metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C104⋊8C4, C8⋊3Dic13, C26.9C42, C26.7M4(2), C13⋊2C8⋊4C4, (C2×C8).8D13, C13⋊4(C8⋊C4), C52.62(C2×C4), C4.21(C4×D13), (C2×C4).92D26, (C2×C104).12C2, C2.4(C4×Dic13), C2.2(C8⋊D13), (C4×Dic13).6C2, (C2×Dic13).4C4, C4.13(C2×Dic13), C22.10(C4×D13), (C2×C52).106C22, (C2×C26).31(C2×C4), (C2×C13⋊2C8).10C2, SmallGroup(416,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C104⋊8C4
G = < a,b | a104=b4=1, bab-1=a77 >
Smallest permutation representation of C104⋊8C4
►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)
(1 319 163 239)(2 396 164 212)(3 369 165 289)(4 342 166 262)(5 315 167 235)(6 392 168 312)(7 365 169 285)(8 338 170 258)(9 415 171 231)(10 388 172 308)(11 361 173 281)(12 334 174 254)(13 411 175 227)(14 384 176 304)(15 357 177 277)(16 330 178 250)(17 407 179 223)(18 380 180 300)(19 353 181 273)(20 326 182 246)(21 403 183 219)(22 376 184 296)(23 349 185 269)(24 322 186 242)(25 399 187 215)(26 372 188 292)(27 345 189 265)(28 318 190 238)(29 395 191 211)(30 368 192 288)(31 341 193 261)(32 314 194 234)(33 391 195 311)(34 364 196 284)(35 337 197 257)(36 414 198 230)(37 387 199 307)(38 360 200 280)(39 333 201 253)(40 410 202 226)(41 383 203 303)(42 356 204 276)(43 329 205 249)(44 406 206 222)(45 379 207 299)(46 352 208 272)(47 325 105 245)(48 402 106 218)(49 375 107 295)(50 348 108 268)(51 321 109 241)(52 398 110 214)(53 371 111 291)(54 344 112 264)(55 317 113 237)(56 394 114 210)(57 367 115 287)(58 340 116 260)(59 313 117 233)(60 390 118 310)(61 363 119 283)(62 336 120 256)(63 413 121 229)(64 386 122 306)(65 359 123 279)(66 332 124 252)(67 409 125 225)(68 382 126 302)(69 355 127 275)(70 328 128 248)(71 405 129 221)(72 378 130 298)(73 351 131 271)(74 324 132 244)(75 401 133 217)(76 374 134 294)(77 347 135 267)(78 320 136 240)(79 397 137 213)(80 370 138 290)(81 343 139 263)(82 316 140 236)(83 393 141 209)(84 366 142 286)(85 339 143 259)(86 416 144 232)(87 389 145 309)(88 362 146 282)(89 335 147 255)(90 412 148 228)(91 385 149 305)(92 358 150 278)(93 331 151 251)(94 408 152 224)(95 381 153 301)(96 354 154 274)(97 327 155 247)(98 404 156 220)(99 377 157 297)(100 350 158 270)(101 323 159 243)(102 400 160 216)(103 373 161 293)(104 346 162 266)
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), (1,319,163,239)(2,396,164,212)(3,369,165,289)(4,342,166,262)(5,315,167,235)(6,392,168,312)(7,365,169,285)(8,338,170,258)(9,415,171,231)(10,388,172,308)(11,361,173,281)(12,334,174,254)(13,411,175,227)(14,384,176,304)(15,357,177,277)(16,330,178,250)(17,407,179,223)(18,380,180,300)(19,353,181,273)(20,326,182,246)(21,403,183,219)(22,376,184,296)(23,349,185,269)(24,322,186,242)(25,399,187,215)(26,372,188,292)(27,345,189,265)(28,318,190,238)(29,395,191,211)(30,368,192,288)(31,341,193,261)(32,314,194,234)(33,391,195,311)(34,364,196,284)(35,337,197,257)(36