direct product, abelian, monomial
Aliases: C14×C28, SmallGroup(392,33)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C14×C28 |
| C1 — C14×C28 |
| C1 — C14×C28 |
Generators and relations for C14×C28
G = < a,b | a14=b28=1, ab=ba >
Subgroups: 80, all normal (8 characteristic)
C1, C2, C2, C4, C22, C7, C2×C4, C14, C28, C2×C14, C72, C2×C28, C7×C14, C7×C14, C7×C28, C142, C14×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, C14, C28, C2×C14, C72, C2×C28, C7×C14, C7×C28, C142, C14×C28
►Regular action on 392 points
Generators in S392
(1 121 181 244 42 271 110 316 69 389 167 292 224 353)(2 122 182 245 43 272 111 317 70 390 168 293 197 354)(3 123 183 246 44 273 112 318 71 391 141 294 198 355)(4 124 184 247 45 274 85 319 72 392 142 295 199 356)(5 125 185 248 46 275 86 320 73 365 143 296 200 357)(6 126 186 249 47 276 87 321 74 366 144 297 201 358)(7 127 187 250 48 277 88 322 75 367 145 298 202 359)(8 128 188 251 49 278 89 323 76 368 146 299 203 360)(9 129 189 252 50 279 90 324 77 369 147 300 204 361)(10 130 190 225 51 280 91 325 78 370 148 301 205 362)(11 131 191 226 52 253 92 326 79 371 149 302 206 363)(12 132 192 227 53 254 93 327 80 372 150 303 207 364)(13 133 193 228 54 255 94 328 81 373 151 304 208 337)(14 134 194 229 55 256 95 329 82 374 152 305 209 338)(15 135 195 230 56 257 96 330 83 375 153 306 210 339)(16 136 196 231 29 258 97 331 84 376 154 307 211 340)(17 137 169 232 30 259 98 332 57 377 155 308 212 341)(18 138 170 233 31 260 99 333 58 378 156 281 213 342)(19 139 171 234 32 261 100 334 59 379 157 282 214 343)(20 140 172 235 33 262 101 335 60 380 158 283 215 344)(21 113 173 236 34 263 102 336 61 381 159 284 216 345)(22 114 174 237 35 264 103 309 62 382 160 285 217 346)(23 115 175 238 36 265 104 310 63 383 161 286 218 347)(24 116 176 239 37 266 105 311 64 384 162 287 219 348)(25 117 177 240 38 267 106 312 65 385 163 288 220 349)(26 118 178 241 39 268 107 313 66 386 164 289 221 350)(27 119 179 242 40 269 108 314 67 387 165 290 222 351)(28 120 180 243 41 270 109 315 68 388 166 291 223 352)
(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)
G:=sub<Sym(392)| (1,121,181,244,42,271,110,316,69,389,167,292,224,353)(2,122,182,245,43,272,111,317,70,390,168,293,197,354)(3,123,183,246,44,273,112,318,71,391,141,294,198,355)(4,124,184,247,45,274,85,319,72,392,142,295,199,356)(5,125,185,248,46,275,86,320,73,365,143,296,200,357)(6,126,186,249,47,276,87,321,74,366,144,297,201,358)(7,127,187,250,48,277,88,322,75,367,145,298,202,359)(8,128,188,251,49,278,89,323,76,368,146,299,203,360)(9,129,189,252,50,279,90,324,77,369,147,300,204,361)(10,130,190,225,51,280,91,325,78,370,148,301,205,362)(11,131,191,226,52,253,92,326,79,371,149,302,206,363)(12,132,192,227,53,254,93,327,80,372,150,303,207,364)(13,133,193,228,54,255,94,328,81,373,151,304,208,337)(14,134,194,229,55,256,95,329,82,374,152,305,209,338)(15,135,195,230,56,257,96,330,83,375,153,306,210,339)(16,136,196,231,29,258,97,331,84,376,154,307,211,340)(17,137,169,232,30,259,98,332,57,377,155,308,212,341)(18,138,170,233,31,260,99,333,58,378,156,281,213,342)(19,139,171,234,32,261,100,334,59,379,157,282,214,343)(20,140,172,235,33,262,101,335,60,380,158,283,215,344)(21,113,173,236,34,263,102,336,61,381,159,284,216,345)(22,114,174,237,35,264,103,309,62,382,160,285,217,346)(23,115,175,238,36,265,104,310,63,383,161,286,218,347)(24,116,176,239,37,266,105,311,64,384,162,287,219,348)(25,117,177,240,38,267,106,312,65,385,163,288,220,349)(26,118,178,241,39,268,107,313,66,386,164,289,221,350)(27,119,179,242,40,269,108,314,67,387,165,290,222,351)(28,120,180,243,41,270,109,315,68,388,166,291,223,352), (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)>;
