direct product, abelian, monomial, 7-elementary
Aliases: C7×C56, SmallGroup(392,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C7×C56 |
| C1 — C7×C56 |
| C1 — C7×C56 |
Generators and relations for C7×C56
G = < a,b | a7=b56=1, ab=ba >
Smallest permutation representation of C7×C56
►Regular action on 392 points
Generators in S392
(1 192 282 385 241 134 87)(2 193 283 386 242 135 88)(3 194 284 387 243 136 89)(4 195 285 388 244 137 90)(5 196 286 389 245 138 91)(6 197 287 390 246 139 92)(7 198 288 391 247 140 93)(8 199 289 392 248 141 94)(9 200 290 337 249 142 95)(10 201 291 338 250 143 96)(11 202 292 339 251 144 97)(12 203 293 340 252 145 98)(13 204 294 341 253 146 99)(14 205 295 342 254 147 100)(15 206 296 343 255 148 101)(16 207 297 344 256 149 102)(17 208 298 345 257 150 103)(18 209 299 346 258 151 104)(19 210 300 347 259 152 105)(20 211 301 348 260 153 106)(21 212 302 349 261 154 107)(22 213 303 350 262 155 108)(23 214 304 351 263 156 109)(24 215 305 352 264 157 110)(25 216 306 353 265 158 111)(26 217 307 354 266 159 112)(27 218 308 355 267 160 57)(28 219 309 356 268 161 58)(29 220 310 357 269 162 59)(30 221 311 358 270 163 60)(31 222 312 359 271 164 61)(32 223 313 360 272 165 62)(33 224 314 361 273 166 63)(34 169 315 362 274 167 64)(35 170 316 363 275 168 65)(36 171 317 364 276 113 66)(37 172 318 365 277 114 67)(38 173 319 366 278 115 68)(39 174 320 367 279 116 69)(40 175 321 368 280 117 70)(41 176 322 369 225 118 71)(42 177 323 370 226 119 72)(43 178 324 371 227 120 73)(44 179 325 372 228 121 74)(45 180 326 373 229 122 75)(46 181 327 374 230 123 76)(47 182 328 375 231 124 77)(48 183 329 376 232 125 78)(49 184 330 377 233 126 79)(50 185 331 378 234 127 80)(51 186 332 379 235 128 81)(52 187 333 380 236 129 82)(53 188 334 381 237 130 83)(54 189 335 382 238 131 84)(55 190 336 383 239 132 85)(56 191 281 384 240 133 86)
(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)| (1,192,282,385,241,134,87)(2,193,283,386,242,135,88)(3,194,284,387,243,136,89)(4,195,285,388,244,137,90)(5,196,286,389,245,138,91)(6,197,287,390,246,139,92)(7,198,288,391,247,140,93)(8,199,289,392,248,141,94)(9,200,290,337,249,142,95)(10,201,291,338,250,143,96)(11,202,292,339,251,144,97)(12,203,293,340,252,145,98)(13,204,294,341,253,146,99)(14,205,295,342,254,147,100)(15,206,296,343,255,148,101)(16,207,297,344,256,149,102)(17,208,298,345,257,150,103)(18,209,299,346,258,151,104)(19,210,300,347,259,152,105)(20,211,301,348,260,153,106)(21,212,302,349,261,154,107)(22,213,303,350,262,155,108)(23,214,304,351,263,156,109)(24,215,305,352,264,157,110)(25,216,306,353,265,158,111)(26,217,307,354,266,159,112)(27,218,308,355,267,160,57)(28,219,309,356,268,161,58)(29,220,310,357,269,162,59)(30,221,311,358,270,163,60)(31,222,312,359,271,164,61)(32,223,313,360,272,165,62)(33,224,314,361,273,166,63)(34,169,315,362,274,167,64)(35,170,316,363,275,168,65)(36,171,317,364,276,113,66)(37,172,318,365,277,114,67)(38,173,319,366,278,115,68)(39,174,320,367,279,116,69)(40,175,321,368,280,117,70)(41,176,322,369,225,118,71)(42,177,323,370,226,119,72)(43,178,324,371,227,120,73)(44,179,325,372,228,121,74)(45,180,326,373,229,122,75)(46,181,327,374,230,123,76)(47,182,328,375,231,124,77)(48,183,329,376,232,125,78)(49,184,330,377,233,126,79)(50,185,331,378,234,127,80)(51,186,332,379,235,128,81)(52,187,333,380,236,129,82)(53,188,334,381,237,130,83)(54,189,335,382,238,131,84)(55,190,336,383,239,132,85)(56,191,281,384,240,133,86), (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( (1,192,282,385,241,134,87)(2,193,283,386,242,135,88)(3,194,284,387,243,136,89)(4,195,285,388,244,137,90)(5,196,286,389,245,138,91)(6,197,287,390,246,139,92)(7,198,288,391,247,140,93)(8,199,289,392,248,141,94)(9,200,290,337,249,142,95)(10,201,291,338,250,143,96)(11,202,292