p-group, metabelian, nilpotent (class 2), monomial
Aliases: C3.1C92, C32.17He3, C33.38C32, C32.93- 1+2, (C3×C9)⋊1C9, C3.1(C9⋊C9), (C32×C9).1C3, C3.1(C32⋊C9), C32.12(C3×C9), SmallGroup(243,2)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C3.C92
G = < a,b,c | a3=b9=c9=1, cbc-1=ab=ba, ac=ca >
Subgroups: 117 in 69 conjugacy classes, 45 normal (5 characteristic)
C1, C3, C3, C9, C32, C32, C3×C9, C3×C9, C33, C32×C9, C3.C92
Quotients: C1, C3, C9, C32, C3×C9, He3, 3- 1+2, C92, C32⋊C9, C9⋊C9, C3.C92
►Regular action on 243 points
Generators in S243
(1 172 91)(2 173 92)(3 174 93)(4 175 94)(5 176 95)(6 177 96)(7 178 97)(8 179 98)(9 180 99)(10 117 30)(11 109 31)(12 110 32)(13 111 33)(14 112 34)(15 113 35)(16 114 36)(17 115 28)(18 116 29)(19 187 106)(20 188 107)(21 189 108)(22 181 100)(23 182 101)(24 183 102)(25 184 103)(26 185 104)(27 186 105)(37 205 124)(38 206 125)(39 207 126)(40 199 118)(41 200 119)(42 201 120)(43 202 121)(44 203 122)(45 204 123)(46 214 133)(47 215 134)(48 216 135)(49 208 127)(50 209 128)(51 210 129)(52 211 130)(53 212 131)(54 213 132)(55 195 142)(56 196 143)(57 197 144)(58 198 136)(59 190 137)(60 191 138)(61 192 139)(62 193 140)(63 194 141)(64 223 151)(65 224 152)(66 225 153)(67 217 145)(68 218 146)(69 219 147)(70 220 148)(71 221 149)(72 222 150)(73 232 160)(74 233 161)(75 234 162)(76 226 154)(77 227 155)(78 228 156)(79 229 157)(80 230 158)(81 231 159)(82 243 169)(83 235 170)(84 236 171)(85 237 163)(86 238 164)(87 239 165)(88 240 166)(89 241 167)(90 242 168)
(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)
(1 242 233 224 56 47 38 31 22)(2 82 162 225 144 216 39 110 182)(3 170 76 217 198 127 40 13 102)(4 236 227 218 59 50 41 34 25)(5 85 156 219 138 210 42 113 185)(6 164 79 220 192 130 43 16 105)(7 239 230 221 62 53 44 28 19)(8 88 159 222 141 213 45 116 188)(9 167 73 223 195 133 37 10 108)(11 181 172 168 161 152 196 215 206)(12 23 92 169 234 66 197 48 126)(14 184 175 171 155 146 190 209 200)(15 26 95 163 228 69 191 51 120)(17 187 178 165 158 149 193 212 203)(18 20 98 166 231 72 194 54 123)(21 180 89 232 151 142 46 205 117)(24 174 83 226 145 136 49 199 111)(27 177 86 229 148 139 52 202 114)(29 107 179 240 81 150 63 132 204)(30 189 99 241 160 64 55 214 124)(32 101 173 243 75 153 57 135 207)(33 183 93 235 154 67 58 208 118)(35 104 176 237 78 147 60 129 201)(36 186 96 238 157 70 61 211 121)(65 143 134 125 109 100 91 90 74)(68 137 128 119 112 103 94 84 77)(71 140 131 122 115 106 97 87 80)
G:=sub<Sym(243)| (1,172,91)(2,173,92)(3,174,93)(4,175,94)(5,176,95)(6,177,96)(7,178,97)(8,179,98)(9,180,99)(10,117,30)(11,109,31)(12,110,32)(13,111,33)(14,112,34)(15,113,35)(16,114,36)(17,115,28)(18,116,29)(19,187,106)(20,188,107)(21,189,108)(22,181,100)(23,182,101)(24,183,102)(25,184,103)(26,185,104)(27,186,105)(37,205,124)(38,206,125)(39,207,126)(40,199,118)(41,200,119)(42,201,120)(43,202,121)(44,203,122)(45,204,123)(46,214,133)(47,215,134)(48,216,135)(49,208,127)(50,209,128)(51,210,129)(52,211,130)(53,212,131)(54,213,132)(55,195,142)(56,196,143)(57,197,144)(58,198,136)(59,190,137)(60,191,138)(61,192,139)(62,193,140)(63,194,141)(64,223,151)(65,224,152)(66,225,153)(67,217,145)(68,218,146)(69,219,147)(70,220,148)(71,221,149)(72,222,150)(73,232,160)(74,233,161)(75,234,162)(76,226,154)(77,227,155)(78,228,156)(79,229,157)(80,230,158)(81,231,159)(82,243,169)(83,235,170)(84,236,171)(85,237,163)(86,238,164)(87,239,165)(88,240,166)(89,241,167)(90,242,168), (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), (1,242,233,224,56,47,38,31,22)(2,82,162,225,144,216,39,110,182)(3,170,76,217,198,127,40,13,102)(4,236,227,218,59,50,41,34,25)(5,85,156,219,138,210,42,113,185)(6,164,79,220,192,130,43,16,105)(7,239,230,221,62,53,44,28,19)(8,88,159,222,141,213,45,116,188)(9,167,73,223,195,133,37,10,108)(11,181,172,168,161,152,196,215,206)(12,23,92,169,234,66,197,48,126)(14,184,175,171,155,146,190,209,200)(15,26,95,163,228,69,191,51,120)(17,187,178,165,158,149,193,212,203)(18,20,98,166,231,72,194,54,123)(21,180,89,232,151,142,46,205,117)(24,174,83,226,145,136,49,199,111)(27,177,86,229,148,139,52,202,114)(29,107,179,240,81,150,63,132,204)(30,189,99,241,160,64,55,214,124)(32,101,173,243,75,153,57,135,207)(33,183,93,235,154,67,58,208,118)(35,104,176,237,78,147,60,129,201)(36,186,96,238,157,70,61,211,121)(65,143,134,125,109,100,91,90,74)(68,137,128,119,112,103,94,84,77)(71,140,131,122,115,106,97,87,80)>;
G:=Group( (1,172,91)(2,173,92)(3,174,93)(4,175,94)(5,176,95)(6,177,96)(7,178,97)(8,179,98)(9,180,99)(10,117,30)(11,109,31)(12,110,32)(13,111,33)(14,112,34)(15,113,35)(16,114,36)(17,115,28)(18,116,29)(19,187,106)(20,188,107)(21,189,108)(22,181,100)(23,182,101)(24,183,102)(25,184,103)(26,185,104)(27,186,105)(37,205,124)(38,206,125)(39,207,126)(40,199,118)(41,200,119)(42,201,120)(43,202,121)(44,203,122)(45,204,123)(46,214,133)(47,215,134)(48,216,135)(49,208,127)(50,209,128)(51,210,129)(52,211,130)(53,212,131)(54,213,132)(55,195,142)(56,196,143)(57,197,144)(58,198,136)(59,190,137)(60,191,138)(61,192,139)(62,193,140)(63,194,141)(64,223,151)(65,224,152)(66,225,153)(67,217,145)(68,218,146)(69,219,147)(70,220,148)(71,221,149)(72,222,150)(73,232,160)(74,233,161)(75,234,162)(76,226,154)(77,227,155)(78,228,156)(79,229,157)(80,230,158)(81,231,159)(82,243,169)(83,235,170)(84,236,171)(85,237,163)(86,238,164)(87,239,165)(88,240,166)(89,241,167)(90,242,168), (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), (1,242,233,224,56,47,38,31,22)(2,82,162,225,144,216,39,110,182)(3,170,76,217,198,127,40,13,102)(4,236,227,218,59,50,41,34,25)(5,85,156,219,138,210,42,113,185)(6,164,79,220,192,130,43,16,105)(7,239,230,221,62,53,44,28,19)(8,88,159,222,141,213,45,116,188)(9,167,73,223,195,133,37,10,108)(11,181,172,168,161,152,196,215,206)(12,23,92,169,234,66,197,48,126)(14,184,175,171,155,146,190,209,200)(15,26,95,163,228,69,191,51,120)(17,187,178,165,158,149,193,212,203)(18,20,98,166,231,72,194,54,123)(21,180,89,232,151,142,46,205,117)(24,174,83,226,145,136,49,199,111)(27,177,86,229,148,139,52,202,114)(29,107,179,240,81,150,63,132,204)(30,189,99,241,160,64,55,214,124)(32,101,173,243,75,153,57,135,207)(33,183,93,235,154,67,58,208,118)(35,104,176,237,78,147,60,129,201)(36,186,96,238,157,70,61,211,121)(65,143,134,125,109,100,91,90,74)(68,137,128,119,112,103,94,84,77)(71,140,131,122,115,106,97,87,80) );
G=PermutationGroup([[(1,172,91),(2,173,92),(3,174,93),(4,175,94),(5,176,95),(6,177,96),(7,178,97),(8,179,98),(9,180,99),(10,117,30),(11,109,31),(12,110,32),(13,111,33),(14,112,34),(15,113,35),(16,114,36),(17,115,28),(18,116,29),(19,187,106),(20,188,107),(21,189,108),(22,181,100),(23,182,101),(24,183,102),(25,184,103),(26,185,104),(27,186,105),(37,205,124),(38,206,125),(39,207,126),(40,199,118),(41,200,119),(42,201,120),(43,202,121),(44,203,122),(45,204,123),(46,214,133),(47,215,134),(48,216,135),(49,208,127),(50,209,128),(51,210,129),(52,211,130),(53,212,131),(54,213,132),(55,195,142),(56,196,143),(57,197,144),(58,198,136),(59,190,137),(60,191,138),(61,192,139),(62,193,140),(63,194,141),(64,223,151),(65,224,152),(66,225,153),(67,217,145),(68,218,146),(69,219,147),(70,220,148),(71,221,149),(72,222,150),(73,232,160),(74,233,161),(75,234,162),(76,226,154),(77,227,155),(78,228,156),(79,229,157),(80,230,158),(81,231,159),(82,243,169),(83,235,170),(84,236,171),(85,237,163),(86,238,164),(87,239,165),(88,240,166),(89,241,167),(90,242,168)], [(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)], [(1,242,233,224,56,47,38,31,22),(2,82,162,225,144,216,39,110,182),(3,170,76,217,198,127,40,13,102),(4,236,227,218,59,50,41,34,25),(5,85,156,219,138,210,42,113,185),(6,164,79,220,192,130,43,16,105),(7,239,230,221,62,53,44,28,19),(8,88,159,222,141,213,45,116,188),(9,167,73,223,195,133,37,10,108),(11,181,172,168,161,152,196,215,206),(12,23,92,169,234,66,197,48,126),(14,184,175,171,155,146,190,209,200),(15,26,95,163,228,69,191,51,120),(17,187,178,165,158,149,193,212,203),(18,20,98,166,231,72,194,54,123),(21,180,89,232,151,142,46,205,117),(24,174,83,226,145,136,49,199,111),(27,177,86,229,148,139,52,202,114),(29,107,179,240,81,150,63,132,204),(30,189,99,241,160,64,55,214,124),(32,101,173,243,75,153,57,135,207),(33,183,93,235,154,67,58,208,118),(35,104,176,237,78,147,60,129,201),(36,186,96,238,157,70,61,211,121),(65,143,134,125,109,100,91,90,74),(68,137,128,119,112,103,94,84,77),(71,140,131,122,115,106,97,87,80)]])
C3.C92 is a maximal subgroup of
C9⋊S3⋊C9 C3.2(C9⋊D9)
99 conjugacy classes
| class | 1 | 3A | ··· | 3Z | 9A | ··· | 9BT |
| order | 1 | 3 | ··· | 3 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 |
99 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C9 | He3 | 3- 1+2 |
| kernel | C3.C92 | C32×C9 | C3×C9 | C32 | C32 |
| # reps | 1 | 8 | 72 | 2 | 16 |
Matrix representation of C3.C92 ►in GL5(𝔽19)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 7 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 16 | 9 |
| 0 | 0 | 4 | 6 | 5 |
| 0 | 0 | 17 | 6 | 9 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[9,0,0,0,0,0,4,0,0,0,0,0,4,4,17,0,0,16,6,6,0,0,9,5,9],[5,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;
C3.C92 in GAP, Magma, Sage, TeX
C_3.C_9^2
% in TeX
G:=Group("C3.C9^2"); // GroupNames label
G:=SmallGroup(243,2);
// by ID
G=gap.SmallGroup(243,2);
# by ID
G:=PCGroup([5,-3,3,-3,3,3,135,121,276]);
// Polycyclic
G:=Group<a,b,c|a^3=b^9=c^9=1,c*b*c^-1=a*b=b*a,a*c=c*a>;
// generators/relations