direct product, p-group, abelian, monomial
Aliases: C3×C92, SmallGroup(243,31)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C3×C92 |
| C1 — C3×C92 |
| C1 — C3×C92 |
Generators and relations for C3×C92
G = < a,b,c | a3=b9=c9=1, ab=ba, ac=ca, bc=cb >
Subgroups: 126, all normal (4 characteristic)
C1, C3, C9, C32, C32, C3×C9, C33, C92, C32×C9, C3×C92
Quotients: C1, C3, C9, C32, C3×C9, C33, C92, C32×C9, C3×C92
►Regular action on 243 points
Generators in S243
(1 48 103)(2 49 104)(3 50 105)(4 51 106)(5 52 107)(6 53 108)(7 54 100)(8 46 101)(9 47 102)(10 145 227)(11 146 228)(12 147 229)(13 148 230)(14 149 231)(15 150 232)(16 151 233)(17 152 234)(18 153 226)(19 154 200)(20 155 201)(21 156 202)(22 157 203)(23 158 204)(24 159 205)(25 160 206)(26 161 207)(27 162 199)(28 83 209)(29 84 210)(30 85 211)(31 86 212)(32 87 213)(33 88 214)(34 89 215)(35 90 216)(36 82 208)(37 92 218)(38 93 219)(39 94 220)(40 95 221)(41 96 222)(42 97 223)(43 98 224)(44 99 225)(45 91 217)(55 110 119)(56 111 120)(57 112 121)(58 113 122)(59 114 123)(60 115 124)(61 116 125)(62 117 126)(63 109 118)(64 182 165)(65 183 166)(66 184 167)(67 185 168)(68 186 169)(69 187 170)(70 188 171)(71 189 163)(72 181 164)(73 128 174)(74 129 175)(75 130 176)(76 131 177)(77 132 178)(78 133 179)(79 134 180)(80 135 172)(81 127 173)(136 191 236)(137 192 237)(138 193 238)(139 194 239)(140 195 240)(141 196 241)(142 197 242)(143 198 243)(144 190 235)
(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 69 128 215 138 94 115 229 160)(2 70 129 216 139 95 116 230 161)(3 71 130 208 140 96 117 231 162)(4 72 131 209 141 97 109 232 154)(5 64 132 210 142 98 110 233 155)(6 65 133 211 143 99 111 234 156)(7 66 134 212 144 91 112 226 157)(8 67 135 213 136 92 113 227 158)(9 68 127 214 137 93 114 228 159)(10 204 46 185 172 32 191 218 122)(11 205 47 186 173 33 192 219 123)(12 206 48 187 174 34 193 220 124)(13 207 49 188 175 35 194 221 125)(14 199 50 189 176 36 195 222 126)(15 200 51 181 177 28 196 223 118)(16 201 52 182 178 29 197 224 119)(17 202 53 183 179 30 198 225 120)(18 203 54 184 180 31 190 217 121)(19 106 164 76 83 241 42 63 150)(20 107 165 77 84 242 43 55 151)(21 108 166 78 85 243 44 56 152)(22 100 167 79 86 235 45 57 153)(23 101 168 80 87 236 37 58 145)(24 102 169 81 88 237 38 59 146)(25 103 170 73 89 238 39 60 147)(26 104 171 74 90 239 40 61 148)(27 105 163 75 82 240 41 62 149)
G:=sub<Sym(243)| (1,48,103)(2,49,104)(3,50,105)(4,51,106)(5,52,107)(6,53,108)(7,54,100)(8,46,101)(9,47,102)(10,145,227)(11,146,228)(12,147,229)(13,148,230)(14,149,231)(15,150,232)(16,151,233)(17,152,234)(18,153,226)(19,154,200)(20,155,201)(21,156,202)(22,157,203)(23,158,204)(24,159,205)(25,160,206)(26,161,207)(27,162,199)(28,83,209)(29,84,210)(30,85,211)(31,86,212)(32,87,213)(33,88,214)(34,89,215)(35,90,216)(36,82,208)(37,92,218)(38,93,219)(39,94,220)(40,95,221)(41,96,222)(42,97,223)(43,98,224)(44,99,225)(45,91,217)(55,110,119)(56,111,120)(57,112,121)(58,113,122)(59,114,123)(60,115,124)(61,116,125)(62,117,126)(63,109,118)(64,182,165)(65,183,166)(66,184,167)(67,185,168)(68,186