direct product, abelian, monomial, 7-elementary
Aliases: C7×C42, SmallGroup(294,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C7×C42 |
| C1 — C7×C42 |
| C1 — C7×C42 |
Generators and relations for C7×C42
G = < a,b | a7=b42=1, ab=ba >
Smallest permutation representation of C7×C42
►Regular action on 294 points
Generators in S294
(1 257 196 135 65 237 100)(2 258 197 136 66 238 101)(3 259 198 137 67 239 102)(4 260 199 138 68 240 103)(5 261 200 139 69 241 104)(6 262 201 140 70 242 105)(7 263 202 141 71 243 106)(8 264 203 142 72 244 107)(9 265 204 143 73 245 108)(10 266 205 144 74 246 109)(11 267 206 145 75 247 110)(12 268 207 146 76 248 111)(13 269 208 147 77 249 112)(14 270 209 148 78 250 113)(15 271 210 149 79 251 114)(16 272 169 150 80 252 115)(17 273 170 151 81 211 116)(18 274 171 152 82 212 117)(19 275 172 153 83 213 118)(20 276 173 154 84 214 119)(21 277 174 155 43 215 120)(22 278 175 156 44 216 121)(23 279 176 157 45 217 122)(24 280 177 158 46 218 123)(25 281 178 159 47 219 124)(26 282 179 160 48 220 125)(27 283 180 161 49 221 126)(28 284 181 162 50 222 85)(29 285 182 163 51 223 86)(30 286 183 164 52 224 87)(31 287 184 165 53 225 88)(32 288 185 166 54 226 89)(33 289 186 167 55 227 90)(34 290 187 168 56 228 91)(35 291 188 127 57 229 92)(36 292 189 128 58 230 93)(37 293 190 129 59 231 94)(38 294 191 130 60 232 95)(39 253 192 131 61 233 96)(40 254 193 132 62 234 97)(41 255 194 133 63 235 98)(42 256 195 134 64 236 99)
(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)
G:=sub<Sym(294)| (1,257,196,135,65,237,100)(2,258,197,136,66,238,101)(3,259,198,137,67,239,102)(4,260,199,138,68,240,103)(5,261,200,139,69,241,104)(6,262,201,140,70,242,105)(7,263,202,141,71,243,106)(8,264,203,142,72,244,107)(9,265,204,143,73,245,108)(10,266,205,144,74,246,109)(11,267,206,145,75,247,110)(12,268,207,146,76,248,111)(13,269,208,147,77,249,112)(14,270,209,148,78,250,113)(15,271,210,149,79,251,114)(16,272,169,150,80,252,115)(17,273,170,151,81,211,116)(18,274,171,152,82,212,117)(19,275,172,153,83,213,118)(20,276,173,154,84,214,119)(21,277,174,155,43,215,120)(22,278,175,156,44,216,121)(23,279,176,157,45,217,122)(24,280,177,158,46,218,123)(25,281,178,159,47,219,124)(26,282,179,160,48,220,125)(27,283,180,161,49,221,126)(28,284,181,162,50,222,85)(29,285,182,163,51,223,86)(30,286,183,164,52,224,87)(31,287,184,165,53,225,88)(32,288,185,166,54,226,89)(33,289,186,167,55,227,90)(34,290,187,168,56,228,91)(35,291,188,127,57,229,92)(36,292,189,128,58,230,93)(37,293,190,129,59,231,94)(38,294,191,130,60,232,95)(39,253,192,131,61,233,96)(40,254,193,132,62,234,97)(41,255,194,133,63,235,98)(42,256,195,134,64,236,99), (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)>;
