metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C104⋊5C4, C26.4D8, C52.5Q8, C8⋊1Dic13, C2.1D104, C26.2Q16, C2.2Dic52, C4.5Dic26, C22.9D52, (C2×C8).3D13, C13⋊3(C2.D8), (C2×C104).5C2, C52.55(C2×C4), (C2×C4).69D26, (C2×C26).14D4, C26.13(C4⋊C4), C52⋊3C4.3C2, C4.7(C2×Dic13), C2.4(C52⋊3C4), (C2×C52).82C22, SmallGroup(416,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C104⋊5C4
G = < a,b | a104=b4=1, bab-1=a-1 >
Smallest permutation representation of C104⋊5C4
►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)
(1 154 276 333)(2 153 277 332)(3 152 278 331)(4 151 279 330)(5 150 280 329)(6 149 281 328)(7 148 282 327)(8 147 283 326)(9 146 284 325)(10 145 285 324)(11 144 286 323)(12 143 287 322)(13 142 288 321)(14 141 289 320)(15 140 290 319)(16 139 291 318)(17 138 292 317)(18 137 293 316)(19 136 294 315)(20 135 295 314)(21 134 296 313)(22 133 297 416)(23 132 298 415)(24 131 299 414)(25 130 300 413)(26 129 301 412)(27 128 302 411)(28 127 303 410)(29 126 304 409)(30 125 305 408)(31 124 306 407)(32 123 307 406)(33 122 308 405)(34 121 309 404)(35 120 310 403)(36 119 311 402)(37 118 312 401)(38 117 209 400)(39 116 210 399)(40 115 211 398)(41 114 212 397)(42 113 213 396)(43 112 214 395)(44 111 215 394)(45 110 216 393)(46 109 217 392)(47 108 218 391)(48 107 219 390)(49 106 220 389)(50 105 221 388)(51 208 222 387)(52 207 223 386)(53 206 224 385)(54 205 225 384)(55 204 226 383)(56 203 227 382)(57 202 228 381)(58 201 229 380)(59 200 230 379)(60 199 231 378)(61 198 232 377)(62 197 233 376)(63 196 234 375)(64 195 235 374)(65 194 236 373)(66 193 237 372)(67 192 238 371)(68 191 239 370)(69 190 240 369)(70 189 241 368)(71 188 242 367)(72 187 243 366)(73 186 244 365)(74 185 245 364)(75 184 246 363)(76 183 247 362)(77 182 248 361)(78 181 249 360)(79 180 250 359)(80 179 251 358)(81 178 252 357)(82 177 253 356)(83 176 254 355)(84 175 255 354)(85 174 256 353)(86 173 257 352)(87 172 258 351)(88 171 259 350)(89 170 260 349)(90 169 261 348)(91 168 262 347)(92 167 263 346)(93 166 264 345)(94 165 265 344)(95 164 266 343)(96 163 267 342)(97 162 268 341)(98 161 269 340)(99 160 270 339)(100 159 271 338)(101 158 272 337)(102 157 273 336)(103 156 274 335)(104 155 275 334)
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), (1,154,276,333)(2,153,277,332)(3,152,278,331)(4,151,279,330)(5,150,280,329)(6,149,281,328)(7,148,282,327)(8,147,283,326)(9,146,284,325)(10,145,285,324)(11,144,286,323)(12,143,287,322)(13,142,288,321)(14,141,289,320)(15,140,290,319)(16,139,291,318)(17,138,292,317)(18,137,293,316)(19,136,294,315)(20,135,295,314)(21,134,296,313)(22,133,297,416)(23,132,298,415)(24,131,299,414)(25,130,300,413)(26,129,301,412)(27,128,302,411)(28,127,303,410)(29,126,304,409)(30,125,305,408)(31,124,306,407)(32,123,307,406)(33,122,308,405)(34,121,309,404)(35,120,310,403)(36,119,311,402)(37,118,312,401)(38,117,209,400)(39,116,210,399)(40,115,211,398)(41,114,212,397)(42,113,213,396)(43,112,214,395)(44,111,215,394)(45,110,216,393)(46,109,217,392)(47,108,218,391)(48,107,219,390)(49,106,220,389)(50,105,221,388)(51,208,222,387)(52,207,223,386)(53,206,224,385)(54,205,225,384)(55,204,226,383)(56,203,227,382)(57,202,228,381)(58,201,229,380)(59,200,230,379)(60,199,231,378)(61,198,232,377)(62,197,233,376)(63,196,234,375)(64,195,235,374)(65,194,236,373)(66,193,237,372)(67,192,238,371)(68,191,239,370)(69,190,240,369)(70,189,241,368)(71,188,242,367)(72,187,243,366)(73,186,244,365)(74,185,245,364)(75,184,246,363)(76,183,247,362)(77,182,248,361)(78,181,249,360)(79,180,250,359)(80,179,251,358)(81,178,252,357)(82,177,253,356)(83,176,254,355)(84,175,255,354)(85,174,256,353)(86,173,257,352)(87,172,258,351)(88,171,259,350)(89,170,260,349)(90,169,261,348)(91,168,262,347)(92,167,263,346)(93,166,264,345)(94,165,265,344)(95,164,266,343)(96,163,267,342)(97,162,268,341)(98,161,269,340)(99,160,270,339)(100,159,271,338)(101,158,272,337)(102,157,273,336)(103,156,274,335)(104,155,275,334)>;