,169)(69,187,170)(70,188,171)(71,189,163)(72,181,164)(73,128,174)(74,129,175)(75,130,176)(76,131,177)(77,132,178)(78,133,179)(79,134,180)(80,135,172)(81,127,173)(136,191,236)(137,192,237)(138,193,238)(139,194,239)(140,195,240)(141,196,241)(142,197,242)(143,198,243)(144,190,235), (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), (1,69,128,215,138,94,115,229,160)(2,70,129,216,139,95,116,230,161)(3,71,130,208,140,96,117,231,162)(4,72,131,209,141,97,109,232,154)(5,64,132,210,142,98,110,233,155)(6,65,133,211,143,99,111,234,156)(7,66,134,212,144,91,112,226,157)(8,67,135,213,136,92,113,227,158)(9,68,127,214,137,93,114,228,159)(10,204,46,185,172,32,191,218,122)(11,205,47,186,173,33,192,219,123)(12,206,48,187,174,34,193,220,124)(13,207,49,188,175,35,194,221,125)(14,199,50,189,176,36,195,222,126)(15,200,51,181,177,28,196,223,118)(16,201,52,182,178,29,197,224,119)(17,202,53,183,179,30,198,225,120)(18,203,54,184,180,31,190,217,121)(19,106,164,76,83,241,42,63,150)(20,107,165,77,84,242,43,55,151)(21,108,166,78,85,243,44,56,152)(22,100,167,79,86,235,45,57,153)(23,101,168,80,87,236,37,58,145)(24,102,169,81,88,237,38,59,146)(25,103,170,73,89,238,39,60,147)(26,104,171,74,90,239,40,61,148)(27,105,163,75,82,240,41,62,149)>;
G:=Group( (1,48,103)(2,49,104)(3,50,105)(4,51,106)(5,52,107)(6,53,108)(7,54,100)(8,46,101)(9,47,102)(10,145,227)(11,146,228)(12,147,229)(13,148,230)(14,149,231)(15,150,232)(16,151,233)(17,152,234)(18,153,226)(19,154,200)(20,155,201)(21,156,202)(22,157,203)(23,158,204)(24,159,205)(25,160,206)(26,161,207)(27,162,199)(28,83,209)(29,84,210)(30,85,211)(31,86,212)(32,87,213)(33,88,214)(34,89,215)(35,90,216)(36,82,208)(37,92,218)(38,93,219)(39,94,220)(40,95,221)(41,96,222)(42,97,223)(43,98,224)(44,99,225)(45,91,217)(55,110,119)(56,111,120)(57,112,121)(58,113,122)(59,114,123)(60,115,124)(61,116,125)(62,117,126)(63,109,118)(64,182,165)(65,183,166)(66,184,167)(67,185,168)(68,186,169)(69,187,170)(70,188,171)(71,189,163)(72,181,164)(73,128,174)(74,129,175)(75,130,176)(76,131,177)(77,132,178)(78,133,179)(79,134,180)(80,135,172)(81,127,173)(136,191,236)(137,192,237)(138,193,238)(139,194,239)(140,195,240)(141,196,241)(142,197,242)(143,198,243)(144,190,235), (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), (1,69,128,215,138,94,115,229,160)(2,70,129,216,139,95,116,230,161)(3,71,130,208,140,96,117,231,162)(4,72,131,209,141,97,109,232,154)(5,64,132,210,142,98,110,233,155)(6,65,133,211,143,99,111,234,156)(7,66,134,212,144,91,112,226,157)(8,67,135,213,136,92,113,227,158)(9,68,127,214,137,93,114,228,159)(10,204,46,185,172,32,191,218,122)(11,205,47,186,173,33,192,219,123)(12,206,48,187,174,34,193,220,124)(13,207,49,188,175,35,194,221,125)(14,199,50,189,176,36,195,222,126)(15,200,51,181,177,28,196,223,118)(16,201,52,182,178,29,197,224,119)(17,202,53,183,179,30,198,225,120)(18,203,54,184,180,31,190,217,121)(19,106,164,76,83,241,42,63,150)(20,107,165,77,84,242,43,55,151)(21,108,166,78,85,243,44,56,152)(22,100,167,79,86,235,45,57,153)(23,101,168,80,87,236,37,58,145)(24,102,169,81,88,237,38,59,146)(25,103,170,73,89,238,39,60,147)(26,104,171,74,90,239,40,61,148)(27,105,163,75,82,240,41,62,149) );
