metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52⋊2Q8, C4.4D52, C4⋊2Dic26, C52.27D4, C42.4D13, C13⋊1(C4⋊Q8), (C4×C52).2C2, C2.4(C2×D52), C26.1(C2×D4), C26.2(C2×Q8), (C2×C4).73D26, C52⋊3C4.4C2, C2.4(C2×Dic26), (C2×C52).85C22, (C2×C26).10C23, (C2×Dic26).3C2, (C2×Dic13).1C22, C22.34(C22×D13), SmallGroup(416,90)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊2Q8
G = < a,b,c | a52=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 408 in 68 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, C2×C4, Q8, C13, C42, C4⋊C4, C2×Q8, C26, C26, C4⋊Q8, Dic13, C52, C2×C26, Dic26, C2×Dic13, C2×C52, C2×C52, C52⋊3C4, C4×C52, C2×Dic26, C52⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, D13, C4⋊Q8, D26, Dic26, D52, C22×D13, C2×Dic26, C2×D52, C52⋊2Q8
►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)
(1 315 122 189)(2 316 123 190)(3 317 124 191)(4 318 125 192)(5 319 126 193)(6 320 127 194)(7 321 128 195)(8 322 129 196)(9 323 130 197)(10 324 131 198)(11 325 132 199)(12 326 133 200)(13 327 134 201)(14 328 135 202)(15 329 136 203)(16 330 137 204)(17 331 138 205)(18 332 139 206)(19 333 140 207)(20 334 141 208)(21 335 142 157)(22 336 143 158)(23 337 144 159)(24 338 145 160)(25 339 146 161)(26 340 147 162)(27 341 148 163)(28 342 149 164)(29 343 150 165)(30 344 151 166)(31 345 152 167)(32 346 153 168)(33 347 154 169)(34 348 155 170)(35 349 156 171)(36 350 105 172)(37 351 106 173)(38 352 107 174)(39 353 108 175)(40 354 109 176)(41 355 110 177)(42 356 111 178)(43 357 112 179)(44 358 113 180)(45 359 114 181)(46 360 115 182)(47 361 116 183)(48 362 117 184)(49 363 118 185)(50 364 119 186)(51 313 120 187)(52 314 121 188)(53 214 296 399)(54 215 297 400)(55 216 298 401)(56 217 299 402)(57 218 300 403)(58 219 301 404)(59 220 302 405)(60 221 303 406)(61 222 304 407)(62 223 305 408)(63 224 306 409)(64 225 307 410)(65 226 308 411)(66 227 309 412)(67 228 310 413)(68 229 311 414)(69 230 312 415)(70 231 261 416)(71 232 262 365)(72 233 263 366)(73 234 264 367)(74 235 265 368)(75 236 266 369)(76 237 267 370)(77 238 268 371)(78 239 269 372)(79 240 270 373)(80 241 271 374)(81 242 272 375)(82 243 273 376)(83 244 274 377)(84 245 275 378)(85 246 276 379)(86 247 277 380)(87 248 278 381)(88 249 279 382)(89 250 280 383)(90 251 281 384)(91 252 282 385)(92 253 283 386)(93 254 284 387)(94 255 285 388)(95 256 286 389)(96 257 287 390)(97 258 288 391)(98 259 289 392)(99 260 290 393)(100 209 291 394)(101 210 292 395)(102 211 293 396)(103 212 294 397)(104 213 295 398)
(1 246 122 379)(2 245 123 378)(3 244 124 377)(4 243 125 376)(5 242 126 375)(6 241 127 374)(7 240 128 373)(8 239 129 372)(9 238 130 371)(10 237 131 370)(11 236 132 369)(12 235 133 368)(13 234 134 367)(14 233 135 366)(15 232 136 365)(16 231 137 416)(17 230 138 415)(18 229 139 414)(19 228 140 413)(20 227 141 412)(21 226 142 411)(22 225 143 410)(23 224 144 409)(24 223 145 408)(25 222 146 407)(26 221 147 406)(27 220 148 405)(28 219 149 404)(29 218 150 403)(30 217 151 402)(31 216 152 401)(32 215 153 400)(33 214 154 399)(34 213 155 398)(35 212 156 397)(36 211 105 396)(37 210 106 395)(38 209 107 394)(39 260 