direct product, p-group, abelian, monomial
Aliases: C9×C27, SmallGroup(243,10)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C9×C27 |
| C1 — C9×C27 |
| C1 — C9×C27 |
Generators and relations for C9×C27
G = < a,b | a9=b27=1, ab=ba >
Smallest permutation representation of C9×C27
►Regular action on 243 points
Generators in S243
(1 129 194 93 63 41 152 235 168)(2 130 195 94 64 42 153 236 169)(3 131 196 95 65 43 154 237 170)(4 132 197 96 66 44 155 238 171)(5 133 198 97 67 45 156 239 172)(6 134 199 98 68 46 157 240 173)(7 135 200 99 69 47 158 241 174)(8 109 201 100 70 48 159 242 175)(9 110 202 101 71 49 160 243 176)(10 111 203 102 72 50 161 217 177)(11 112 204 103 73 51 162 218 178)(12 113 205 104 74 52 136 219 179)(13 114 206 105 75 53 137 220 180)(14 115 207 106 76 54 138 221 181)(15 116 208 107 77 28 139 222 182)(16 117 209 108 78 29 140 223 183)(17 118 210 82 79 30 141 224 184)(18 119 211 83 80 31 142 225 185)(19 120 212 84 81 32 143 226 186)(20 121 213 85 55 33 144 227 187)(21 122 214 86 56 34 145 228 188)(22 123 215 87 57 35 146 229 189)(23 124 216 88 58 36 147 230 163)(24 125 190 89 59 37 148 231 164)(25 126 191 90 60 38 149 232 165)(26 127 192 91 61 39 150 233 166)(27 128 193 92 62 40 151 234 167)
(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)
G:=sub<Sym(243)| (1,129,194,93,63,41,152,235,168)(2,130,195,94,64,42,153,236,169)(3,131,196,95,65,43,154,237,170)(4,132,197,96,66,44,155,238,171)(5,133,198,97,67,45,156,239,172)(6,134,199,98,68,46,157,240,173)(7,135,200,99,69,47,158,241,174)(8,109,201,100,70,48,159,242,175)(9,110,202,101,71,49,160,243,176)(10,111,203,102,72,50,161,217,177)(11,112,204,103,73,51,162,218,178)(12,113,205,104,74,52,136,219,179)(13,114,206,105,75,53,137,220,180)(14,115,207,106,76,54,138,221,181)(15,116,208,107,77,28,139,222,182)(16,117,209,108,78,29,140,223,183)(17,118,210,82,79,30,141,224,184)(18,119,211,83,80,31,142,225,185)(19,120,212,84,81,32,143,226,186)(20,121,213,85,55,33,144,227,187)(21,122,214,86,56,34,145,228,188)(22,123,215,87,57,35,146,229,189)(23,124,216,88,58,36,147,230,163)(24,125,190,89,59,37,148,231,164)(25,126,191,90,60,38,149,232,165)(26,127,192,91,61,39,150,233,166)(27,128,193,92,62,40,151,234,167), (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)>;
G:=Group( (1,129,194,93,63,41,152,235,168)(2,130,195,94,64,42,153,236,169)(3,131,196,95,65,43,154,237,170)(4,132,197,96,66,44,155,238,171)(5,133,198,97,67,45,156,239,172)(6,134,199,98,68,46,157,240,173)(7,135,200,99,69,47,158,241,174)(8,109,201,100,70,48,159,242,175)(9,110,202,101,71,49,160,243,176)(10,111,203,102,72,50,161,217,177)(11,112,204,103,73,51,162,218,178)(12,113,205,104,74,52,136,219,179)(13,114,206,105,75,53,137,220,180)(14,115,207,106,76,54,138,221,181)(15,116,208,107,77,28,139,222,182)(16,117,209,108,78,29,140,223,183)(17,118,210,82,79,30,141,224,184)(18,119,211,83,80,31,142,225,185)(19,120,212,84,81,32,143,226,186)(20,121,213,85,55,33,144,227,187)(21,122,214,86,56,34,145,228,188)(22,123,215,87,57,35,146,229,189)(23,124,216,88,58,36,147,230,163)(24,125,190,89,59,37,148,231,164)(25,126,191,90,60,38,149,232,165)(26,127,192,91,61,39,150,233,166)(27,128,193,92,62,40,151,234,167), (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) );
G=PermutationGroup([[(1,129,194,93,63,41,152,235,168),(2,130,195,94,64,42,153,236,169),(3,131,196,95,65,43,154,237,170),(4,132,197,96,66,44,155,238,171),(5,133,198,97,67,45,156,239,172),(6,134,199,98,68,46,157,240,173),(7,135,200,99,69,47,158,241,174),(8,109,201,100,70,48,159,242,175),(9,110,202,101,71,49,160,243,176),(10,111,203,102,72,50,161,217,177),(11,112,204,103,73,51,162,218,178),(12,113,205,104,74,52,136,219,179),(13,114,206,105,75,53,137,220,180),(14,115,207,106,76,54,138,221,181),(15,116,208,107,77,28,139,222,182),(16,117,209,108,78,29,140,223,183),(17,118,210,82,79,30,141,224,184),(18,119,211,83,80,31,142,225,185),(19,120,212,84,81,32,143,226,186),(20,121,213,85,55,33,144,227,187),(21,122,214,86,56,34,145,228,188),(22,123,215,87,57,35,146,229,189),(23,124,216,88,58,36,147,230,163),(24,125,190,89,59,37,148,231,164),(25,126,191,90,60,38,149,232,165),(26,127,192,91,61,39,150,233,166),(27,128,193,92,62,40,151,234,167)], [(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)]])
C9×C27 is a maximal subgroup of
C9⋊D27
243 conjugacy classes
| class | 1 | 3A | ··· | 3H | 9A | ··· | 9BT | 27A | ··· | 27FF |
| order | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 27 | ··· | 27 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
243 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||
| image | C1 | C3 | C3 | C9 | C9 | C27 |
| kernel | C9×C27 | C92 | C3×C27 | C27 | C3×C9 | C9 |
| # reps | 1 | 2 | 6 | 54 | 18 | 162 |
Matrix representation of C9×C27 ►in GL2(𝔽109) generated by
| 1 | 0 |
| 0 | 66 |
| 25 | 0 |
| 0 | 21 |
G:=sub<GL(2,GF(109))| [1,0,0,66],[25,0,0,21] >;
C9×C27 in GAP, Magma, Sage, TeX
C_9\times C_{27} % in TeX
G:=Group("C9xC27"); // GroupNames label
G:=SmallGroup(243,10);
// by ID
G=gap.SmallGroup(243,10);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,45,96,147]);
// Polycyclic
G:=Group<a,b|a^9=b^27=1,a*b=b*a>;
// generators/relations
Export