metabelian, supersoluble, monomial, A-group
Aliases: C52⋊4C16, C20.8F5, C5⋊1(C5⋊C16), C10.3(C5⋊C8), (C5×C10).4C8, (C5×C20).7C4, C2.(C52⋊4C8), C52⋊7C8.3C2, C4.2(C5⋊F5), SmallGroup(400,58)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊4C16 |
Generators and relations for C52⋊4C16
G = < a,b,c | a5=b5=c16=1, ab=ba, cac-1=a3, cbc-1=b3 >
Smallest permutation representation of C52⋊4C16
►Regular action on 400 points
Generators in S400
(1 367 132 95 385)(2 96 368 386 133)(3 387 81 134 353)(4 135 388 354 82)(5 355 136 83 389)(6 84 356 390 137)(7 391 85 138 357)(8 139 392 358 86)(9 359 140 87 393)(10 88 360 394 141)(11 395 89 142 361)(12 143 396 362 90)(13 363 144 91 397)(14 92 364 398 129)(15 399 93 130 365)(16 131 400 366 94)(17 215 121 311 77)(18 312 216 78 122)(19 79 313 123 217)(20 124 80 218 314)(21 219 125 315 65)(22 316 220 66 126)(23 67 317 127 221)(24 128 68 222 318)(25 223 113 319 69)(26 320 224 70 114)(27 71 305 115 209)(28 116 72 210 306)(29 211 117 307 73)(30 308 212 74 118)(31 75 309 119 213)(32 120 76 214 310)(33 106 56 149 269)(34 150 107 270 57)(35 271 151 58 108)(36 59 272 109 152)(37 110 60 153 257)(38 154 111 258 61)(39 259 155 62 112)(40 63 260 97 156)(41 98 64 157 261)(42 158 99 262 49)(43 263 159 50 100)(44 51 264 101 160)(45 102 52 145 265)(46 146 103 266 53)(47 267 147 54 104)(48 55 268 105 148)(161 237 344 195 296)(162 196 238 297 345)(163 298 197 346 239)(164 347 299 240 198)(165 225 348 199 300)(166 200 226 301 349)(167 302 201 350 227)(168 351 303 228 202)(169 229 352 203 304)(170 204 230 289 337)(171 290 205 338 231)(172 339 291 232 206)(173 233 340 207 292)(174 208 234 293 341)(175 294 193 342 235)(176 343 295 236 194)(177 243 382 328 277)(178 329 244 278 383)(179 279 330 384 245)(180 369 280 246 331)(181 247 370 332 281)(182 333 248 282 371)(183 283 334 372 249)(184 373 284 250 335)(185 251 374 336 285)(186 321 252 286 375)(187 287 322 376 253)(188 377 288 254 323)(189 255 378 324 273)(190 325 256 274 379)(191 275 326 380 241)(192 381 276 242 327)
(1 154 233 80 384)(2 65 155 369 234)(3 370 66 235 156)(4 236 371 157 67)(5 158 237 68 372)(6 69 159 373 238)(7 374 70 239 160)(8 240 375 145 71)(9 146 225 72 376)(10 73 147 377 226)(11 378 74 227 148)(12 228 379 149 75)(13 150 229 76 380)(14 77 151 381 230)(15 382 78 231 152)(16 232 383 153 79)(17 58 276 289 92)(18 290 59 93 277)(19 94 291 278 60)(20 279 95 61 292)(21 62 280 293 96)(22 294 63 81 281)(23 82 295 282 64)(24 283 83 49 296)(25 50 284 297 84)(26 298 51 85 285)(27 86 299 286 52)(28 287 87 53 300)(29 54 288 301 88)(30 302 55 89 273)(31 90 303 274 56)(32 275 91 57 304)(33 119 396 168 325)(34 169 120 326 397)(35 327 170 398 121)(36 399 328 122 171)(37 123 400 172 329)(38 173 124 330 385)(39 331 174 386 125)(40 387 332 126 175)(41 127 388 176 333)(42 161 128 334 389)(43 335 162 390 113)(44 391 336 114 163)(45 115 392 164 321)(46 165 116 322 393)(47 323 166 394 117)(48 395 324 118 167)(97 353 247 220 342)(98 221 354 343 248)(99 344 222 249 355)(100 250 345 356 223)(101 357 251 224 346)(102 209 358 347 252)(103 348 210 253 359)(104 254 349 360 211)(105 361 255 212 350)(106 213 362 351 256)(107 352 214 241 363)(108 242 337 364 215)(109 365 243 216 338)(110 217 366 339 244)(111 340 218 245 367)(112 246 341 368 219)(129 311 271 192 204)(130 177 312 205 272)(131 206 178 257 313)(132 258 207 314 179)(133 315 259 180 208)(134 181 316 193 260)(135 194 182 261 317)(136 262 195 318 183)(137 319 263 184 196)(138 185 320 197 264)(139 198 186 265 305)(140 266 199 306 187)(141 307 267 188 200)(142 189 308 201 268)(143 202 190 269 309)(144 270 203 310 191)
(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)
G:=sub<Sym(400)| (1,367,132,95,385)(2,96,368,386,133)(3,387,81,134,353)(4,135,388,354,82)(5,355,136,83,389)(6,84,356,390,137)(7,391,85,138,357)(8,139,392,358,86)(9,359,140,87,393)(10,88,360,394,141)(11,395,89,142,361)(12,143,396,362,90)(13,363,144,91,397)(14,92,364,398,129)(15,399,93,130,365)(16,131,400,366,94)(17,215,121,311,77)(18,312,216,78,122)(19,79,313,123,217)(20,124,80,218,314)(21,219,125,315,65)(22,316,220,66,126)(23,67,317,127,221)(24,128,68,222,318)(25,223,113,319,69)(26,320,224,70,114)(27,71,305,115,209)(28,116,72,210,306)(29,211,117,307,73)(30,308,212,74,118)(31,75,309,119,213)(32,120,76,214,310)(33,106,56,149,269)(34,150,107,270,57)(35,271,151,58,108)(36,59,272,109,152)(37,110,60,153,257)(38,154,111,258,61)(39,259,155,62,112)(40,63,260,97,156)(41,98,64,157,261)(42,158,99,262,49)(43,263,159,50,100)(44,51,264,101,160)(45,102,52,145,265)(46,146,103,266,53)(47,267,147,54,104)(48,55,268,105,148)(161,237,344,195,296)(162,196,238,297,345)(163,298,197,346,239)(164,347,299,240,198)(165,225,348,199,300)(166,200,226,301,349)(167,302,201,350,227)(168,351,303,228,202)(169,229,352,203,304)(170,204,230,289,337)(171,290,205,338,231)(172,339,291,232,206)(173,233,340,207,292)(174,208,234,293,341)(175,294,193,342,235)(176,343,295,236,194)(177,243,382,328,277)(178,329,244,278,383)(179,279,330,384,245)(180,369,280,246,331)(181,247,370,332,281)(182,333,248,282,371)(183,283,334,372,249)(184,373,284,250,335)(185,251,374,336,285)(186,321,252,286,375)(187,287,322,376,253)(188,377,288,254,323)(189,255,378,324,273)(190,325,256,274,379)(191,275,326,380,241)(192,381,276,242,327), (1,154,233,80,384)(2,65,155,369,234)(3,370,66,235,156)(4,236,371,157,67)(5,158,237,68,372)(6,69,159,373,238)(7,374,70,239,160)(8,240,375,145,71)(9,146,225,72,376)(10,73,147,377,226)(11,378,74,227,148)(12,228,379,149,75)(13,150,229,76,380)(14,77,151,381,230)(15,382,78,231,152)(16,232,383,153,79)(17,58,276,289,92)(18,290,59,93,277)(19,94,291,278,60)(20,279,95,61,292)(21,62,280,293,96)(22,294,63,81,281)(23,82,295,282,64)(24,283,83,49,296)(25,50,284,297,84)(26,298,51,85,285)(27,86,299,286,52)(28,287,87,53,300)(29,54,288,301,88)(30,302,55,89,273)(31,90,303,274,56)(32,275,91,57,304)(33,119,396,168,325)(34,169,120,326,397)(35,327,170,398,121)(36,399,328,122,171)(37,123,400,172,329)(38,173,124,330,385)(39,331,174,386,125)(40,387,332,126,175)(41,127,388,176,333)(42,161,128,334,389)(43,335,162,390,113)(44,391,336,114,163)(45,115,392,164,321)(46,165,116,322,393)(47,323,166,394,117)(48,395,324,118