108 393)(40 259 109 392)(41 258 110 391)(42 257 111 390)(43 256 112 389)(44 255 113 388)(45 254 114 387)(46 253 115 386)(47 252 116 385)(48 251 117 384)(49 250 118 383)(50 249 119 382)(51 248 120 381)(52 247 121 380)(53 169 296 347)(54 168 297 346)(55 167 298 345)(56 166 299 344)(57 165 300 343)(58 164 301 342)(59 163 302 341)(60 162 303 340)(61 161 304 339)(62 160 305 338)(63 159 306 337)(64 158 307 336)(65 157 308 335)(66 208 309 334)(67 207 310 333)(68 206 311 332)(69 205 312 331)(70 204 261 330)(71 203 262 329)(72 202 263 328)(73 201 264 327)(74 200 265 326)(75 199 266 325)(76 198 267 324)(77 197 268 323)(78 196 269 322)(79 195 270 321)(80 194 271 320)(81 193 272 319)(82 192 273 318)(83 191 274 317)(84 190 275 316)(85 189 276 315)(86 188 277 314)(87 187 278 313)(88 186 279 364)(89 185 280 363)(90 184 281 362)(91 183 282 361)(92 182 283 360)(93 181 284 359)(94 180 285 358)(95 179 286 357)(96 178 287 356)(97 177 288 355)(98 176 289 354)(99 175 290 353)(100 174 291 352)(101 173 292 351)(102 172 293 350)(103 171 294 349)(104 170 295 348)
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), (1,315,122,189)(2,316,123,190)(3,317,124,191)(4,318,125,192)(5,319,126,193)(6,320,127,194)(7,321,128,195)(8,322,129,196)(9,323,130,197)(10,324,131,198)(11,325,132,199)(12,326,133,200)(13,327,134,201)(14,328,135,202)(15,329,136,203)(16,330,137,204)(17,331,138,205)(18,332,139,206)(19,333,140,207)(20,334,141,208)(21,335,142,157)(22,336,143,158)(23,337,144,159)(24,338,145,160)(25,339,146,161)(26,340,147,162)(27,341,148,163)(28,342,149,164)(29,343,150,165)(30,344,151,166)(31,345,152,167)(32,346,153,168)(33,347,154,169)(34,348,155,170)(35,349,156,171)(36,350,105,172)(37,351,106,173)(38,352,107,174)(39,353,108,175)(40,354,109,176)(41,355,110,177)(42,356,111,178)(43,357,112,179)(44,358,113,180)(45,359,114,181)(46,360,115,182)(47,361,116,183)(48,362,117,184)(49,363,118,185)(50,364,119,186)(51,313,120,187)(52,314,121,188)(53,214,296,399)(54,215,297,400)(55,216,298,401)(56,217,299,402)(57,218,300,403)(58,219,301,404)(59,220,302,405)(60,221,303,406)(61,222,304,407)(62,223,305,408)(63,224,306,409)(64,225,307,410)(65,226,308,411)(66,227,309,412)(67,228,310,413)(68,229,311,414)(69,230,312,415)(70,231,261,416)(71,232,262,365)(72,233,263,366)(73,234,264,367)(74,235,265,368)(75,236,266,369)(76,237,267,370)(77,238,268,371)(78,239,269,372)(79,240,270,373)(80,241,271,374)(81,242,272,375)(82,243,273,376)(83,244,274,377)(84,245,275,378)(85,246,276,379)(86,247,277,380)(87,248,278,381)(88,249,279,382)(89,250,280,383)(90,251,281,384)(91,252,282,385)(92,253,283,386)(93,254,284,387)(94,255,285,388)(95,256,286,389)(96,257,287,390)(97,258,288,391)(98,259,289,392)(99,260,290,393)(100,209,291,394)(101,210,292,395)(102,211,293,396)(103,212,294,397)(104,213,295,398), (1,246,122,379)(2,245,123,378)(3,244,124,377)(4,243,125,376)(5,242,126,375)(6,241,127,374)(7,240,128,373)(8,239,129,372)(9,238,130,371)(10,237,131,370)(11,236,132,369)(12,235,133,368)(13,234,134,367)(14,233,135,366)(15,232,136,365)(16,231,137,416)(17,230,138,415)(18,229,139,414)(19,228,140,413)(20,227,141,412)(21,226,142