direct product, abelian, monomial
Aliases: C6×C42, SmallGroup(252,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C6×C42 |
| C1 — C6×C42 |
| C1 — C6×C42 |
Generators and relations for C6×C42
G = < a,b | a6=b42=1, ab=ba >
Subgroups: 60, all normal (8 characteristic)
C1, C2, C3, C22, C6, C7, C32, C2×C6, C14, C3×C6, C21, C2×C14, C62, C42, C3×C21, C2×C42, C3×C42, C6×C42
Quotients: C1, C2, C3, C22, C6, C7, C32, C2×C6, C14, C3×C6, C21, C2×C14, C62, C42, C3×C21, C2×C42, C3×C42, C6×C42
►Regular action on 252 points
Generators in S252
(1 235 81 169 89 130)(2 236 82 170 90 131)(3 237 83 171 91 132)(4 238 84 172 92 133)(5 239 43 173 93 134)(6 240 44 174 94 135)(7 241 45 175 95 136)(8 242 46 176 96 137)(9 243 47 177 97 138)(10 244 48 178 98 139)(11 245 49 179 99 140)(12 246 50 180 100 141)(13 247 51 181 101 142)(14 248 52 182 102 143)(15 249 53 183 103 144)(16 250 54 184 104 145)(17 251 55 185 105 146)(18 252 56 186 106 147)(19 211 57 187 107 148)(20 212 58 188 108 149)(21 213 59 189 109 150)(22 214 60 190 110 151)(23 215 61 191 111 152)(24 216 62 192 112 153)(25 217 63 193 113 154)(26 218 64 194 114 155)(27 219 65 195 115 156)(28 220 66 196 116 157)(29 221 67 197 117 158)(30 222 68 198 118 159)(31 223 69 199 119 160)(32 224 70 200 120 161)(33 225 71 201 121 162)(34 226 72 202 122 163)(35 227 73 203 123 164)(36 228 74 204 124 165)(37 229 75 205 125 166)(38 230 76 206 126 167)(39 231 77 207 85 168)(40 232 78 208 86 127)(41 233 79 209 87 128)(42 234 80 210 88 129)
(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)
G:=sub<Sym(252)| (1,235,81,169,89,130)(2,236,82,170,90,131)(3,237,83,171,91,132)(4,238,84,172,92,133)(5,239,43,173,93,134)(6,240,44,174,94,135)(7,241,45,175,95,136)(8,242,46,176,96,137)(9,243,47,177,97,138)(10,244,48,178,98,139)(11,245,49,179,99,140)(12,246,50,180,100,141)(13,247,51,181,101,142)(14,248,52,182,102,143)(15,249,53,183,103,144)(16,250,54,184,104,145)(17,251,55,185,105,146)(18,252,56,186,106,147)(19,211,57,187,107,148)(20,212,58,188,108,149)(21,213,59,189,109,150)(22,214,60,190,110,151)(23,215,61,191,111,152)(24,216,62,192,112,153)(25,217,63,193,113,154)(26,218,64,194,114,155)(27,219,65,195,115,156)(28,220,66,196,116,157)(29,221,67,197,117,158)(30,222,68,198,118,159)(31,223,69,199,119,160)(32,224,70,200,120,161)(33,225,71,201,121,162)(34,226,72,202,122,163)(35,227,73,203,123,164)(36,228,74,204,124,165)(37,229,75,205,125,166)(38,230,76,206,126,167)(39,231,77,207,85,168)(40,232,78,208,86,127)(41,233,79,209,87,128)(42,234,80,210,88,129), (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)>;
G:=Group( (1,235,81,169,89,130)(2,236,82,170,90,131)(3,237,83,171,91,132)(4,238,84,172,92,133)(5,239,43,173,93,134)(6,240,44,174,94,135)(7,241,45,175,95,136)(8,242,46,176,96,137)(9,243,47,177,97,138)(10,244,48,178,98,139)(11,245,49,179,99,140)(12,246,50,180,100,141)(13,247,51,181,101,142)(14,248,52,182,102,143)(15,249,53,183,103,144)(16,250,54,184,104,145)(17,251,55,185,105,146)(18,252,56,186,106,147)(19,211,57,187,107,148)(20,212,58,188,108,149)(21,213,59,189,109,150)(22,214,60,190,110,151)(23,215,61,191,111,152)(24,216,62,192,112,153)(25,217,63,193,113,154)(26,218,64,194,114,155)(27,219,65,195,115,156)(28,220,66,196,116,157)(29,221,67,197,117,158)(30,222,68,198,118,159)(31,223,69,199,119,160)(32,224,70,200,120,161)(33,225,71,201,121,162)(34,226,72,202,122,163)(35,227,73,203,123,164)(36,228,74,204,124,165)(37,229,75,205,125,166)(38,230,76,206,126,167)(39,231,77,207,85,168)(40,232,78,208,86,127)(41,233,79,209,87,128)(42,234,80,210,88,129), (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) );
G=PermutationGroup([[(1,235,81,169,89,130),(2,236,82,170,90,131),(3,237,83,171,91,132),(4,238,84,172,92,133),(5,239,43,173,93,134),(6,240,44,174,94,135),(7,241,45,175,95,136),(8,242,46,176,96,137),(9,243,47,177,97,138),(10,244,48,178,98,139),(11,245,49,179,99,140),(12,246,50,180,100,141),(13,247,51,181,101,142),(14,248,52,182,102,143),(15,249,53,183,103,144),(16,250,54,184,104,145),(17,251,55,185,105,146),(18,252,56,186,106,147),(19,211,57,187,107,148),(20,212,58,188,108,149),(21,213,59,189,109,150),(22,214,60,190,110,151),(23,215,61,191,111,152),(24,216,62,192,112,153),(25,217,63,193,113,154),(26,218,64,194,114,155),(27,219,65,195,115,156),(28,220,66,196,116,157),(29,221,67,197,117,158),(30,222,68,198,118,159),(31,223,69,199,119,160),(32,224,70,200,120,161),(33,225,71,201,121,162),(34,226,72,202,122,163),(35,227,73,203,123,164),(36,228,74,204,124,165),(37,229,75,205,125,166),(38,230,76,206,126,167),(39,231,77,207,85,168),(40,232,78,208,86,127),(41,233,79,209,87,128),(42,234,80,210,88,129)], [(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)]])
252 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 6A | ··· | 6X | 7A | ··· | 7F | 14A | ··· | 14R | 21A | ··· | 21AV | 42A | ··· | 42EN |
| order | 1 | 2 | 2 | 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 | 1 | ··· | 1 |
252 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 |
| kernel | C6×C42 | C3×C42 | C2×C42 | C42 | C62 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 3 | 8 | 24 | 6 | 18 | 48 | 144 |
Matrix representation of C6×C42 ►in GL2(𝔽43) generated by
| 7 | 0 |
| 0 | 42 |
| 30 | 0 |
| 0 | 23 |
G:=sub<GL(2,GF(43))| [7,0,0,42],[30,0,0,23] >;
C6×C42 in GAP, Magma, Sage, TeX
C_6\times C_{42} % in TeX
G:=Group("C6xC42"); // GroupNames label
G:=SmallGroup(252,46);
// by ID
G=gap.SmallGroup(252,46);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-7]);
// Polycyclic
G:=Group<a,b|a^6=b^42=1,a*b=b*a>;
// generators/relations