direct product, abelian, monomial, 2-elementary
Aliases: C4×C104, SmallGroup(416,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C104 |
| C1 — C4×C104 |
| C1 — C4×C104 |
Generators and relations for C4×C104
G = < a,b | a4=b104=1, ab=ba >
Smallest permutation representation of C4×C104
►Regular action on 416 points
Generators in S416
(1 250 406 130)(2 251 407 131)(3 252 408 132)(4 253 409 133)(5 254 410 134)(6 255 411 135)(7 256 412 136)(8 257 413 137)(9 258 414 138)(10 259 415 139)(11 260 416 140)(12 261 313 141)(13 262 314 142)(14 263 315 143)(15 264 316 144)(16 265 317 145)(17 266 318 146)(18 267 319 147)(19 268 320 148)(20 269 321 149)(21 270 322 150)(22 271 323 151)(23 272 324 152)(24 273 325 153)(25 274 326 154)(26 275 327 155)(27 276 328 156)(28 277 329 157)(29 278 330 158)(30 279 331 159)(31 280 332 160)(32 281 333 161)(33 282 334 162)(34 283 335 163)(35 284 336 164)(36 285 337 165)(37 286 338 166)(38 287 339 167)(39 288 340 168)(40 289 341 169)(41 290 342 170)(42 291 343 171)(43 292 344 172)(44 293 345 173)(45 294 346 174)(46 295 347 175)(47 296 348 176)(48 297 349 177)(49 298 350 178)(50 299 351 179)(51 300 352 180)(52 301 353 181)(53 302 354 182)(54 303 355 183)(55 304 356 184)(56 305 357 185)(57 306 358 186)(58 307 359 187)(59 308 360 188)(60 309 361 189)(61 310 362 190)(62 311 363 191)(63 312 364 192)(64 209 365 193)(65 210 366 194)(66 211 367 195)(67 212 368 196)(68 213 369 197)(69 214 370 198)(70 215 371 199)(71 216 372 200)(72 217 373 201)(73 218 374 202)(74 219 375 203)(75 220 376 204)(76 221 377 205)(77 222 378 206)(78 223 379 207)(79 224 380 208)(80 225 381 105)(81 226 382 106)(82 227 383 107)(83 228 384 108)(84 229 385 109)(85 230 386 110)(86 231 387 111)(87 232 388 112)(88 233 389 113)(89 234 390 114)(90 235 391 115)(91 236 392 116)(92 237 393 117)(93 238 394 118)(94 239 395 119)(95 240 396 120)(96 241 397 121)(97 242 398 122)(98 243 399 123)(99 244 400 124)(100 245 401 125)(101 246 402 126)(102 247 403 127)(103 248 404 128)(104 249 405 129)
(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)
G:=sub<Sym(416)| (1,250,406,130)(2,251,407,131)(3,252,408,132)(4,253,409,133)(5,254,410,134)(6,255,411,135)(7,256,412,136)(8,257,413,137)(9,258,414,138)(10,259,415,139)(11,260,416,140)(12,261,313,141)(13,262,314,142)(14,263,315,143)(15,264,316,144)(16,265,317,145)(17,266,318,146)(18,267,319,147)(19,268,320,148)(20,269,321,149)(21,270,322,150)(22,271,323,151)(23,272,324,152)(24,273,325,153)(25,274,326,154)(26,275,327,155)(27,276,328,156)(28,277,329,157)(29,278,330,158)(30,279,331,159)(31,280,332,160)(32,281,333,161)(33,282,334,162)(34,283,335,163)(35,284,336,164)(36,285,337,165)(37,286,338,166)(38,287,339,167)(39,288,340,168)(40,289,341,169)(41,290,342,170)(42,291,343,171)(43,292,344,172)(44,293,345,173)(45,294,346,174)(46,295,347,175)(47,296,348,176)(48,297,349,177)(49,298,350,178)(50,299,351,179)(51,300,352,180)(52,301,353,181)(53,302,354,182)(54,303,355,183)(55,304,356,184)(56,305,357,185)(57,306,358,186)(58,307,359,187)(59,308,360,188)(60,309,361,189)(61,310,362,190)(62,311,363,191)(63,312,364,192)(64,209,365,193)(65,210,366,194)(66,211,367,195)(67,212,368,196)(68,213,369,197)(69,214,370,198)(70,215,371,199)(71,216,372,200)(72,217,373,201)(73,218,374,202)(74,219,375,203)(75,220,376,204)(76,221,377,205)(77,222,378,206)(78,223,379,207)(79,224,380,208)(80,225,381,105)(81,226,382,106)(82,227,383,107)(83,228,384,108)(84,229,385,109)(85,230,386,110)(86,231,387,111)(87,232,388,112)(88,233,389,113)(89,234,390,114)(90,235,391,115)(91,236,392,116)(92,237,393,117)(93,238,394,118)(94,239,395,119)(95,240,396,120)(96,241,397,121)(97,242,398,122)(98,243,399,123)(99,244,400,124)(100,245,401,125)(101,246,402,126)(102,247,403,127)(103,248,404,128)(104,249,405,129), (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)>;