,411)(22,225,143,410)(23,224,144,409)(24,223,145,408)(25,222,146,407)(26,221,147,406)(27,220,148,405)(28,219,149,404)(29,218,150,403)(30,217,151,402)(31,216,152,401)(32,215,153,400)(33,214,154,399)(34,213,155,398)(35,212,156,397)(36,211,105,396)(37,210,106,395)(38,209,107,394)(39,260,108,393)(40,259,109,392)(41,258,110,391)(42,257,111,390)(43,256,112,389)(44,255,113,388)(45,254,114,387)(46,253,115,386)(47,252,116,385)(48,251,117,384)(49,250,118,383)(50,249,119,382)(51,248,120,381)(52,247,121,380)(53,169,296,347)(54,168,297,346)(55,167,298,345)(56,166,299,344)(57,165,300,343)(58,164,301,342)(59,163,302,341)(60,162,303,340)(61,161,304,339)(62,160,305,338)(63,159,306,337)(64,158,307,336)(65,157,308,335)(66,208,309,334)(67,207,310,333)(68,206,311,332)(69,205,312,331)(70,204,261,330)(71,203,262,329)(72,202,263,328)(73,201,264,327)(74,200,265,326)(75,199,266,325)(76,198,267,324)(77,197,268,323)(78,196,269,322)(79,195,270,321)(80,194,271,320)(81,193,272,319)(82,192,273,318)(83,191,274,317)(84,190,275,316)(85,189,276,315)(86,188,277,314)(87,187,278,313)(88,186,279,364)(89,185,280,363)(90,184,281,362)(91,183,282,361)(92,182,283,360)(93,181,284,359)(94,180,285,358)(95,179,286,357)(96,178,287,356)(97,177,288,355)(98,176,289,354)(99,175,290,353)(100,174,291,352)(101,173,292,351)(102,172,293,350)(103,171,294,349)(104,170,295,348)>;
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), (1,315,122,189)(2,316,123,190)(3,317,124,191)(4,318,125,192)(5,319,126,193)(6,320,127,194)(7,321,128,195)(8,322,129,196)(9,323,130,197)(10,324,131,198)(11,325,132,199)(12,326,133,200)(13,327,134,201)(14,328,135,202)(15,329,136,203)(16,330,137,204)(17,331,138,205)(18,332,139,206)(19,333,140,207)(20,334,141,208)(21,335,142,157)(22,336,143,158)(23,337,144,159)(24,338,145,160)(25,339,146,161)(26,340,147,162)(27,341,148,163)(28,342,149,164)(29,343,150,165)(30,344,151,166)(31,345,152,167)(32,346,153,168)(33,347,154,169)(34,348,155,170)(35,349,156,171)(36,350,105,172)(37,351,106,173)(38,352,107,174)(39,353,108,175)(40,354,109,176)(41,355,110,177)(42,356,111,178)(43,357,112,179)(44,358,113,180)(45,359,114,181)(46,360,115,182)(47,361,116,183)(48,362,117,184)(49,363,118,185)(50,364,119,186)(51,313,120,187)(52,314,121,188)(53,214,296,399)(54,215,297,400)(55,216,298,401)(56,217,299,402)(57,218,300,403)(58,219,301,404)(59,220,302,405)(60,221,303,406)(61,222,304,407)(62,223,305,408)(63,224,306,409)(64,225,307,410)(65,226,308,411)(66,227,309,412)(67,228,310,413)(68,229,311,414)(69,230,312,415)(70,231,261,416)(71,232,262,365)(72,233,263,366)(73,234,264,367)(74,235,265,368)(75,236,266,369)(76,237,267,370)(77,238,268,371)(78,239,269,372)(79,240,270,373)(80,241,271,374)(81,242,272,375)(82,243,273,376)(83,244,274,377)(84,245,275,378)(85,246,276,379)(86,247,277,380)(87,248,278,381)(88,249,279,382)(89,250,280,383)(90,251,281,384)(91,252,282,385)(92,253,283,386)(93,254,284,387)(94,255,285,388)(95,256,286,389)(96,257,287,390)(97,258,288,391)(98,259,289,392)(99,260,290,393)(100,209,291,394)(101,210,292,395)(102,211,293,396)(103,212,294,397)(104,213,295,398), (1,246,122,379)(2,245,123,378)(3,