direct product, p-group, abelian, monomial
Aliases: C7×C49, SmallGroup(343,2)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C7×C49 |
| C1 — C7×C49 |
| C1 — C7×C49 |
Generators and relations for C7×C49
G = < a,b | a7=b49=1, ab=ba >
Smallest permutation representation of C7×C49
►Regular action on 343 points
Generators in S343
(1 228 325 83 259 162 118)(2 229 326 84 260 163 119)(3 230 327 85 261 164 120)(4 231 328 86 262 165 121)(5 232 329 87 263 166 122)(6 233 330 88 264 167 123)(7 234 331 89 265 168 124)(8 235 332 90 266 169 125)(9 236 333 91 267 170 126)(10 237 334 92 268 171 127)(11 238 335 93 269 172 128)(12 239 336 94 270 173 129)(13 240 337 95 271 174 130)(14 241 338 96 272 175 131)(15 242 339 97 273 176 132)(16 243 340 98 274 177 133)(17 244 341 50 275 178 134)(18 245 342 51 276 179 135)(19 197 343 52 277 180 136)(20 198 295 53 278 181 137)(21 199 296 54 279 182 138)(22 200 297 55 280 183 139)(23 201 298 56 281 184 140)(24 202 299 57 282 185 141)(25 203 300 58 283 186 142)(26 204 301 59 284 187 143)(27 205 302 60 285 188 144)(28 206 303 61 286 189 145)(29 207 304 62 287 190 146)(30 208 305 63 288 191 147)(31 209 306 64 289 192 99)(32 210 307 65 290 193 100)(33 211 308 66 291 194 101)(34 212 309 67 292 195 102)(35 213 310 68 293 196 103)(36 214 311 69 294 148 104)(37 215 312 70 246 149 105)(38 216 313 71 247 150 106)(39 217 314 72 248 151 107)(40 218 315 73 249 152 108)(41 219 316 74 250 153 109)(42 220 317 75 251 154 110)(43 221 318 76 252 155 111)(44 222 319 77 253 156 112)(45 223 320 78 254 157 113)(46 224 321 79 255 158 114)(47 225 322 80 256 159 115)(48 226 323 81 257 160 116)(49 227 324 82 258 161 117)
(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)
G:=sub<Sym(343)| (1,228,325,83,259,162,118)(2,229,326,84,260,163,119)(3,230,327,85,261,164,120)(4,231,328,86,262,165,121)(5,232,329,87,263,166,122)(6,233,330,88,264,167,123)(7,234,331,89,265,168,124)(8,235,332,90,266,169,125)(9,236,333,91,267,170,126)(10,237,334,92,268,171,127)(11,238,335,93,269,172,128)(12,239,336,94,270,173,129)(13,240,337,95,271,174,130)(14,241,338,96,272,175,131)(15,242,339,97,273,176,132)(16,243,340,98,274,177,133)(17,244,341,50,275,178,134)(18,245,342,51,276,179,135)(19,197,343,52,277,180,136)(20,198,295,53,278,181,137)(21,199,296,54,279,182,138)(22,200,297,55,280,183,139)(23,201,298,56,281,184,140)(24,202,299,57,282,185,141)(25,203,300,58,283,186,142)(26,204,301,59,284,187,143)(27,205,302,60,285,188,144)(28,206,303,61,286,189,145)(29,207,304,62,287,190,146)(30,208,305,63,288,191,147)(31,209,306,64,289,192,99)(32,210,307,65,290,193,100)(33,211,308,66,291,194,101)(34,212,309,67,292,195,102)(35,213,310,68,293,196,103)(36,214,311,69,294,148,104)(37,215,312,70,246,149,105)(38,216,313,71,247,150,106)(39,217,314,72,248,151,107)(40,218,315,73,249,152,108)(41,219,316,74,250,153,109)(42,220,317,75,251,154,110)(43,221,318,76,252,155,111)(44,222,319,77,253,156,112)(45,223,320,78,254,157,113)(46,224,321,79,255,158,114)(47,225,322,80,256,159,115)(48,226,323,81,257,160,116)(49,227,324,82,258,161,117), (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)>;
