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G = C9×C27order 243 = 35

Abelian group of type [9,27]

direct product, p-group, abelian, monomial

Aliases: C9×C27, SmallGroup(243,10)

Series: Derived Chief Lower central Upper central Jennings

C1 — C9×C27
C1C3C32C3×C9C92 — C9×C27
C1 — C9×C27
C1 — C9×C27
C1C3C3C3C3C3C3C3×C9C3×C9 — 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
(1 131 83 185 47 239 154 207 67)(2 132 84 186 48 240 155 208 68)(3 133 85 187 49 241 156 209 69)(4 134 86 188 50 242 157 210 70)(5 135 87 189 51 243 158 211 71)(6 109 88 163 52 217 159 212 72)(7 110 89 164 53 218 160 213 73)(8 111 90 165 54 219 161 214 74)(9 112 91 166 28 220 162 215 75)(10 113 92 167 29 221 136 216 76)(11 114 93 168 30 222 137 190 77)(12 115 94 169 31 223 138 191 78)(13 116 95 170 32 224 139 192 79)(14 117 96 171 33 225 140 193 80)(15 118 97 172 34 226 141 194 81)(16 119 98 173 35 227 142 195 55)(17 120 99 174 36 228 143 196 56)(18 121 100 175 37 229 144 197 57)(19 122 101 176 38 230 145 198 58)(20 123 102 177 39 231 146 199 59)(21 124 103 178 40 232 147 200 60)(22 125 104 179 41 233 148 201 61)(23 126 105 180 42 234 149 202 62)(24 127 106 181 43 235 150 203 63)(25 128 107 182 44 236 151 204 64)(26 129 108 183 45 237 152 205 65)(27 130 82 184 46 238 153 206 66)
(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)| (1,131,83,185,47,239,154,207,67)(2,132,84,186,48,240,155,208,68)(3,133,85,187,49,241,156,209,69)(4,134,86,188,50,242,157,210,70)(5,135,87,189,51,243,158,211,71)(6,109,88,163,52,217,159,212,72)(7,110,89,164,53,218,160,213,73)(8,111,90,165,54,219,161,214,74)(9,112,91,166,28,220,162,215,75)(10,113,92,167,29,221,136,216,76)(11,114,93,168,30,222,137,190,77)(12,115,94,169,31,223,138,191,78)(13,116,95,170,32,224,139,192,79)(14,117,96,171,33,225,140,193,80)(15,118,97,172,34,226,141,194,81)(16,119,98,173,35,227,142,195,55)(17,120,99,174,36,228,143,196,56)(18,121,100,175,37,229,144,197,57)(19,122,101,176,38,230,145,198,58)(20,123,102,177,39,231,146,199,59)(21,124,103,178,40,232,147,200,60)(22,125,104,179,41,233,148,201,61)(23,126,105,180,42,234,149,202,62)(24,127,106,181,43,235,150,203,63)(25,128,107,182,44,236,151,204,64)(26,129,108,183,45,237,152,205,65)(27,130,82,184,46,238,153,206,66), (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( (1,131,83,185,47,239,154,207,67)(2,132,84,186,48,240,155,208,68)(3,133,85,187,49,241,156,209,69)(4,134,86,188,50,242,157,210,70)(5,135,87,189,51,243,158,211,71)(6,109,88,163,52,217,159,212,72)(7,110,89,164,53,218,160,213,73)(8,111,90,165,54,219,161,214,74)(9,112,91,166,28,220,162,215,75)(10,113,92,167,29,221,136,216,76)(11,114,93,168,30,222,137,190,77)(12,115,94,169,31,223,138,191,78)(13,116,95,170,32,224,139,192,79)(14,117,96,171,33,225,140,193,80)(15,118,97,172,34,226,141,194,81)(16,119,98,173,35,227,142,195,55)(17,120,99,174,36,228,143,196,56)(18,121,100,175,37,229,144,197,57)(19,122,101,176,38,230,145,198,58)(20,123,102,177,39,231,146,199,59)(21,124,103,178,40,232,147,200,60)(22,125,104,179,41,233,148,201,61)(23,126,105,180,42,234,149,202,62)(24,127,106,181,43,235,150,203,63)(25,128,107,182,44,236,151,204,64)(26,129,108,183,45,237,152,205,65)(27,130,82,184,46,238,153,206,66), (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([(1,131,83,185,47,239,154,207,67),(2,132,84,186,48,240,155,208,68),(3,133,85,187,49,241,156,209,69),(4,134,86,188,50,242,157,210,70),(5,135,87,189,51,243,158,211,71),(6,109,88,163,52,217,159,212,72),(7,110,89,164,53,218,160,213,73),(8,111,90,165,54,219,161,214,74),(9,112,91,166,28,220,162,215,75),(10,113,92,167,29,221,136,216,76),(11,114,93,168,30,222,137,190,77),(12,115,94,169,31,223,138,191,78),(13,116,95,170,32,224,139,192,79),(14,117,96,171,33,225,140,193,80),(15,118,97,172,34,226,141,194,81),(16,119,98,173,35,227,142,195,55),(17,120,99,174,36,228,143,196,56),(18,121,100,175,37,229,144,197,57),(19,122,101,176,38,230,145,198,58),(20,123,102,177,39,231,146,199,59),(21,124,103,178,40,232,147,200,60),(22,125,104,179,41,233,148,201,61),(23,126,105,180,42,234,149,202,62),(24,127,106,181,43,235,150,203,63),(25,128,107,182,44,236,151,204,64),(26,129,108,183,45,237,152,205,65),(27,130,82,184,46,238,153,206,66)], [(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···3H9A···9BT27A···27FF
order13···39···927···27
size11···11···11···1

243 irreducible representations

dim111111
type+
imageC1C3C3C9C9C27
kernelC9×C27C92C3×C27C27C3×C9C9
# reps1265418162

Matrix representation of C9×C27 in GL2(𝔽109) generated by

10
066
,
250
021
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

Subgroup lattice of C9×C27 in TeX

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