p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C9⋊C27, C92.3C3, C9.73- 1+2, (C3×C9).4C9, C3.2(C9⋊C9), C3.2(C3×C27), (C3×C27).1C3, C3.3(C27⋊C3), C32.15(C3×C9), (C3×C9).32C32, SmallGroup(243,21)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C9⋊C27
G = < a,b | a9=b27=1, bab-1=a7 >
Smallest permutation representation of C9⋊C27
►Regular action on 243 points
Generators in S243
(1 133 209 166 74 143 91 220 45)(2 221 144 167 134 46 92 75 210)(3 76 47 168 222 211 93 135 145)(4 109 212 169 77 146 94 223 48)(5 224 147 170 110 49 95 78 213)(6 79 50 171 225 214 96 111 148)(7 112 215 172 80 149 97 226 51)(8 227 150 173 113 52 98 81 216)(9 55 53 174 228 190 99 114 151)(10 115 191 175 56 152 100 229 54)(11 230 153 176 116 28 101 57 192)(12 58 29 177 231 193 102 117 154)(13 118 194 178 59 155 103 232 30)(14 233 156 179 119 31 104 60 195)(15 61 32 180 234 196 105 120 157)(16 121 197 181 62 158 106 235 33)(17 236 159 182 122 34 107 63 198)(18 64 35 183 237 199 108 123 160)(19 124 200 184 65 161 82 238 36)(20 239 162 185 125 37 83 66 201)(21 67 38 186 240 202 84 126 136)(22 127 203 187 68 137 85 241 39)(23 242 138 188 128 40 86 69 204)(24 70 41 189 243 205 87 129 139)(25 130 206 163 71 140 88 217 42)(26 218 141 164 131 43 89 72 207)(27 73 44 165 219 208 90 132 142)
(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)| (1,133,209,166,74,143,91,220,45)(2,221,144,167,134,46,92,75,210)(3,76,47,168,222,211,93,135,145)(4,109,212,169,77,146,94,223,48)(5,224,147,170,110,49,95,78,213)(6,79,50,171,225,214,96,111,148)(7,112,215,172,80,149,97,226,51)(8,227,150,173,113,52,98,81,216)(9,55,53,174,228,190,99,114,151)(10,115,191,175,56,152,100,229,54)(11,230,153,176,116,28,101,57,192)(12,58,29,177,231,193,102,117,154)(13,118,194,178,59,155,103,232,30)(14,233,156,179,119,31,104,60,195)(15,61,32,180,234,196,105,120,157)(16,121,197,181,62,158,106,235,33)(17,236,159,182,122,34,107,63,198)(18,64,35,183,237,199,108,123,160)(19,124,200,184,65,161,82,238,36)(20,239,162,185,125,37,83,66,201)(21,67,38,186,240,202,84,126,136)(22,127,203,187,68,137,85,241,39)(23,242,138,188,128,40,86,69,204)(24,70,41,189,243,205,87,129,139)(25,130,206,163,71,140,88,217,42)(26,218,141,164,131,43,89,72,207)(27,73,44,165,219,208,90,132,142), (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( (1,133,209,166,74,143,91,220,45)(2,221,144,167,134,46,92,75,210)(3,76,47,168,222,211,93,135,145)(4,109,212,169,77,146,94,223,48)(5,224,147,170,110,49,95,78,213)(6,79,50,171,225,214,96,111,148)(7,112,215,172,80,149,97,226,51)(8,227,150,173,113,52,98,81,216)(9,55,53,174,228,190,99,114,151)(10,115,191,175,56,152,100,229,54)(11,230,153,176,116,28,101,57,192)(12,58,29,177,231,193,102,117,154)(13,118,194,178,59,155,103,232,30)(14,233,156,179,119,31,104,60,195)(15,61,32,180,234,196,105,120,157)(16,121,197,181,62,158,106,235,33)(17,236,159,182,122,34,107,63,198)(18,64,35,183,237,199,108,123,160)(19,124,200,184,65,161,82,238,36)(20,239,162,185,125,37,83,66,201)(21,67,38,186,240,202,84,126,136)(22,127,203,187,68,137,85,241,39)(23,242,138,188,128,40,86,69,204)(24,70,41,189,243,205,87,129,139)(25,130,206,163,71,140,88,217,42)(26,218,141,164,131,43,89,72,207)(27,73,44,165,219,208,90,132,142), (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([[(1,133,209,166,74,143,91,220,45),(2,221,144,167,134,46,92,75,210),(3,76,47,168,222,211,93,135,145),(4,109,212,169,77,146,94,223,48),(5,224,147,170,110,49,95,78,213),(6,79,50,171,225,214,96,111,148),(7,112,215,172,80,149,97,226,51),(8,227,150,173,113,52,98,81,216),(9,55,53,174,228,190,99,114,151),(10,115,191,175,56,152,100,229,54),(11,230,153,176,116,28,101,57,192),(12,58,29,177,231,193,102,117,154),(13,118,194,178,59,155,103,232,30),(14,233,156,179,119,31,104,60,195),(15,61,32,180,234,196,105,120,157),(16,121,197,181,62,158,106,235,33),(17,236,159,182,122,34,107,63,198),(18,64,35,183,237,199,108,123,160),(19,124,200,184,65,161,82,238,36),(20,239,162,185,125,37,83,66,201),(21,67,38,186,240,202,84,126,136),(22,127,203,187,68,137,85,241,39),(23,242,138,188,128,40,86,69,204),(24,70,41,189,243,205,87,129,139),(25,130,206,163,71,140,88,217,42),(26,218,141,164,131,43,89,72,207),(27,73,44,165,219,208,90,132,142)], [(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⋊C54
99 conjugacy classes
| class | 1 | 3A | ··· | 3H | 9A | ··· | 9R | 9S | ··· | 9AJ | 27A | ··· | 27BB |
| order | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 | 27 | ··· | 27 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | C9 | C27 | 3- 1+2 | C27⋊C3 |
| kernel | C9⋊C27 | C92 | C3×C27 | C3×C9 | C9 | C9 | C3 |
| # reps | 1 | 2 | 6 | 18 | 54 | 6 | 12 |
Matrix representation of C9⋊C27 ►in GL4(𝔽109) generated by
| 45 | 0 | 0 | 0 |
| 0 | 1 | 62 | 0 |
| 0 | 0 | 108 | 1 |
| 0 | 1 | 108 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 60 | 45 | 23 |
| 0 | 54 | 106 | 78 |
| 0 | 108 | 23 | 52 |
G:=sub<GL(4,GF(109))| [45,0,0,0,0,1,0,1,0,62,108,108,0,0,1,0],[5,0,0,0,0,60,54,108,0,45,106,23,0,23,78,52] >;
C9⋊C27 in GAP, Magma, Sage, TeX
C_9\rtimes C_{27} % in TeX
G:=Group("C9:C27"); // GroupNames label
G:=SmallGroup(243,21);
// by ID
G=gap.SmallGroup(243,21);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,135,121,36,78]);
// Polycyclic
G:=Group<a,b|a^9=b^27=1,b*a*b^-1=a^7>;
// generators/relations
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