,167)(97,353,247,220,342)(98,221,354,343,248)(99,344,222,249,355)(100,250,345,356,223)(101,357,251,224,346)(102,209,358,347,252)(103,348,210,253,359)(104,254,349,360,211)(105,361,255,212,350)(106,213,362,351,256)(107,352,214,241,363)(108,242,337,364,215)(109,365,243,216,338)(110,217,366,339,244)(111,340,218,245,367)(112,246,341,368,219)(129,311,271,192,204)(130,177,312,205,272)(131,206,178,257,313)(132,258,207,314,179)(133,315,259,180,208)(134,181,316,193,260)(135,194,182,261,317)(136,262,195,318,183)(137,319,263,184,196)(138,185,320,197,264)(139,198,186,265,305)(140,266,199,306,187)(141,307,267,188,200)(142,189,308,201,268)(143,202,190,269,309)(144,270,203,310,191), (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)>;
G:=Group( (1,367,132,95,385)(2,96,368,386,133)(3,387,81,134,353)(4,135,388,354,82)(5,355,136,83,389)(6,84,356,390,137)(7,391,85,138,357)(8,139,392,358,86)(9,359,140,87,393)(10,88,360,394,141)(11,395,89,142,361)(12,143,396,362,90)(13,363,144,91,397)(14,92,364,398,129)(15,399,93,130,365)(16,131,400,366,94)(17,215,121,311,77)(18,312,216,78,122)(19,79,313,123,217)(20,124,80,218,314)(21,219,125,315,65)(22,316,220,66,126)(23,67,317,127,221)(24,128,68,222,318)(25,223,113,319,69)(26,320,224,70,114)(27,71,305,115,209)(28,116,72,210,306)(29,211,117,307,73)(30,308,212,74,118)(31,75,309,119,213)(32,120,76,214,310)(33,106,56,149,269)(34,150,107,270,57)(35,271,151,58,108)(36,59,272,109,152)(37,110,60,153,257)(38,154,111,258,61)(39,259,155,62,112)(40,63,260,97,156)(41,98,64,157,261)(42,158,99,262,49)(43,263,159,50,100)(44,51,264,101,160)(45,102,52,145,265)(46,146,103,266,53)(47,267,147,54,104)(48,55,268,105,148)(161,237,344,195,296)(162,196,238,297,345)(163,298,197,346,239)(164,347,299,240,198)(165,225,348,199,300)(166,200,226,301,349)(167,302,201,350,227)(168,351,303,228,202)(169,229,352,203,304)(170,204,230,289,337)(171,290,205,338,231)(172,339,291,232,206)(173,233,340,207,292)(174,208,234,293,341)(175,294,193,342,235)(176,343,295,236,194)(177,243,382,328,277)(178,329,244,278,383)(179,279,330,384,245)(180,369,280,246,331)(181,247,370,332,281)(182,333,248,282,371)(183,283,334,372,249)(184,373,284,250,335)(185,251,374,336,285)(186,321,252,286,375)(187,287,322,376,253)(188,377,288,254,323)(189,255,378,324,273)(190,325,256,274,379)(191,275,326,380,241)(192,381,276,242,327), (1,154,233,80,384)(2,65,155,369,234)(3,370,66,235,156)(4,236,371,157,67)(5,158,237,68,372)(6,69,159,373,238)(7,374,70,239,160)(8,240,375,145,71)(9,146,225,72,376)(10,73,147,377,226)(11,378,74,227,148)(12,228,379,149,75)(13,150,229,76,380)(14,77,151,381,230)(15,382,78,231,152)(16,232,383,153,79)(17,58,276,289,92)(18,290,59,93,277)(19,94,291,278,60)(20,279,95,61,292)(21,62,280,293,96)(22,294,63,81,281)(23,82,295,282,64)(24,283,83,49,296)(25,50,284,297,84)(26,298,51,85,285)(27,86,299,286,52)(28,287,87,53,300)(29,54,288,301,88)(30,302,55,89,273)(31,90,303,274,56)(32,275,91,57,304)(33,119,396,168,325)(34,169,120,326,397)(35,327,170,398,121)(36,399,328,122,171)(37,123,400,172,329)(38,173,124,330,385)(39,331,174,386,125)(40,387,332,126,175)(