,414,198,230)(37,387,199,307)(38,360,200,280)(39,333,201,253)(40,410,202,226)(41,383,203,303)(42,356,204,276)(43,329,205,249)(44,406,206,222)(45,379,207,299)(46,352,208,272)(47,325,105,245)(48,402,106,218)(49,375,107,295)(50,348,108,268)(51,321,109,241)(52,398,110,214)(53,371,111,291)(54,344,112,264)(55,317,113,237)(56,394,114,210)(57,367,115,287)(58,340,116,260)(59,313,117,233)(60,390,118,310)(61,363,119,283)(62,336,120,256)(63,413,121,229)(64,386,122,306)(65,359,123,279)(66,332,124,252)(67,409,125,225)(68,382,126,302)(69,355,127,275)(70,328,128,248)(71,405,129,221)(72,378,130,298)(73,351,131,271)(74,324,132,244)(75,401,133,217)(76,374,134,294)(77,347,135,267)(78,320,136,240)(79,397,137,213)(80,370,138,290)(81,343,139,263)(82,316,140,236)(83,393,141,209)(84,366,142,286)(85,339,143,259)(86,416,144,232)(87,389,145,309)(88,362,146,282)(89,335,147,255)(90,412,148,228)(91,385,149,305)(92,358,150,278)(93,331,151,251)(94,408,152,224)(95,381,153,301)(96,354,154,274)(97,327,155,247)(98,404,156,220)(99,377,157,297)(100,350,158,270)(101,323,159,243)(102,400,160,216)(103,373,161,293)(104,346,162,266)>;
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), (1,319,163,239)(2,396,164,212)(3,369,165,289)(4,342,166,262)(5,315,167,235)(6,392,168,312)(7,365,169,285)(8,338,170,258)(9,415,171,231)(10,388,172,308)(11,361,173,281)(12,334,174,254)(13,411,175,227)(14,384,176,304)(15,357,177,277)(16,330,178,250)(17,407,179,223)(18,380,180,300)(19,353,181,273)(20,326,182,246)(21,403,183,219)(22,376,184,296)(23,349,185,269)(24,322,186,242)(25,399,187,215)(26,372,188,292)(27,345,189,265)(28,318,190,238)(29,395,191,211)(30,368,192,288)(31,341,193,261)(32,314,194,234)(33,391,195,311)(34,364,196,284)(35,337,197,257)(36,414,198,230)(37,387,199,307)(38,360,200,280)(39,333,201,253)(40,410,202,226)(41,383,203,303)(42,356,204,276)(43,329,205,249)(44,406,206,222)(45,379,207,299)(46,352,208,272)(47,325,105,245)(48,402,106,218)(49,375,107,295)(50,348,108,268)(51,321,109,241)(52,398,110,214)(53,371,111,291)(54,344,112,264)(55,317,113,237)(56,394,114,210)(57,367,115,287)(58,340,116,260)(59,313,117,233)(60,390,118,310)(61,363,119,283)(62,336,120,256)(63,413,121,229)(64,386,122,306)(65,359,123,279)(66,332,124,252)(67,409,125,225)(68,382,126,302)(69,355,127,275)(70,328,128,248)(71,405,129,221)(72,378,130,298)(73,351,131,271)(74,324,132,244)(75,401,133,217)(76,374,134,294)(77,347,135,267)(78,320,136,240)(79,397,137,213)(80,370,138,290)(81,343,139,263)(82,316,140,236)(83,393,141,209)(84,366,142,286)(85,339,143,259)(86,416,144,232)(87,389,145,309)(88,362,146,282)(89,335,147,255)(90,412,148,228)(91,385,149,305)(92,358,150,278)(93,331,151,251)(94,408,152,224)(95,381,153,301)(96,354,154,274)(97,327,155,247)(98,404,156,220)(99,377,157,297)(100,350,158,270)(101,323,159,243)(102,400,160,216)(103,373,161,293)(104,346,162,266) );