244,124,377)(4,243,125,376)(5,242,126,375)(6,241,127,374)(7,240,128,373)(8,239,129,372)(9,238,130,371)(10,237,131,370)(11,236,132,369)(12,235,133,368)(13,234,134,367)(14,233,135,366)(15,232,136,365)(16,231,137,416)(17,230,138,415)(18,229,139,414)(19,228,140,413)(20,227,141,412)(21,226,142,411)(22,225,143,410)(23,224,144,409)(24,223,145,408)(25,222,146,407)(26,221,147,406)(27,220,148,405)(28,219,149,404)(29,218,150,403)(30,217,151,402)(31,216,152,401)(32,215,153,400)(33,214,154,399)(34,213,155,398)(35,212,156,397)(36,211,105,396)(37,210,106,395)(38,209,107,394)(39,260,108,393)(40,259,109,392)(41,258,110,391)(42,257,111,390)(43,256,112,389)(44,255,113,388)(45,254,114,387)(46,253,115,386)(47,252,116,385)(48,251,117,384)(49,250,118,383)(50,249,119,382)(51,248,120,381)(52,247,121,380)(53,169,296,347)(54,168,297,346)(55,167,298,345)(56,166,299,344)(57,165,300,343)(58,164,301,342)(59,163,302,341)(60,162,303,340)(61,161,304,339)(62,160,305,338)(63,159,306,337)(64,158,307,336)(65,157,308,335)(66,208,309,334)(67,207,310,333)(68,206,311,332)(69,205,312,331)(70,204,261,330)(71,203,262,329)(72,202,263,328)(73,201,264,327)(74,200,265,326)(75,199,266,325)(76,198,267,324)(77,197,268,323)(78,196,269,322)(79,195,270,321)(80,194,271,320)(81,193,272,319)(82,192,273,318)(83,191,274,317)(84,190,275,316)(85,189,276,315)(86,188,277,314)(87,187,278,313)(88,186,279,364)(89,185,280,363)(90,184,281,362)(91,183,282,361)(92,182,283,360)(93,181,284,359)(94,180,285,358)(95,179,286,357)(96,178,287,356)(97,177,288,355)(98,176,289,354)(99,175,290,353)(100,174,291,352)(101,173,292,351)(102,172,293,350)(103,171,294,349)(104,170,295,348) );
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)], [(1,315,122,189),(2,316,123,190),(3,317,124,191),(4,318,125,192),(5,319,126,193),(6,320,127,194),(7,321,128,195),(8,322,129,196),(9,323,130,197),(10,324,131,198),(11,325,132,199),(12,326,133,200),(13,327,134,201),(14,328,135,202),(15,329,136,203),(16,330,137,204),(17,331,138,205),(18,332,139,206),(19,333,140,207),(20,334,141,208),(21,335,142,157),(22,336,143,158),(23,337,144,159),(24,338,145,160),(25,339,146,161),(26,340,147,162),(27,341,148,163),(28,342,149,164),(29,343,150,165),(30,344,151,166),(31,345,152,167),(32,346,153,168),(33,347,154,169),(34,348,155,170),(35,349,156,171),(36,350,105,172),(37,351,106,173),(38,352,107,174),(39,353,108,175),(40,354,109,176),(41,355,110,177),(42,356,111,178),(43,357,112,179),(44,358,113,180),(45,359,114,181),(46,360,115,182),(47,361,116,183),(48,362,117,184),(49,363,118,185),(50,364,119,186),(51,313,120,187),(52,314,121,188),(53,214,296,399),(54,215,297,400),(55,216,298,401),(56,217,299,402),(57,218,300,403),(58,219,301,404),(59,220,302,405),(60,221,303,406),(61,222,304,407),(62,223,305,408),(63,224,306,409),(64,225,307,410),(65,226,308,411),(66,227,309,412),(67,228,310,413),(68,229,311,414),(69,230,312,415),(70,231,261,416),(71,232,262,365),(72,233,263,366),(73,234,264,367),(74,235,265,368),(75,236,266,369),(76,237,267,370),(77,238,268,371),(78,239,269,372),(79,240,270,373),(80,241,271,374),(81,242,272,375),(82,243,273,376),(83,244,274,377),(84