41,127,388,176,333)(42,161,128,334,389)(43,335,162,390,113)(44,391,336,114,163)(45,115,392,164,321)(46,165,116,322,393)(47,323,166,394,117)(48,395,324,118,167)(97,353,247,220,342)(98,221,354,343,248)(99,344,222,249,355)(100,250,345,356,223)(101,357,251,224,346)(102,209,358,347,252)(103,348,210,253,359)(104,254,349,360,211)(105,361,255,212,350)(106,213,362,351,256)(107,352,214,241,363)(108,242,337,364,215)(109,365,243,216,338)(110,217,366,339,244)(111,340,218,245,367)(112,246,341,368,219)(129,311,271,192,204)(130,177,312,205,272)(131,206,178,257,313)(132,258,207,314,179)(133,315,259,180,208)(134,181,316,193,260)(135,194,182,261,317)(136,262,195,318,183)(137,319,263,184,196)(138,185,320,197,264)(139,198,186,265,305)(140,266,199,306,187)(141,307,267,188,200)(142,189,308,201,268)(143,202,190,269,309)(144,270,203,310,191), (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) );
G=PermutationGroup([[(1,367,132,95,385),(2,96,368,386,133),(3,387,81,134,353),(4,135,388,354,82),(5,355,136,83,389),(6,84,356,390,137),(7,391,85,138,357),(8,139,392,358,86),(9,359,140,87,393),(10,88,360,394,141),(11,395,89,142,361),(12,143,396,362,90),(13,363,144,91,397),(14,92,364,398,129),(15,399,93,130,365),(16,131,400,366,94),(17,215,121,311,77),(18,312,216,78,122),(19,79,313,123,217),(20,124,80,218,314),(21,219,125,315,65),(22,316,220,66,126),(23,67,317,127,221),(24,128,68,222,318),(25,223,113,319,69),(26,320,224,70,114),(27,71,305,115,209),(28,116,72,210,306),(29,211,117,307,73),(30,308,212,74,118),(31,75,309,119,213),(32,120,76,214,310),(33,106,56,149,269),(34,150,107,270,57),(35,271,151,58,108),(36,59,272,109,152),(37,110,60,153,257),(38,154,111,258,61),(39,259,155,62,112),(40,63,260,97,156),(41,98,64,157,261),(42,158,99,262,49),(43,263,159,50,100),(44,51,264,101,160),(45,102,52,145,265),(46,146,103,266,53),(47,267,147,54,104),(48,55,268,105,148),(161,237,344,195,296),(162,196,238,297,345),(163,298,197,346,239),(164,347,299,240,198),(165,225,348,199,300),(166,200,226,301,349),(167,302,201,350,227),(168,351,303,228,202),(169,229,352,203,304),(170,204,230,289,337),(171,290,205,338,231),(172,339,291,232,206),(173,233,340,207,292),(174,208,234,293,341),(175,294,193,342,235),(176,343,295,236,194),(177,243,382,328,277),(178,329,244,278,383),(179,279,330,384,245),(180,369,280,246,331),(181,247,370,332,281),(182,333,248,282,371),(183,283,334,372,249),(184,373,284,250,335),(185,251,374,336,285),(186,321,252,286,375),(187,287,322,376,253),(188,377,288,254,323),(189,255,378,324,273),(190,325,256,274,379),(191,275,326,380,241),(192,381,276,242,327)], [(1,154,233,80,384),(2,65,155,369,234),(3,370,66,235,156),(4,236,371,157,67),(5,158,237,68,372),(6,69,159,373,238),(7,374,70,239,160),(8,240,375,145,71),(9,146,225,72,376),(10,73,147,377,226),(11,378,74,227,148),(12,228,379,149,75),(13,150,229,76,380),(14,77,151,381,230),(15,382,78,231,152),(16,232,383,153,79),(17,58,276,289,92),(18,290,59,93,277),(19,94,291,278,60),(20,279,95,61,292),(21,62,280,293,96),(22,294,63,81,281),(23,82,295,282,64),(24,283,83,49,296),(25,50,284,297,84),(26,298,51,85,285