metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C104⋊6C4, C52.4Q8, C8⋊2Dic13, C4.4Dic26, C26.2SD16, C22.8D52, (C2×C8).6D13, C13⋊3(C4.Q8), C52.54(C2×C4), (C2×C104).8C2, (C2×C26).13D4, (C2×C4).68D26, C26.12(C4⋊C4), C52⋊3C4.2C2, C4.6(C2×Dic13), C2.3(C52⋊3C4), C2.2(C104⋊C2), (C2×C52).81C22, SmallGroup(416,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C104⋊6C4
G = < a,b | a104=b4=1, bab-1=a51 >
Smallest permutation representation of C104⋊6C4
►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)
(1 226 347 204)(2 277 348 151)(3 224 349 202)(4 275 350 149)(5 222 351 200)(6 273 352 147)(7 220 353 198)(8 271 354 145)(9 218 355 196)(10 269 356 143)(11 216 357 194)(12 267 358 141)(13 214 359 192)(14 265 360 139)(15 212 361 190)(16 263 362 137)(17 210 363 188)(18 261 364 135)(19 312 365 186)(20 259 366 133)(21 310 367 184)(22 257 368 131)(23 308 369 182)(24 255 370 129)(25 306 371 180)(26 253 372 127)(27 304 373 178)(28 251 374 125)(29 302 375 176)(30 249 376 123)(31 300 377 174)(32 247 378 121)(33 298 379 172)(34 245 380 119)(35 296 381 170)(36 243 382 117)(37 294 383 168)(38 241 384 115)(39 292 385 166)(40 239 386 113)(41 290 387 164)(42 237 388 111)(43 288 389 162)(44 235 390 109)(45 286 391 160)(46 233 392 107)(47 284 393 158)(48 231 394 105)(49 282 395 156)(50 229 396 207)(51 280 397 154)(52 227 398 205)(53 278 399 152)(54 225 400 203)(55 276 401 150)(56 223 402 201)(57 274 403 148)(58 221 404 199)(59 272 405 146)(60 219 406 197)(61 270 407 144)(62 217 408 195)(63 268 409 142)(64 215 410 193)(65 266 411 140)(66 213 412 191)(67 264 413 138)(68 211 414 189)(69 262 415 136)(70 209 416 187)(71 260 313 134)(72 311 314 185)(73 258 315 132)(74 309 316 183)(75 256 317 130)(76 307 318 181)(77 254 319 128)(78 305 320 179)(79 252 321 126)(80 303 322 177)(81 250 323 124)(82 301 324 175)(83 248 325 122)(84 299 326 173)(85 246 327 120)(86 297 328 171)(87 244 329 118)(88 295 330 169)(89 242 331 116)(90 293 332 167)(91 240 333 114)(92 291 334 165)(93 238 335 112)(94 289 336 163)(95 236 337 110)(96 287 338 161)(97 234 339 108)(98 285 340 159)(99 232 341 106)(100 283 342 157)(101 230 343 208)(102 281 344 155)(103 228 345 206)(104 279 346 153)
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), (1,226,347,204)(2,277,348,151)(3,224,349,202)(4,275,350,149)(5,222,351,200)(6,273,352,147)(7,220,353,198)(8,271,354,145)(9,218,355,196)(10,269,356,143)(11,216,357,194)(12,267,358,141)(13,214,359,192)(14,265,360,139)(15,212,361,190)(16,263,362,137)(17,210,363,188)(18,261,364,135)(19,312,365,186)(20,259,366,133)(21,310,367,184)(22,257,368,131)(23,308,369,182)(24,255,370,129)(25,306,371,180)(26,253,372,127)(27,304,373,178)(28,251,374,125)(29,302,375,176)(30,249,376,123)(31,300,377,174)(32,247,378,121)(33,298,379,172)(34,245,380,119)(35,296,381,170)(36,243,382,117)(37,294,383,168)(38,241,384,115)(39,292,385,166)(40,239,386,113)(41,290,387,164)(42,237,388,111)(43,288,389,162)(44,235,390,109)(45,286,391,160)(46,233,392,107)(47,284,393,158)(48,231,394,105)(49,282,395,156)(50,229,396,207)(51,280,397,154)(52,227,398,205)(53,278,399,152)(54,225,400,203)(55,276,401,150)(56,223,402,201)(57,274,403,148)(58,221,404,199)(59,272,405,146)(60,219,406,197)(61,270,407,144)(62,217,408,195)(63,268,409,142)(64,215,410,193)(65,266,411,140)(66,213,412,191)(67,264,413,138)(68,211,414,189)(69,262,415,136)(70,209,416,187)(71,260,313,134)(72,311,314,185)(73,258,315,132)(74,309,316,183)(75,256,317,130)(76,307,318,181)(77,254,319,128)(78,305,320,179)(79,252,321,126)(80,303,322,177)(81,250,323,124)(82,301,324,175)(83,248,325,122)(84,299,326,173)(85,246,327,120)(86,297,328,171)(87,244,329,118)(88,295,330,169)(89,242,331,116)(90,293,332,167)(91,240,333,114)(92,291,334,165)(93,238,335,112)(94,289,336,163)(95,236,337,110)(96,287,338,161)(97,234,339,108)(98,285,340,159)(99,232,341,106)(100,283,342,157)(101,230,343,208)(102,281,344,155)(103,228,345,206)(104,279,346,153)>;