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)], [(1,319,163,239),(2,396,164,212),(3,369,165,289),(4,342,166,262),(5,315,167,235),(6,392,168,312),(7,365,169,285),(8,338,170,258),(9,415,171,231),(10,388,172,308),(11,361,173,281),(12,334,174,254),(13,411,175,227),(14,384,176,304),(15,357,177,277),(16,330,178,250),(17,407,179,223),(18,380,180,300),(19,353,181,273),(20,326,182,246),(21,403,183,219),(22,376,184,296),(23,349,185,269),(24,322,186,242),(25,399,187,215),(26,372,188,292),(27,345,189,265),(28,318,190,238),(29,395,191,211),(30,368,192,288),(31,341,193,261),(32,314,194,234),(33,391,195,311),(34,364,196,284),(35,337,197,257),(36,414,198,230),(37,387,199,307),(38,360,200,280),(39,333,201,253),(40,410,202,226),(41,383,203,303),(42,356,204,276),(43,329,205,249),(44,406,206,222),(45,379,207,299),(46,352,208,272),(47,325,105,245),(48,402,106,218),(49,375,107,295),(50,348,108,268),(51,321,109,241),(52,398,110,214),(53,371,111,291),(54,344,112,264),(55,317,113,237),(56,394,114,210),(57,367,115,287),(58,340,116,260),(59,313,117,233),(60,390,118,310),(61,363,119,283),(62,336,120,256),(63,413,121,229),(64,386,122,306),(65,359,123,279),(66,332,124,252),(67,409,125,225),(68,382,126,302),(69,355,127,275),(70,328,128,248),(71,405,129,221),(72,378,130,298),(73,351,131,271),(74,324,132,244),(75,401,133,217),(76,374,134,294),(77,347,135,267),(78,320,136,240),(79,397,137,213),(80,370,138,290),(81,343,139,263),(82,316,140,236),(83,393,141,209),(84,366,142,286),(85,339,143,259),(86,416,144,232),(87,389,145,309),(88,362,146,282),(89,335,147,255),(90,412,148,228),(91,385,149,305),(92,358,150,278),(93,331,151,251),(94,408,152,224),(95,381,153,301),(96,354,154,274),(97,327,155,247),(98,404,156,220),(99,377,157,297),(100,350,158,270),(101,323,159,243),(102,400,160,216),(103,373,161,293),(104,346,162,266)]])
116 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 26 | 26 | 26 | 26 | 2 | 2 | 2 | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | D13 | Dic13 | D26 | C4×D13 | C4×D13 | C8⋊D13 |
| kernel | C104⋊8C4 | C2×C13⋊2C8 | C4×Dic13 | C2×C104 | C13⋊2C8 | C104 | C2×Dic13 | C26 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 6 | 12 | 6 | 12 | 12 | 48 |
Matrix representation of C104⋊8C4 ►in GL4(𝔽313) generated by
| 294 | 49 | 0 | 0 |
| 264 | 77 | 0 | 0 |
| 0 | 0 | 24 | 186 |
| 0 | 0 | 168 | 62 |
| 1 | 247 | 0 | 0 |
| 19 | 312 | 0 | 0 |
| 0 | 0 | 3 | 235 |
| 0 | 0 | 273 | 310 |
G:=sub<GL(4,GF(313))| [294,264,0,0,49,77,0,0,0,0,24,168,0,0,186,62],[1,19,0,0,247,312,0,0,0,0,3,273,0,0,235,310] >;
C104⋊8C4 in GAP, Magma, Sage, TeX
C_{104}\rtimes_8C_4 % in TeX
G:=Group("C104:8C4"); // GroupNames label
G:=SmallGroup(416,22);
// by ID
G=gap.SmallGroup(416,22);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,55,69,13829]);
// Polycyclic
G:=Group<a,b|a^104=b^4=1,b*a*b^-1=a^77>;
// generators/relations
Export