G:=Group( (1,121,181,244,42,271,110,316,69,389,167,292,224,353)(2,122,182,245,43,272,111,317,70,390,168,293,197,354)(3,123,183,246,44,273,112,318,71,391,141,294,198,355)(4,124,184,247,45,274,85,319,72,392,142,295,199,356)(5,125,185,248,46,275,86,320,73,365,143,296,200,357)(6,126,186,249,47,276,87,321,74,366,144,297,201,358)(7,127,187,250,48,277,88,322,75,367,145,298,202,359)(8,128,188,251,49,278,89,323,76,368,146,299,203,360)(9,129,189,252,50,279,90,324,77,369,147,300,204,361)(10,130,190,225,51,280,91,325,78,370,148,301,205,362)(11,131,191,226,52,253,92,326,79,371,149,302,206,363)(12,132,192,227,53,254,93,327,80,372,150,303,207,364)(13,133,193,228,54,255,94,328,81,373,151,304,208,337)(14,134,194,229,55,256,95,329,82,374,152,305,209,338)(15,135,195,230,56,257,96,330,83,375,153,306,210,339)(16,136,196,231,29,258,97,331,84,376,154,307,211,340)(17,137,169,232,30,259,98,332,57,377,155,308,212,341)(18,138,170,233,31,260,99,333,58,378,156,281,213,342)(19,139,171,234,32,261,100,334,59,379,157,282,214,343)(20,140,172,235,33,262,101,335,60,380,158,283,215,344)(21,113,173,236,34,263,102,336,61,381,159,284,216,345)(22,114,174,237,35,264,103,309,62,382,160,285,217,346)(23,115,175,238,36,265,104,310,63,383,161,286,218,347)(24,116,176,239,37,266,105,311,64,384,162,287,219,348)(25,117,177,240,38,267,106,312,65,385,163,288,220,349)(26,118,178,241,39,268,107,313,66,386,164,289,221,350)(27,119,179,242,40,269,108,314,67,387,165,290,222,351)(28,120,180,243,41,270,109,315,68,388,166,291,223,352), (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) );
G=PermutationGroup([[(1,121,181,244,42,271,110,316,69,389,167,292,224,353),(2,122,182,245,43,272,111,317,70,390,168,293,197,354),(3,123,183,246,44,273,112,318,71,391,141,294,198,355),(4,124,184,247,45,274,85,319,72,392,142,295,199,356),(5,125,185,248,46,275,86,320,73,365,143,296,200,357),(6,126,186,249,47,276,87,321,74,366,144,297,201,358),(7,127,187,250,48,277,88,322,75,367,145,298,202,359),(8,128,188,251,49,278,89,323,76,368,146,299,203,360),(9,129,189,252,50,279,90,324,77,369,147,300,204,361),(10,130,190,225,51,280,91,325,78,370,148,301,205,362),(11,131,191,226,52,253,92,326,79,371,149,302,206,363),(12,132,192,227,53,254,93,327,80,372,150,303,207,364),(13,133,193,228,54,255,94,328,81,373,151,304,208,337),(14,134,194,229,55,256,95,329,82,374,152,305,209,338),(15,135,195,230,56,257,96,330,83,375,153,306,210,339),(16,136,196,231,29,258,97,331,84,376,154,307,211,340),(17,137,169,232,30,259,98,332,57,377,155,308,212,341),(18,138,170,233,31,260,99,333,58,378,156,281,213,342),(19,139,171,234,32,261,100,334,59,379,157,282,214,343),(20,140,172,235,33,262,101,335,60,380,158,283,215,344),(21,113,173,236,34,263,102,336,61,381,159,284,216,345),(22,114,174,237,35,264,103,309,62,382,160,285,217,346),(23,115,175,238,36,265,104,310,63,383,161,286,218,347),(24,116,176,239,37,266,105,311,64,384,162,287,219,348),(25,117,177,240,38,267,106,312,65,385,163,288,220,349),(26,118,178,241,39,268,107,313,66,386,164,289,221,350),(27,119,179,242,40,269,108,314,67,387,165,290,222,351),(28,120,180,243,41,270,109,315,68,388,166,291,223,352)], [(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)]])
392 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | ··· | 7AV | 14A | ··· | 14EN | 28A | ··· | 28GJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
392 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 |
| kernel | C14×C28 | C7×C28 | C142 | C7×C14 | C2×C28 | C28 | C2×C14 | C14 |
| # reps | 1 | 2 | 1 | 4 | 48 | 96 | 48 | 192 |
Matrix representation of C14×C28 ►in GL2(𝔽29) generated by
| 5 | 0 |
| 0 | 1 |
| 19 | 0 |
| 0 | 19 |
G:=sub<GL(2,GF(29))| [5,0,0,1],[19,0,0,19] >;
C14×C28 in GAP, Magma, Sage, TeX
C_{14}\times C_{28} % in TeX
G:=Group("C14xC28"); // GroupNames label
G:=SmallGroup(392,33);
// by ID
G=gap.SmallGroup(392,33);
# by ID
G:=PCGroup([5,-2,-2,-7,-7,-2,980]);
// Polycyclic
G:=Group<a,b|a^14=b^28=1,a*b=b*a>;
// generators/relations