G:=Group( (1,250,406,130)(2,251,407,131)(3,252,408,132)(4,253,409,133)(5,254,410,134)(6,255,411,135)(7,256,412,136)(8,257,413,137)(9,258,414,138)(10,259,415,139)(11,260,416,140)(12,261,313,141)(13,262,314,142)(14,263,315,143)(15,264,316,144)(16,265,317,145)(17,266,318,146)(18,267,319,147)(19,268,320,148)(20,269,321,149)(21,270,322,150)(22,271,323,151)(23,272,324,152)(24,273,325,153)(25,274,326,154)(26,275,327,155)(27,276,328,156)(28,277,329,157)(29,278,330,158)(30,279,331,159)(31,280,332,160)(32,281,333,161)(33,282,334,162)(34,283,335,163)(35,284,336,164)(36,285,337,165)(37,286,338,166)(38,287,339,167)(39,288,340,168)(40,289,341,169)(41,290,342,170)(42,291,343,171)(43,292,344,172)(44,293,345,173)(45,294,346,174)(46,295,347,175)(47,296,348,176)(48,297,349,177)(49,298,350,178)(50,299,351,179)(51,300,352,180)(52,301,353,181)(53,302,354,182)(54,303,355,183)(55,304,356,184)(56,305,357,185)(57,306,358,186)(58,307,359,187)(59,308,360,188)(60,309,361,189)(61,310,362,190)(62,311,363,191)(63,312,364,192)(64,209,365,193)(65,210,366,194)(66,211,367,195)(67,212,368,196)(68,213,369,197)(69,214,370,198)(70,215,371,199)(71,216,372,200)(72,217,373,201)(73,218,374,202)(74,219,375,203)(75,220,376,204)(76,221,377,205)(77,222,378,206)(78,223,379,207)(79,224,380,208)(80,225,381,105)(81,226,382,106)(82,227,383,107)(83,228,384,108)(84,229,385,109)(85,230,386,110)(86,231,387,111)(87,232,388,112)(88,233,389,113)(89,234,390,114)(90,235,391,115)(91,236,392,116)(92,237,393,117)(93,238,394,118)(94,239,395,119)(95,240,396,120)(96,241,397,121)(97,242,398,122)(98,243,399,123)(99,244,400,124)(100,245,401,125)(101,246,402,126)(102,247,403,127)(103,248,404,128)(104,249,405,129), (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) );
G=PermutationGroup([[(1,250,406,130),(2,251,407,131),(3,252,408,132),(4,253,409,133),(5,254,410,134),(6,255,411,135),(7,256,412,136),(8,257,413,137),(9,258,414,138),(10,259,415,139),(11,260,416,140),(12,261,313,141),(13,262,314,142),(14,263,315,143),(15,264,316,144),(16,265,317,145),(17,266,318,146),(18,267,319,147),(19,268,320,148),(20,269,321,149),(21,270,322,150),(22,271,323,151),(23,272,324,152),(24,273,325,153),(25,274,326,154),(26,275,327,155),(27,276,328,156),(28,277,329,157),(29,278,330,158),(30,279,331,159),(31,280,332,160),(32,281,333,161),(33,282,334,162),(34,283,335,163),(35,284,336,164),(36,285,337,165),(37,286,338,166),(38,287,339,167),(39,288,340,168),(40,289,341,169),(41,290,342,170),(42,291,343,171),(43,292,344,172),(44,293,345,173),(45,294,346,174),(46,295,347,175),(47,296,348,176),(48,297,349,177),(49,298,350,178),(50,299,351,179),(51,300,352,180),(52,301,353,181),(53,302,354,182),(54,303,355,183),(55,304,356,184),(56,305,357,185),(57,306,358,186),(58,307,359,187),(59,308,360,188),(60,309,361,189),(61,310,362,190),(62,311,363,191),(63,312,364,192),(64,209,365,193),(65,210,366,194),(66,211,367,195),(67,212,368,196),(68,213,369,197),(69,214,370,198),(70,215,371,199),(71,216,372,200),(72,217,373,201),(73,218,374,202),(74,219,375,203),(75,220,376,204),(76,221,377,205),(77,222,378,206),(78,223,379,207),(79,224,380,208),(80,225,381,105),(81,226,382,106),(82,227,383,107),(83,228,384,108),(84,229,385,109),(85,230,386,110),(86,231,387,111),(87,232,388,112),(88,233,389,113),(89,234,390,114),(90,235,391,115),(91,236,392,116),(92,237,393,117),(93,238,394,118),(94,239,395,119),(95,240,396,120),(96,241,397,121),(97,242,398,122),(98,243,399,123),(99,244,400,124),(100,245,401,125),(101,246,402,126),(102,247,403,127),(103,248,404,128),(104,249,405,129)], [(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)]])
416 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 8A | ··· | 8P | 13A | ··· | 13L | 26A | ··· | 26AJ | 52A | ··· | 52EN | 104A | ··· | 104GJ |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
416 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C13 | C26 | C26 | C52 | C52 | C104 |
| kernel | C4×C104 | C4×C52 | C2×C104 | C104 | C2×C52 | C52 | C4×C8 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 8 | 4 | 16 | 12 | 12 | 24 | 96 | 48 | 192 |
Matrix representation of C4×C104 ►in GL2(𝔽313) generated by
| 25 | 0 |
| 0 | 312 |
| 163 | 0 |
| 0 | 133 |
G:=sub<GL(2,GF(313))| [25,0,0,312],[163,0,0,133] >;
C4×C104 in GAP, Magma, Sage, TeX
C_4\times C_{104} % in TeX
G:=Group("C4xC104"); // GroupNames label
G:=SmallGroup(416,46);
// by ID
G=gap.SmallGroup(416,46);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,312,631,117]);
// Polycyclic
G:=Group<a,b|a^4=b^104=1,a*b=b*a>;
// generators/relations
Export