,339,251,144,97)(12,203,293,340,252,145,98)(13,204,294,341,253,146,99)(14,205,295,342,254,147,100)(15,206,296,343,255,148,101)(16,207,297,344,256,149,102)(17,208,298,345,257,150,103)(18,209,299,346,258,151,104)(19,210,300,347,259,152,105)(20,211,301,348,260,153,106)(21,212,302,349,261,154,107)(22,213,303,350,262,155,108)(23,214,304,351,263,156,109)(24,215,305,352,264,157,110)(25,216,306,353,265,158,111)(26,217,307,354,266,159,112)(27,218,308,355,267,160,57)(28,219,309,356,268,161,58)(29,220,310,357,269,162,59)(30,221,311,358,270,163,60)(31,222,312,359,271,164,61)(32,223,313,360,272,165,62)(33,224,314,361,273,166,63)(34,169,315,362,274,167,64)(35,170,316,363,275,168,65)(36,171,317,364,276,113,66)(37,172,318,365,277,114,67)(38,173,319,366,278,115,68)(39,174,320,367,279,116,69)(40,175,321,368,280,117,70)(41,176,322,369,225,118,71)(42,177,323,370,226,119,72)(43,178,324,371,227,120,73)(44,179,325,372,228,121,74)(45,180,326,373,229,122,75)(46,181,327,374,230,123,76)(47,182,328,375,231,124,77)(48,183,329,376,232,125,78)(49,184,330,377,233,126,79)(50,185,331,378,234,127,80)(51,186,332,379,235,128,81)(52,187,333,380,236,129,82)(53,188,334,381,237,130,83)(54,189,335,382,238,131,84)(55,190,336,383,239,132,85)(56,191,281,384,240,133,86), (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([[(1,192,282,385,241,134,87),(2,193,283,386,242,135,88),(3,194,284,387,243,136,89),(4,195,285,388,244,137,90),(5,196,286,389,245,138,91),(6,197,287,390,246,139,92),(7,198,288,391,247,140,93),(8,199,289,392,248,141,94),(9,200,290,337,249,142,95),(10,201,291,338,250,143,96),(11,202,292,339,251,144,97),(12,203,293,340,252,145,98),(13,204,294,341,253,146,99),(14,205,295,342,254,147,100),(15,206,296,343,255,148,101),(16,207,297,344,256,149,102),(17,208,298,345,257,150,103),(18,209,299,346,258,151,104),(19,210,300,347,259,152,105),(20,211,301,348,260,153,106),(21,212,302,349,261,154,107),(22,213,303,350,262,155,108),(23,214,304,351,263,156,109),(24,215,305,352,264,157,110),(25,216,306,353,265,158,111),(26,217,307,354,266,159,112),(27,218,308,355,267,160,57),(28,219,309,356,268,161,58),(29,220,310,357,269,162,59),(30,221,311,358,270,163,60),(31,222,312,359,271,164,61),(32,223,313,360,272,165,62),(33,224,314,361,273,166,63),(34,169,315,362,274,167,64),(35,170,316,363,275,168,65),(36,171,317,364,276,113,66),(37,172,318,365,277,114,67),(38,173,319,366,278,115,68),(39,174,320,367,279,116,69),(40,175,321,368,280,117,70),(41,176,322,369,225,118,71),(42,177,323,370,226,119,72),(43,178,324,371,227,120,73),(44,179,325,372,228,121,74),(45,180,326,373,229,122,75),(46,181,327,374,230,123,76),(47,182,328,375,231,124,77),(48,183,329,376,232,125,78),(49,184,330,377,233,126,79),(50,185,331,378,234,127,80),(51,186,332,379,235,128,81),(52,187,333,380,236,129,82),(53,188,334,381,237,130,83),(54,189,335,382,238,131,84),(55,190,336,383,239,132,85),(56,191,281,384,240,133,86)], [(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 | 2 | 4A | 4B | 7A | ··· | 7AV | 8A | 8B | 8C | 8D | 14A | ··· | 14AV | 28A | ··· | 28CR | 56A | ··· | 56GJ |
| order | 1 | 2 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 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 | C4 | C7 | C8 | C14 | C28 | C56 |
| kernel | C7×C56 | C7×C28 | C7×C14 | C56 | C72 | C28 | C14 | C7 |
| # reps | 1 | 1 | 2 | 48 | 4 | 48 | 96 | 192 |
Matrix representation of C7×C56 ►in GL2(𝔽113) generated by
| 106 | 0 |
| 0 | 49 |
| 44 | 0 |
| 0 | 14 |
G:=sub<GL(2,GF(113))| [106,0,0,49],[44,0,0,14] >;
C7×C56 in GAP, Magma, Sage, TeX
C_7\times C_{56} % in TeX
G:=Group("C7xC56"); // GroupNames label
G:=SmallGroup(392,16);
// by ID
G=gap.SmallGroup(392,16);
# by ID
G:=PCGroup([5,-2,-7,-7,-2,-2,490,58]);
// Polycyclic
G:=Group<a,b|a^7=b^56=1,a*b=b*a>;
// generators/relations
Export