G=PermutationGroup([[(1,48,103),(2,49,104),(3,50,105),(4,51,106),(5,52,107),(6,53,108),(7,54,100),(8,46,101),(9,47,102),(10,145,227),(11,146,228),(12,147,229),(13,148,230),(14,149,231),(15,150,232),(16,151,233),(17,152,234),(18,153,226),(19,154,200),(20,155,201),(21,156,202),(22,157,203),(23,158,204),(24,159,205),(25,160,206),(26,161,207),(27,162,199),(28,83,209),(29,84,210),(30,85,211),(31,86,212),(32,87,213),(33,88,214),(34,89,215),(35,90,216),(36,82,208),(37,92,218),(38,93,219),(39,94,220),(40,95,221),(41,96,222),(42,97,223),(43,98,224),(44,99,225),(45,91,217),(55,110,119),(56,111,120),(57,112,121),(58,113,122),(59,114,123),(60,115,124),(61,116,125),(62,117,126),(63,109,118),(64,182,165),(65,183,166),(66,184,167),(67,185,168),(68,186,169),(69,187,170),(70,188,171),(71,189,163),(72,181,164),(73,128,174),(74,129,175),(75,130,176),(76,131,177),(77,132,178),(78,133,179),(79,134,180),(80,135,172),(81,127,173),(136,191,236),(137,192,237),(138,193,238),(139,194,239),(140,195,240),(141,196,241),(142,197,242),(143,198,243),(144,190,235)], [(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)], [(1,69,128,215,138,94,115,229,160),(2,70,129,216,139,95,116,230,161),(3,71,130,208,140,96,117,231,162),(4,72,131,209,141,97,109,232,154),(5,64,132,210,142,98,110,233,155),(6,65,133,211,143,99,111,234,156),(7,66,134,212,144,91,112,226,157),(8,67,135,213,136,92,113,227,158),(9,68,127,214,137,93,114,228,159),(10,204,46,185,172,32,191,218,122),(11,205,47,186,173,33,192,219,123),(12,206,48,187,174,34,193,220,124),(13,207,49,188,175,35,194,221,125),(14,199,50,189,176,36,195,222,126),(15,200,51,181,177,28,196,223,118),(16,201,52,182,178,29,197,224,119),(17,202,53,183,179,30,198,225,120),(18,203,54,184,180,31,190,217,121),(19,106,164,76,83,241,42,63,150),(20,107,165,77,84,242,43,55,151),(21,108,166,78,85,243,44,56,152),(22,100,167,79,86,235,45,57,153),(23,101,168,80,87,236,37,58,145),(24,102,169,81,88,237,38,59,146),(25,103,170,73,89,238,39,60,147),(26,104,171,74,90,239,40,61,148),(27,105,163,75,82,240,41,62,149)]])
C3×C92 is a maximal subgroup of
C92⋊8S3
243 conjugacy classes
| class | 1 | 3A | ··· | 3Z | 9A | ··· | 9HH |
| order | 1 | 3 | ··· | 3 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
243 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C3 | C9 |
| kernel | C3×C92 | C92 | C32×C9 | C3×C9 |
| # reps | 1 | 18 | 8 | 216 |
Matrix representation of C3×C92 ►in GL3(𝔽19) generated by
| 11 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 7 |
| 1 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 17 |
G:=sub<GL(3,GF(19))| [11,0,0,0,11,0,0,0,7],[1,0,0,0,16,0,0,0,1],[1,0,0,0,5,0,0,0,17] >;
C3×C92 in GAP, Magma, Sage, TeX
C_3\times C_9^2
% in TeX
G:=Group("C3xC9^2"); // GroupNames label
G:=SmallGroup(243,31);
// by ID
G=gap.SmallGroup(243,31);
# by ID
G:=PCGroup([5,-3,3,3,-3,3,135,276]);
// Polycyclic
G:=Group<a,b,c|a^3=b^9=c^9=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations