direct product, abelian, monomial, 2-elementary
Aliases: C4×C108, SmallGroup(432,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C108 |
| C1 — C4×C108 |
| C1 — C4×C108 |
Generators and relations for C4×C108
G = < a,b | a4=b108=1, ab=ba >
Subgroups: 60, all normal (12 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C9, C12, C2×C6, C42, C18, C2×C12, C27, C36, C2×C18, C4×C12, C54, C2×C36, C108, C2×C54, C4×C36, C2×C108, C4×C108
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C9, C12, C2×C6, C42, C18, C2×C12, C27, C36, C2×C18, C4×C12, C54, C2×C36, C108, C2×C54, C4×C36, C2×C108, C4×C108
►Regular action on 432 points
Generators in S432
(1 253 359 176)(2 254 360 177)(3 255 361 178)(4 256 362 179)(5 257 363 180)(6 258 364 181)(7 259 365 182)(8 260 366 183)(9 261 367 184)(10 262 368 185)(11 263 369 186)(12 264 370 187)(13 265 371 188)(14 266 372 189)(15 267 373 190)(16 268 374 191)(17 269 375 192)(18 270 376 193)(19 271 377 194)(20 272 378 195)(21 273 379 196)(22 274 380 197)(23 275 381 198)(24 276 382 199)(25 277 383 200)(26 278 384 201)(27 279 385 202)(28 280 386 203)(29 281 387 204)(30 282 388 205)(31 283 389 206)(32 284 390 207)(33 285 391 208)(34 286 392 209)(35 287 393 210)(36 288 394 211)(37 289 395 212)(38 290 396 213)(39 291 397 214)(40 292 398 215)(41 293 399 216)(42 294 400 109)(43 295 401 110)(44 296 402 111)(45 297 403 112)(46 298 404 113)(47 299 405 114)(48 300 406 115)(49 301 407 116)(50 302 408 117)(51 303 409 118)(52 304 410 119)(53 305 411 120)(54 306 412 121)(55 307 413 122)(56 308 414 123)(57 309 415 124)(58 310 416 125)(59 311 417 126)(60 312 418 127)(61 313 419 128)(62 314 420 129)(63 315 421 130)(64 316 422 131)(65 317 423 132)(66 318 424 133)(67 319 425 134)(68 320 426 135)(69 321 427 136)(70 322 428 137)(71 323 429 138)(72 324 430 139)(73 217 431 140)(74 218 432 141)(75 219 325 142)(76 220 326 143)(77 221 327 144)(78 222 328 145)(79 223 329 146)(80 224 330 147)(81 225 331 148)(82 226 332 149)(83 227 333 150)(84 228 334 151)(85 229 335 152)(86 230 336 153)(87 231 337 154)(88 232 338 155)(89 233 339 156)(90 234 340 157)(91 235 341 158)(92 236 342 159)(93 237 343 160)(94 238 344 161)(95 239 345 162)(96 240 346 163)(97 241 347 164)(98 242 348 165)(99 243 349 166)(100 244 350 167)(101 245 351 168)(102 246 352 169)(103 247 353 170)(104 248 354 171)(105 249 355 172)(106 250 356 173)(107 251 357 174)(108 252 358 175)
(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)(325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432)
G:=sub<Sym(432)| (1,253,359,176)(2,254,360,177)(3,255,361,178)(4,256,362,179)(5,257,363,180)(6,258,364,181)(7,259,365,182)(8,260,366,183)(9,261,367,184)(10,262,368,185)(11,263,369,186)(12,264,370,187)(13,265,371,188)(14,266,372,189)(15,267,373,190)(16,268,374,191)(17,269,375,192)(18,270,376,193)(19,271,377,194)(20,272,378,195)(21,273,379,196)(22,274,380,197)(23,275,381,198)(24,276,382,199)(25,277,383,200)(26,278,384,201)(27,279,385,202)(28,280,386,203)(29,281,387,204)(30,282,388,205)(31,283,389,206)(32,284,390,207)(33,285,391,208)(34,286,392,209