G:=Group( (1,228,325,83,259,162,118)(2,229,326,84,260,163,119)(3,230,327,85,261,164,120)(4,231,328,86,262,165,121)(5,232,329,87,263,166,122)(6,233,330,88,264,167,123)(7,234,331,89,265,168,124)(8,235,332,90,266,169,125)(9,236,333,91,267,170,126)(10,237,334,92,268,171,127)(11,238,335,93,269,172,128)(12,239,336,94,270,173,129)(13,240,337,95,271,174,130)(14,241,338,96,272,175,131)(15,242,339,97,273,176,132)(16,243,340,98,274,177,133)(17,244,341,50,275,178,134)(18,245,342,51,276,179,135)(19,197,343,52,277,180,136)(20,198,295,53,278,181,137)(21,199,296,54,279,182,138)(22,200,297,55,280,183,139)(23,201,298,56,281,184,140)(24,202,299,57,282,185,141)(25,203,300,58,283,186,142)(26,204,301,59,284,187,143)(27,205,302,60,285,188,144)(28,206,303,61,286,189,145)(29,207,304,62,287,190,146)(30,208,305,63,288,191,147)(31,209,306,64,289,192,99)(32,210,307,65,290,193,100)(33,211,308,66,291,194,101)(34,212,309,67,292,195,102)(35,213,310,68,293,196,103)(36,214,311,69,294,148,104)(37,215,312,70,246,149,105)(38,216,313,71,247,150,106)(39,217,314,72,248,151,107)(40,218,315,73,249,152,108)(41,219,316,74,250,153,109)(42,220,317,75,251,154,110)(43,221,318,76,252,155,111)(44,222,319,77,253,156,112)(45,223,320,78,254,157,113)(46,224,321,79,255,158,114)(47,225,322,80,256,159,115)(48,226,323,81,257,160,116)(49,227,324,82,258,161,117), (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) );
G=PermutationGroup([[(1,228,325,83,259,162,118),(2,229,326,84,260,163,119),(3,230,327,85,261,164,120),(4,231,328,86,262,165,121),(5,232,329,87,263,166,122),(6,233,330,88,264,167,123),(7,234,331,89,265,168,124),(8,235,332,90,266,169,125),(9,236,333,91,267,170,126),(10,237,334,92,268,171,127),(11,238,335,93,269,172,128),(12,239,336,94,270,173,129),(13,240,337,95,271,174,130),(14,241,338,96,272,175,131),(15,242,339,97,273,176,132),(16,243,340,98,274,177,133),(17,244,341,50,275,178,134),(18,245,342,51,276,179,135),(19,197,343,52,277,180,136),(20,198,295,53,278,181,137),(21,199,296,54,279,182,138),(22,200,297,55,280,183,139),(23,201,298,56,281,184,140),(24,202,299,57,282,185,141),(25,203,300,58,283,186,142),(26,204,301,59,284,187,143),(27,205,302,60,285,188,144),(28,206,303,61,286,189,145),(29,207,304,62,287,190,146),(30,208,305,63,288,191,147),(31,209,306,64,289,192,99),(32,210,307,65,290,193,100),(33,211,308,66,291,194,101),(34,212,309,67,292,195,102),(35,213,310,68,293,196,103),(36,214,311,69,294,148,104),(37,215,312,70,246,149,105),(38,216,313,71,247,150,106),(39,217,314,72,248,151,107),(40,218,315,73,249,152,108),(41,219,316,74,250,153,109),(42,220,317,75,251,154,110),(43,221,318,76,252,155,111),(44,222,319,77,253,156,112),(45,223,320,78,254,157,113),(46,224,321,79,255,158,114),(47,225,322,80,256,159,115),(48,226,323,81,257,160,116),(49,227,324,82,258,161,117)], [(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)]])
343 conjugacy classes
| class | 1 | 7A | ··· | 7AV | 49A | ··· | 49KH |
| order | 1 | 7 | ··· | 7 | 49 | ··· | 49 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
343 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C7 | C7 | C49 |
| kernel | C7×C49 | C49 | C72 | C7 |
| # reps | 1 | 42 | 6 | 294 |
Matrix representation of C7×C49 ►in GL2(𝔽197) generated by
| 1 | 0 |
| 0 | 178 |
| 59 | 0 |
| 0 | 154 |
G:=sub<GL(2,GF(197))| [1,0,0,178],[59,0,0,154] >;
C7×C49 in GAP, Magma, Sage, TeX
C_7\times C_{49} % in TeX
G:=Group("C7xC49"); // GroupNames label
G:=SmallGroup(343,2);
// by ID
G=gap.SmallGroup(343,2);
# by ID
G:=PCGroup([3,-7,7,-7,147]);
// Polycyclic
G:=Group<a,b|a^7=b^49=1,a*b=b*a>;
// generators/relations
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