G:=Group( (1,257,196,135,65,237,100)(2,258,197,136,66,238,101)(3,259,198,137,67,239,102)(4,260,199,138,68,240,103)(5,261,200,139,69,241,104)(6,262,201,140,70,242,105)(7,263,202,141,71,243,106)(8,264,203,142,72,244,107)(9,265,204,143,73,245,108)(10,266,205,144,74,246,109)(11,267,206,145,75,247,110)(12,268,207,146,76,248,111)(13,269,208,147,77,249,112)(14,270,209,148,78,250,113)(15,271,210,149,79,251,114)(16,272,169,150,80,252,115)(17,273,170,151,81,211,116)(18,274,171,152,82,212,117)(19,275,172,153,83,213,118)(20,276,173,154,84,214,119)(21,277,174,155,43,215,120)(22,278,175,156,44,216,121)(23,279,176,157,45,217,122)(24,280,177,158,46,218,123)(25,281,178,159,47,219,124)(26,282,179,160,48,220,125)(27,283,180,161,49,221,126)(28,284,181,162,50,222,85)(29,285,182,163,51,223,86)(30,286,183,164,52,224,87)(31,287,184,165,53,225,88)(32,288,185,166,54,226,89)(33,289,186,167,55,227,90)(34,290,187,168,56,228,91)(35,291,188,127,57,229,92)(36,292,189,128,58,230,93)(37,293,190,129,59,231,94)(38,294,191,130,60,232,95)(39,253,192,131,61,233,96)(40,254,193,132,62,234,97)(41,255,194,133,63,235,98)(42,256,195,134,64,236,99), (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) );
G=PermutationGroup([[(1,257,196,135,65,237,100),(2,258,197,136,66,238,101),(3,259,198,137,67,239,102),(4,260,199,138,68,240,103),(5,261,200,139,69,241,104),(6,262,201,140,70,242,105),(7,263,202,141,71,243,106),(8,264,203,142,72,244,107),(9,265,204,143,73,245,108),(10,266,205,144,74,246,109),(11,267,206,145,75,247,110),(12,268,207,146,76,248,111),(13,269,208,147,77,249,112),(14,270,209,148,78,250,113),(15,271,210,149,79,251,114),(16,272,169,150,80,252,115),(17,273,170,151,81,211,116),(18,274,171,152,82,212,117),(19,275,172,153,83,213,118),(20,276,173,154,84,214,119),(21,277,174,155,43,215,120),(22,278,175,156,44,216,121),(23,279,176,157,45,217,122),(24,280,177,158,46,218,123),(25,281,178,159,47,219,124),(26,282,179,160,48,220,125),(27,283,180,161,49,221,126),(28,284,181,162,50,222,85),(29,285,182,163,51,223,86),(30,286,183,164,52,224,87),(31,287,184,165,53,225,88),(32,288,185,166,54,226,89),(33,289,186,167,55,227,90),(34,290,187,168,56,228,91),(35,291,188,127,57,229,92),(36,292,189,128,58,230,93),(37,293,190,129,59,231,94),(38,294,191,130,60,232,95),(39,253,192,131,61,233,96),(40,254,193,132,62,234,97),(41,255,194,133,63,235,98),(42,256,195,134,64,236,99)], [(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)]])
294 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | ··· | 7AV | 14A | ··· | 14AV | 21A | ··· | 21CR | 42A | ··· | 42CR |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
294 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 |
| kernel | C7×C42 | C7×C21 | C7×C14 | C72 | C42 | C21 | C14 | C7 |
| # reps | 1 | 1 | 2 | 2 | 48 | 48 | 96 | 96 |
Matrix representation of C7×C42 ►in GL2(𝔽43) generated by
| 16 | 0 |
| 0 | 4 |
| 7 | 0 |
| 0 | 2 |
G:=sub<GL(2,GF(43))| [16,0,0,4],[7,0,0,2] >;
C7×C42 in GAP, Magma, Sage, TeX
C_7\times C_{42} % in TeX
G:=Group("C7xC42"); // GroupNames label
G:=SmallGroup(294,23);
// by ID
G=gap.SmallGroup(294,23);
# by ID
G:=PCGroup([4,-2,-3,-7,-7]);
// Polycyclic
G:=Group<a,b|a^7=b^42=1,a*b=b*a>;
// generators/relations
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