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), (1,154,276,333)(2,153,277,332)(3,152,278,331)(4,151,279,330)(5,150,280,329)(6,149,281,328)(7,148,282,327)(8,147,283,326)(9,146,284,325)(10,145,285,324)(11,144,286,323)(12,143,287,322)(13,142,288,321)(14,141,289,320)(15,140,290,319)(16,139,291,318)(17,138,292,317)(18,137,293,316)(19,136,294,315)(20,135,295,314)(21,134,296,313)(22,133,297,416)(23,132,298,415)(24,131,299,414)(25,130,300,413)(26,129,301,412)(27,128,302,411)(28,127,303,410)(29,126,304,409)(30,125,305,408)(31,124,306,407)(32,123,307,406)(33,122,308,405)(34,121,309,404)(35,120,310,403)(36,119,311,402)(37,118,312,401)(38,117,209,400)(39,116,210,399)(40,115,211,398)(41,114,212,397)(42,113,213,396)(43,112,214,395)(44,111,215,394)(45,110,216,393)(46,109,217,392)(47,108,218,391)(48,107,219,390)(49,106,220,389)(50,105,221,388)(51,208,222,387)(52,207,223,386)(53,206,224,385)(54,205,225,384)(55,204,226,383)(56,203,227,382)(57,202,228,381)(58,201,229,380)(59,200,230,379)(60,199,231,378)(61,198,232,377)(62,197,233,376)(63,196,234,375)(64,195,235,374)(65,194,236,373)(66,193,237,372)(67,192,238,371)(68,191,239,370)(69,190,240,369)(70,189,241,368)(71,188,242,367)(72,187,243,366)(73,186,244,365)(74,185,245,364)(75,184,246,363)(76,183,247,362)(77,182,248,361)(78,181,249,360)(79,180,250,359)(80,179,251,358)(81,178,252,357)(82,177,253,356)(83,176,254,355)(84,175,255,354)(85,174,256,353)(86,173,257,352)(87,172,258,351)(88,171,259,350)(89,170,260,349)(90,169,261,348)(91,168,262,347)(92,167,263,346)(93,166,264,345)(94,165,265,344)(95,164,266,343)(96,163,267,342)(97,162,268,341)(98,161,269,340)(99,160,270,339)(100,159,271,338)(101,158,272,337)(102,157,273,336)(103,156,274,335)(104,155,275,334) );
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)], [(1,154,276,333),(2,153,277,332),(3,152,278,331),(4,151,279,330),(5,150,280,329),(6,149,281,328),(7,148,282,327),(8,147,283,326),(9,146,284,325),(10,145,285,324),(11,144,286,323),(12,143,287,322),(13,142,288,321),(14,141,289,320),(15,140,290,319),(16,139,291,318),(17,138,292,317),(18,137,293,316),(19,136,294,315),(20,135,295,314),(21,134,296,313),(22,133,297,416),(23,132,298,415),(24,131,299,414),(25,130,300,413),(26,129,301,412),(27,128,302,411),(28,127,303,410),(29,126,304,409),(30,125,305,408),(31,124,306,407),(32,123,307,406),(33,122,308,405),(34,121,309,404),(35,120,310,403),(36,119,311,402),(37,118,312,401),(38,117,209,400),(39,116,210,399),(40,115,211,398),(41,114,212,397),(42,113,213,396),(43,112,214,395),(44,111,215,394),(45,110,216,393),(46,109,217,392),(47,108,218,391),(48,107,219,390),(49,106,220,389),(50,105,221,388),(51,208,222,387),(52,207,223,386),(53,206,224,385),(54,205,225,384),(55,204,226,383),(56,203,227,382),(57,202,228,381),(58,201,229,380),(59,200,230,379),(60,199,231,378),(61,198,232,377),(62,197,233,376),(63,196,234,375),(64,195,235,374),(65,194,236,373),(66,193,237,372),(67,192,238,371),(68,191,239,370),(69,190,240,369),(70,189,241,368),(71,188,242,367),(72,187,243,366),(73,186,244,365),(74,185,245,364),(75,184,246,363),(76,183,247,362),(77,182,248,361),(78,181,249,360),(79,180,250,359),(80,179,251,358),(81,178,252,357),(82,177,253,356),(83,176,254,355),(84,175,255,354),(85,174,256,353),(86,173,257,352),(87,172,258,351),(88,171,259,350),(89,170,260,349),(90,169,261,348),(91,168,262,347),(92,167,263,346),(93,166,264,345),(94,165,265,344),(95,164,266,343),(96,163,267,342),(97,162,268,341),(98,161,269,340),(99,160,270,339),(100,159,271,338),(101,158,272,337),(102,157,273,336),(103,156,274,335),(104,155,275,334)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 52 | 52 | 52 | 52 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | + | - | + | - | + | + | - | |
| image | C1 | C2 | C2 | C4 | Q8 | D4 | D8 | Q16 | D13 | Dic13 | D26 | Dic26 | D52 | D104 | Dic52 |
| kernel | C104⋊5C4 | C52⋊3C4 | C2×C104 | C104 | C52 | C2×C26 | C26 | C26 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 2 | 2 | 6 | 12 | 6 | 12 | 12 | 24 | 24 |
Matrix representation of C104⋊5C4 ►in GL3(𝔽313) generated by
| 1 | 0 | 0 |
| 0 | 33 | 4 |
| 0 | 309 | 28 |
| 25 | 0 | 0 |
| 0 | 194 | 175 |
| 0 | 89 | 119 |
G:=sub<GL(3,GF(313))| [1,0,0,0,33,309,0,4,28],[25,0,0,0,194,89,0,175,119] >;
C104⋊5C4 in GAP, Magma, Sage, TeX
C_{104}\rtimes_5C_4 % in TeX
G:=Group("C104:5C4"); // GroupNames label
G:=SmallGroup(416,25);
// by ID
G=gap.SmallGroup(416,25);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,151,579,69,13829]);
// Polycyclic
G:=Group<a,b|a^104=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
Export