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), (1,226,347,204)(2,277,348,151)(3,224,349,202)(4,275,350,149)(5,222,351,200)(6,273,352,147)(7,220,353,198)(8,271,354,145)(9,218,355,196)(10,269,356,143)(11,216,357,194)(12,267,358,141)(13,214,359,192)(14,265,360,139)(15,212,361,190)(16,263,362,137)(17,210,363,188)(18,261,364,135)(19,312,365,186)(20,259,366,133)(21,310,367,184)(22,257,368,131)(23,308,369,182)(24,255,370,129)(25,306,371,180)(26,253,372,127)(27,304,373,178)(28,251,374,125)(29,302,375,176)(30,249,376,123)(31,300,377,174)(32,247,378,121)(33,298,379,172)(34,245,380,119)(35,296,381,170)(36,243,382,117)(37,294,383,168)(38,241,384,115)(39,292,385,166)(40,239,386,113)(41,290,387,164)(42,237,388,111)(43,288,389,162)(44,235,390,109)(45,286,391,160)(46,233,392,107)(47,284,393,158)(48,231,394,105)(49,282,395,156)(50,229,396,207)(51,280,397,154)(52,227,398,205)(53,278,399,152)(54,225,400,203)(55,276,401,150)(56,223,402,201)(57,274,403,148)(58,221,404,199)(59,272,405,146)(60,219,406,197)(61,270,407,144)(62,217,408,195)(63,268,409,142)(64,215,410,193)(65,266,411,140)(66,213,412,191)(67,264,413,138)(68,211,414,189)(69,262,415,136)(70,209,416,187)(71,260,313,134)(72,311,314,185)(73,258,315,132)(74,309,316,183)(75,256,317,130)(76,307,318,181)(77,254,319,128)(78,305,320,179)(79,252,321,126)(80,303,322,177)(81,250,323,124)(82,301,324,175)(83,248,325,122)(84,299,326,173)(85,246,327,120)(86,297,328,171)(87,244,329,118)(88,295,330,169)(89,242,331,116)(90,293,332,167)(91,240,333,114)(92,291,334,165)(93,238,335,112)(94,289,336,163)(95,236,337,110)(96,287,338,161)(97,234,339,108)(98,285,340,159)(99,232,341,106)(100,283,342,157)(101,230,343,208)(102,281,344,155)(103,228,345,206)(104,279,346,153) );
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)], [(1,226,347,204),(2,277,348,151),(3,224,349,202),(4,275,350,149),(5,222,351,200),(6,273,352,147),(7,220,353,198),(8,271,354,145),(9,218,355,196),(10,269,356,143),(11,216,357,194),(12,267,358,141),(13,214,359,192),(14,265,360,139),(15,212,361,190),(16,263,362,137),(17,210,363,188),(18,261,364,135),(19,312,365,186),(20,259,366,133),(21,310,367,184),(22,257,368,131),(23,308,369,182),(24,255,370,129),(25,306,371,180),(26,253,372,127),(27,304,373,178),(28,251,374,125),(29,302,375,176),(30,249,376,123),(31,300,377,174),(32,247,378,121),(33,298,379,172),(34,245,380,119),(35,296,381,170),(36,243,382,117),(37,294,383,168),(38,241,384,115),(39,292,385,166),(40,239,386,113),(41,290,387,164),(42,237,388,111),(43,288,389,162),(44,235,390,109),(45,286,391,160),(46,233,392,107),(47,284,393,158),(48,231,394,105),(49,282,395,156),(50,229,396,207),(51,280,397,154),(52,227,398,205),(53,278,399,152),(54,225,400,203),(55,276,401,150),(56,223,402,201),(57,274,403,148),(58,221,404,199),(59,272,405,146),(60,219,406,197),(61,270,407,144),(62,217,408,195),(63,268,409,142),(64,215,410,193),(65,266,411,140),(66,213,412,191),(67,264,413,138),(68,211,414,189),(69,262,415,136),(70,209,416,187),(71,260,313,134),(72,311,314,185),(73,258,315,132),(74,309,316,183),(75,256,317,130),(76,307,318,181),(77,254,319,128),(78,305,320,179),(79,252,321,126),(80,303,322,177),(81,250,323,124),(82,301,324,175),(83,248,325,122),(84,299,326,173),(85,246,327,120),(86,297,328,171),(87,244,329,118),(88,295,330,169),(89,242,331,116),(90,293,332,167),(91,240,333,114),(92,291,334,165),(93,238,335,112),(94,289,336,163),(95,236,337,110),(96,287,338,161),(97,234,339,108),(98,285,340,159),(99,232,341,106),(100,283,342,157),(101,230,343,208),(102,281,344,155),(103,228,345,206),(104,279,346,153)]])
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 |
| type | + | + | + | - | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C4 | Q8 | D4 | SD16 | D13 | Dic13 | D26 | Dic26 | D52 | C104⋊C2 |
| kernel | C104⋊6C4 | C52⋊3C4 | C2×C104 | C104 | C52 | C2×C26 | C26 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 4 | 6 | 12 | 6 | 12 | 12 | 48 |
Matrix representation of C104⋊6C4 ►in GL3(𝔽313) generated by
| 312 | 0 | 0 |
| 0 | 130 | 28 |
| 0 | 285 | 6 |
| 25 | 0 | 0 |
| 0 | 200 | 41 |
| 0 | 139 | 113 |
G:=sub<GL(3,GF(313))| [312,0,0,0,130,285,0,28,6],[25,0,0,0,200,139,0,41,113] >;
C104⋊6C4 in GAP, Magma, Sage, TeX
C_{104}\rtimes_6C_4 % in TeX
G:=Group("C104:6C4"); // GroupNames label
G:=SmallGroup(416,24);
// by ID
G=gap.SmallGroup(416,24);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,579,69,13829]);
// Polycyclic
G:=Group<a,b|a^104=b^4=1,b*a*b^-1=a^51>;
// generators/relations
Export