,245,275,378),(85,246,276,379),(86,247,277,380),(87,248,278,381),(88,249,279,382),(89,250,280,383),(90,251,281,384),(91,252,282,385),(92,253,283,386),(93,254,284,387),(94,255,285,388),(95,256,286,389),(96,257,287,390),(97,258,288,391),(98,259,289,392),(99,260,290,393),(100,209,291,394),(101,210,292,395),(102,211,293,396),(103,212,294,397),(104,213,295,398)], [(1,246,122,379),(2,245,123,378),(3,244,124,377),(4,243,125,376),(5,242,126,375),(6,241,127,374),(7,240,128,373),(8,239,129,372),(9,238,130,371),(10,237,131,370),(11,236,132,369),(12,235,133,368),(13,234,134,367),(14,233,135,366),(15,232,136,365),(16,231,137,416),(17,230,138,415),(18,229,139,414),(19,228,140,413),(20,227,141,412),(21,226,142,411),(22,225,143,410),(23,224,144,409),(24,223,145,408),(25,222,146,407),(26,221,147,406),(27,220,148,405),(28,219,149,404),(29,218,150,403),(30,217,151,402),(31,216,152,401),(32,215,153,400),(33,214,154,399),(34,213,155,398),(35,212,156,397),(36,211,105,396),(37,210,106,395),(38,209,107,394),(39,260,108,393),(40,259,109,392),(41,258,110,391),(42,257,111,390),(43,256,112,389),(44,255,113,388),(45,254,114,387),(46,253,115,386),(47,252,116,385),(48,251,117,384),(49,250,118,383),(50,249,119,382),(51,248,120,381),(52,247,121,380),(53,169,296,347),(54,168,297,346),(55,167,298,345),(56,166,299,344),(57,165,300,343),(58,164,301,342),(59,163,302,341),(60,162,303,340),(61,161,304,339),(62,160,305,338),(63,159,306,337),(64,158,307,336),(65,157,308,335),(66,208,309,334),(67,207,310,333),(68,206,311,332),(69,205,312,331),(70,204,261,330),(71,203,262,329),(72,202,263,328),(73,201,264,327),(74,200,265,326),(75,199,266,325),(76,198,267,324),(77,197,268,323),(78,196,269,322),(79,195,270,321),(80,194,271,320),(81,193,272,319),(82,192,273,318),(83,191,274,317),(84,190,275,316),(85,189,276,315),(86,188,277,314),(87,187,278,313),(88,186,279,364),(89,185,280,363),(90,184,281,362),(91,183,282,361),(92,182,283,360),(93,181,284,359),(94,180,285,358),(95,179,286,357),(96,178,287,356),(97,177,288,355),(98,176,289,354),(99,175,290,353),(100,174,291,352),(101,173,292,351),(102,172,293,350),(103,171,294,349),(104,170,295,348)]])
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 | D4 | Q8 | D13 | D26 | Dic26 | D52 |
| kernel | C52⋊2Q8 | C52⋊3C4 | C4×C52 | C2×Dic26 | C52 | C52 | C42 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 2 | 4 | 6 | 18 | 48 | 24 |
Matrix representation of C52⋊2Q8 ►in GL4(𝔽53) generated by
| 23 | 18 | 0 | 0 |
| 35 | 9 | 0 | 0 |
| 0 | 0 | 7 | 16 |
| 0 | 0 | 52 | 13 |
| 25 | 34 | 0 | 0 |
| 19 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 41 | 30 | 0 | 0 |
| 4 | 12 | 0 | 0 |
| 0 | 0 | 42 | 12 |
| 0 | 0 | 43 | 11 |
G:=sub<GL(4,GF(53))| [23,35,0,0,18,9,0,0,0,0,7,52,0,0,16,13],[25,19,0,0,34,28,0,0,0,0,1,0,0,0,0,1],[41,4,0,0,30,12,0,0,0,0,42,43,0,0,12,11] >;
C52⋊2Q8 in GAP, Magma, Sage, TeX
C_{52}\rtimes_2Q_8 % in TeX
G:=Group("C52:2Q8"); // GroupNames label
G:=SmallGroup(416,90);
// by ID
G=gap.SmallGroup(416,90);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,217,103,218,50,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations