub<GL(6,GF(53))| [6,14,0,0,0,0,39,47,0,0,0,0,0,0,4,21,0,0,0,0,27,36,0,0,0,0,0,0,49,6,0,0,0,0,6,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,0,52,0,0,0,0,1,0],[52,0,0,0,0,0,0,52,0,0,0,0,0,0,1,0,0,0,0,0,52,52,0,0,0,0,0,0,49,6,0,0,0,0,6,4] >;
C52⋊Q8 in GAP, Magma, Sage, TeX
C_{52}\rtimes Q_8 % in TeX
G:=Group("C52:Q8"); // GroupNames label
G:=SmallGroup(416,109);
// by ID
G=gap.SmallGroup(416,109);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,103,218,188,50,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^4=1,c^2=b^2,b*a*b^-1=a^27,c*a*c^-1=a^25,c*b*c^-1=b^-1>;
// generators/relations