p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C27⋊2C9, C92.1C3, C3.2C92, (C3×C9).1C9, C9.5(C3×C9), (C3×C27).2C3, C3.1(C27⋊C3), C32.13(C3×C9), (C3×C9).30C32, SmallGroup(243,11)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C27⋊2C9
G = < a,b | a27=b9=1, bab-1=a10 >
Smallest permutation representation of C27⋊2C9
►Regular action on 243 points
Generators in S243
(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)
(1 106 38 123 236 175 62 143 207)(2 98 48 124 228 185 63 162 190)(3 90 31 125 220 168 64 154 200)(4 82 41 126 239 178 65 146 210)(5 101 51 127 231 188 66 138 193)(6 93 34 128 223 171 67 157 203)(7 85 44 129 242 181 68 149 213)(8 104 54 130 234 164 69 141 196)(9 96 37 131 226 174 70 160 206)(10 88 47 132 218 184 71 152 216)(11 107 30 133 237 167 72 144 199)(12 99 40 134 229 177 73 136 209)(13 91 50 135 221 187 74 155 192)(14 83 33 109 240 170 75 147 202)(15 102 43 110 232 180 76 139 212)(16 94 53 111 224 163 77 158 195)(17 86 36 112 243 173 78 150 205)(18 105 46 113 235 183 79 142 215)(19 97 29 114 227 166 80 161 198)(20 89 39 115 219 176 81 153 208)(21 108 49 116 238 186 55 145 191)(22 100 32 117 230 169 56 137 201)(23 92 42 118 222 179 57 156 211)(24 84 52 119 241 189 58 148 194)(25 103 35 120 233 172 59 140 204)(26 95 45 121 225 182 60 159 214)(27 87 28 122 217 165 61 151 197)
G:=sub<Sym(243)| (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), (1,106,38,123,236,175,62,143,207)(2,98,48,124,228,185,63,162,190)(3,90,31,125,220,168,64,154,200)(4,82,41,126,239,178,65,146,210)(5,101,51,127,231,188,66,138,193)(6,93,34,128,223,171,67,157,203)(7,85,44,129,242,181,68,149,213)(8,104,54,130,234,164,69,141,196)(9,96,37,131,226,174,70,160,206)(10,88,47,132,218,184,71,152,216)(11,107,30,133,237,167,72,144,199)(12,99,40,134,229,177,73,136,209)(13,91,50,135,221,187,74,155,192)(14,83,33,109,240,170,75,147,202)(15,102,43,110,232,180,76,139,212)(16,94,53,111,224,163,77,158,195)(17,86,36,112,243,173,78,150,205)(18,105,46,113,235,183,79,142,215)(19,97,29,114,227,166,80,161,198)(20,89,39,115,219,176,81,153,208)(21,108,49,116,238,186,55,145,191)(22,100,32,117,230,169,56,137,201)(23,92,42,118,222,179,57,156,211)(24,84,52,119,241,189,58,148,194)(25,103,35,120,233,172,59,140,204)(26,95,45,121,225,182,60,159,214)(27,87,28,122,217,165,61,151,197)>;
G:=Group( (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), (1,106,38,123,236,175,62,143,207)(2,98,48,124,228,185,63,162,190)(3,90,31,125,220,168,64,154,200)(4,82,41,126,239,178,65,146,210)(5,101,51,127,231,188,66,138,193)(6,93,34,128,223,171,67,157,203)(7,85,44,129,242,181,68,149,213)(8,104,54,130,234,164,69,141,196)(9,96,37,131,226,174,70,160,206)(10,88,47,132,218,184,71,152,216)(11,107,30,133,237,167,72,144,199)(12,99,40,134,229,177,73,136,209)(13,91,50,135,221,187,74,155,192)(14,83,33,109,240,170,75,147,202)(15,102,43,110,232,180,76,139,212)(16,94,53,111,224,163,77,158,195)(17,86,36,112,243,173,78,150,205)(18,105,46,113,235,183,79,142,215)(19,97,29,114,227,166,80,161,198)(20,89,39,115,219,176,81,153,208)(21,108,49,116,238,186,55,145,191)(22,100,32,117,230,169,56,137,201)(23,92,42,118,222,179,57,156,211)(24,84,52,119,241,189,58,148,194)(25,103,35,120,233,172,59,140,204)(26,95,45,121,225,182,60,159,214)(27,87,28,122,217,165,61,151,197) );
G=PermutationGroup([[(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)], [(1,106,38,123,236,175,62,143,207),(2,98,48,124,228,185,63,162,190),(3,90,31,125,220,168,64,154,200),(4,82,41,126,239,178,65,146,210),(5,101,51,127,231,188,66,138,193),(6,93,34,128,223,171,67,157,203),(7,85,44,129,242,181,68,149,213),(8,104,54,130,234,164,69,141,196),(9,96,37,131,226,174,70,160,206),(10,88,47,132,218,184,71,152,216),(11,107,30,133,237,167,72,144,199),(12,99,40,134,229,177,73,136,209),(13,91,50,135,221,187,74,155,192),(14,83,33,109,240,170,75,147,202),(15,102,43,110,232,180,76,139,212),(16,94,53,111,224,163,77,158,195),(17,86,36,112,243,173,78,150,205),(18,105,46,113,235,183,79,142,215),(19,97,29,114,227,166,80,161,198),(20,89,39,115,219,176,81,153,208),(21,108,49,116,238,186,55,145,191),(22,100,32,117,230,169,56,137,201),(23,92,42,118,222,179,57,156,211),(24,84,52,119,241,189,58,148,194),(25,103,35,120,233,172,59,140,204),(26,95,45,121,225,182,60,159,214),(27,87,28,122,217,165,61,151,197)]])
C27⋊2C9 is a maximal subgroup of
C27⋊3C18
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 |
| type | + | |||||
| image | C1 | C3 | C3 | C9 | C9 | C27⋊C3 |
| kernel | C27⋊2C9 | C92 | C3×C27 | C27 | C3×C9 | C3 |
| # reps | 1 | 2 | 6 | 54 | 18 | 18 |
Matrix representation of C27⋊2C9 ►in GL4(𝔽109) generated by
| 1 | 0 | 0 | 0 |
| 0 | 71 | 45 | 38 |
| 0 | 27 | 34 | 16 |
| 0 | 67 | 17 | 4 |
| 105 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 82 | 71 | 0 |
| 0 | 71 | 45 | 38 |
G:=sub<GL(4,GF(109))| [1,0,0,0,0,71,27,67,0,45,34,17,0,38,16,4],[105,0,0,0,0,0,82,71,0,1,71,45,0,0,0,38] >;
C27⋊2C9 in GAP, Magma, Sage, TeX
C_{27}\rtimes_2C_9 % in TeX
G:=Group("C27:2C9"); // GroupNames label
G:=SmallGroup(243,11);
// by ID
G=gap.SmallGroup(243,11);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,45,841,96,147]);
// Polycyclic
G:=Group<a,b|a^27=b^9=1,b*a*b^-1=a^10>;
// generators/relations
Export