)(35,287,393,210)(36,288,394,211)(37,289,395,212)(38,290,396,213)(39,291,397,214)(40,292,398,215)(41,293,399,216)(42,294,400,109)(43,295,401,110)(44,296,402,111)(45,297,403,112)(46,298,404,113)(47,299,405,114)(48,300,406,115)(49,301,407,116)(50,302,408,117)(51,303,409,118)(52,304,410,119)(53,305,411,120)(54,306,412,121)(55,307,413,122)(56,308,414,123)(57,309,415,124)(58,310,416,125)(59,311,417,126)(60,312,418,127)(61,313,419,128)(62,314,420,129)(63,315,421,130)(64,316,422,131)(65,317,423,132)(66,318,424,133)(67,319,425,134)(68,320,426,135)(69,321,427,136)(70,322,428,137)(71,323,429,138)(72,324,430,139)(73,217,431,140)(74,218,432,141)(75,219,325,142)(76,220,326,143)(77,221,327,144)(78,222,328,145)(79,223,329,146)(80,224,330,147)(81,225,331,148)(82,226,332,149)(83,227,333,150)(84,228,334,151)(85,229,335,152)(86,230,336,153)(87,231,337,154)(88,232,338,155)(89,233,339,156)(90,234,340,157)(91,235,341,158)(92,236,342,159)(93,237,343,160)(94,238,344,161)(95,239,345,162)(96,240,346,163)(97,241,347,164)(98,242,348,165)(99,243,349,166)(100,244,350,167)(101,245,351,168)(102,246,352,169)(103,247,353,170)(104,248,354,171)(105,249,355,172)(106,250,356,173)(107,251,357,174)(108,252,358,175), (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)(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)>;
G:=Group( (1,253,359,176)(2,254,360,177)(3,255,361,178)(4,256,362,179)(5,257,363,180)(6,258,364,181)(7,259,365,182)(8,260,366,183)(9,261,367,184)(10,262,368,185)(11,263,369,186)(12,264,370,187)(13,265,371,188)(14,266,372,189)(15,267,373,190)(16,268,374,191)(17,269,375,192)(18,270,376,193)(19,271,377,194)(20,272,378,195)(21,273,379,196)(22,274,380,197)(23,275,381,198)(24,276,382,199)(25,277,383,200)(26,278,384,201)(27,279,385,202)(28,280,386,203)(29,281,387,204)(30,282,388,205)(31,283,389,206)(32,284,390,207)(33,285,391,208)(34,286,392,209)(35,287,393,210)(36,288,394,211)(37,289,395,212)(38,290,396,213)(39,291,397,214)(40,292,398,215)(41,293,399,216)(42,294,400,109)(43,295,401,110)(44,296,402,111)(45,297,403,112)(46,298,404,113)(47,299,405,114)(48,300,406,115)(49,301,407,116)(50,302,408,117)(51,303,409,118)(52,304,410,119)(53,305,411,120)(54,306,412,121)(55,307,413,122)(56,308,414,123)(57,309,415,124)(58,310,416,125)(59,311,417,126)(60,312,418,127)(61,313,419,128)(62,314,420,129)(63,315,421,130)(64,316,422,131)(65,317,423,132)(66,318,424,133)(67,319,425,134)(68,320,426,135)(69,321,427,136)(70,322,428,137)(71,323,429,138)(72,324,430,139)(73,217,431,140)(74,218,432,141)(75,219,325,142)(76,220,326,143)(77,221,327,144)(78,222,328,145)(79,223,329,146)(80,224,330,147)(81,225,331,148)(82,226,332,149)(83,227,333,150)(84,228,334,151)(85,229,335,152)(86,230,336,153)(87,231,337,154)(88,232,338,155)(89,233,339,156)(90,234,340,157)(91,235,341,158)(92,236,342,159)(93,237,343,160)(94,238,344,161)(95,239,345,162)(96,240,346,163)(97,241,347,164)(98,242,348,165)(99,243,349,166)(100,244,350,167)(101,245,351,168)(102,246,352,169)(103,247,353,170)(104,248,354,171)(105,249,355,172)(106,250,356,173)(107,251,357,174)(108,252,358,175), (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)(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432) );
G=PermutationGroup([[(1,253,359,176),(2,254,360,177),(3,255,361,178),(4,256,362,179),(5,257,363,180),(6,258,364,181),(7,259,365,182),(8,260,366,183),(9,261,367,184),(10,262,368,185),(11,263,369,186),(12,264,370,187),(13,265,371,188),(14,266,372,189),(15,267,373,190),(16,268,374,191),(17,269,375,192),(18,270,376,193),(19,271,377,194),(20,272,378,195),(21,273,379,196),(22,274,380,197),(23,275,381,198),(24,276,382,199),(25,277,383,200),(26,278,384,201),(27,279,385,202),(28,280,386,203),(29,281,387,204),(30,282,388,205),(31,283,389,206),(32,284,390,207),(33,285,391,208),(34,286,392,209),(35,287,393,210),(36,288,394,211),(37,289,395,212),(38,290,396,213),(39,291,397,214),(40,292,398,215),(41,293,399,216),(42,294,400,109),(43,295,401,110),(44,296,402,111),(45,297,403,112),(46,298,404,113),(47,299,405,114),(48,300,406,115),(49,301,407,116),(50,302,408,117),(51,303,409,118),(52,304,410,119),(53,305,411,120),(54,306,412,121),(55,307,413,122),(56,308,414,123),(57,309,415,124),(58,310,416,125),(59,311,417,126),(60,312,418,127),(61,313,419,128),(62,314,420,129),(63,315,421,130),(64,316,422,131),(65,317,423,132),(66,318,424,133),(67,319,425,134),(68,320,426,135),(69,321,427,136),(70,322,428,137),(71,323,429,138),(72,324,430,139),(73,217,431,140),(74,218,432,141),(75,219,325,142),(76,220,326,143),(77,221,327,144),(78,222,328,145),(79,223,329,146),(80,224,330,147),(81,225,331,148),(82,226,332,149),(83,227,333,150),(84,228,334,151),(85,229,335,152),(86,230,336,153),(87,231,337,154),(88,232,338,155),(89,233,339,156),(90,234,340,157),(91,235,341,158),(92,236,342,159),(93,237,343,160),(94,238,344,161),(95,239,345,162),(96,240,346,163),(97,241,347,164),(98,242,348,165),(99,243,349,166),(100,244,350,167),(101,245,351,168),(102,246,352,169),(103,247,353,170),(104,248,354,171),(105,249,355,172),(106,250,356,173),(107,251,357,174),(108,252,358,175)], [(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),(325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)]])
432 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6F | 9A | ··· | 9F | 12A | ··· | 12X | 18A | ··· | 18R | 27A | ··· | 27R | 36A | ··· | 36BT | 54A | ··· | 54BB | 108A | ··· | 108HH |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
432 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C27 | C36 | C54 | C108 |
| kernel | C4×C108 | C2×C108 | C4×C36 | C108 | C2×C36 | C4×C12 | C36 | C2×C12 | C42 | C12 | C2×C4 | C4 |
| # reps | 1 | 3 | 2 | 12 | 6 | 6 | 24 | 18 | 18 | 72 | 54 | 216 |
Matrix representation of C4×C108 ►in GL2(𝔽109) generated by
| 1 | 0 |
| 0 | 76 |
| 77 | 0 |
| 0 | 29 |
G:=sub<GL(2,GF(109))| [1,0,0,76],[77,0,0,29] >;
C4×C108 in GAP, Magma, Sage, TeX
C_4\times C_{108} % in TeX
G:=Group("C4xC108"); // GroupNames label
G:=SmallGroup(432,20);
// by ID
G=gap.SmallGroup(432,20);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,176,268,166]);
// Polycyclic
G:=Group<a,b|a^4=b^108=1,a*b=b*a>;
// generators/relations