),(27,86,299,286,52),(28,287,87,53,300),(29,54,288,301,88),(30,302,55,89,273),(31,90,303,274,56),(32,275,91,57,304),(33,119,396,168,325),(34,169,120,326,397),(35,327,170,398,121),(36,399,328,122,171),(37,123,400,172,329),(38,173,124,330,385),(39,331,174,386,125),(40,387,332,126,175),(41,127,388,176,333),(42,161,128,334,389),(43,335,162,390,113),(44,391,336,114,163),(45,115,392,164,321),(46,165,116,322,393),(47,323,166,394,117),(48,395,324,118,167),(97,353,247,220,342),(98,221,354,343,248),(99,344,222,249,355),(100,250,345,356,223),(101,357,251,224,346),(102,209,358,347,252),(103,348,210,253,359),(104,254,349,360,211),(105,361,255,212,350),(106,213,362,351,256),(107,352,214,241,363),(108,242,337,364,215),(109,365,243,216,338),(110,217,366,339,244),(111,340,218,245,367),(112,246,341,368,219),(129,311,271,192,204),(130,177,312,205,272),(131,206,178,257,313),(132,258,207,314,179),(133,315,259,180,208),(134,181,316,193,260),(135,194,182,261,317),(136,262,195,318,183),(137,319,263,184,196),(138,185,320,197,264),(139,198,186,265,305),(140,266,199,306,187),(141,307,267,188,200),(142,189,308,201,268),(143,202,190,269,309),(144,270,203,310,191)], [(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)]])
40 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | ··· | 5F | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 16A | ··· | 16H | 20A | ··· | 20L |
| order | 1 | 2 | 4 | 4 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 25 | 25 | 25 | 25 | 4 | ··· | 4 | 25 | ··· | 25 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | - | ||||
| image | C1 | C2 | C4 | C8 | C16 | F5 | C5⋊C8 | C5⋊C16 |
| kernel | C52⋊4C16 | C52⋊7C8 | C5×C20 | C5×C10 | C52 | C20 | C10 | C5 |
| # reps | 1 | 1 | 2 | 4 | 8 | 6 | 6 | 12 |
Matrix representation of C52⋊4C16 ►in GL8(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 240 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 240 | 240 | 240 |
| 142 | 4 | 185 | 101 | 0 | 0 | 0 | 0 |
| 181 | 97 | 237 | 138 | 0 | 0 | 0 | 0 |
| 140 | 41 | 144 | 84 | 0 | 0 | 0 | 0 |
| 103 | 43 | 200 | 99 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 90 | 227 | 94 | 27 |
| 0 | 0 | 0 | 0 | 108 | 41 | 14 | 104 |
| 0 | 0 | 0 | 0 | 214 | 63 | 200 | 67 |
| 0 | 0 | 0 | 0 | 137 | 4 | 178 | 151 |
G:=sub<GL(8,GF(241))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,1,0,0,0,0,0,240,0,0,1,0,0,0,0,240,0,0,0],[0,0,0,240,0,0,0,0,1,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240],[142,181,140,103,0,0,0,0,4,97,41,43,0,0,0,0,185,237,144,200,0,0,0,0,101,138,84,99,0,0,0,0,0,0,0,0,90,108,214,137,0,0,0,0,227,41,63,4,0,0,0,0,94,14,200,178,0,0,0,0,27,104,67,151] >;
C52⋊4C16 in GAP, Magma, Sage, TeX
C_5^2\rtimes_4C_{16} % in TeX
G:=Group("C5^2:4C16"); // GroupNames label
G:=SmallGroup(400,58);
// by ID
G=gap.SmallGroup(400,58);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,12,31,50,964,970,5765,5771]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^16=1,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^3>;
// generators/relations
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