direct product, p-group, abelian, monomial
Aliases: C33×C9, SmallGroup(243,61)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C33×C9 |
| C1 — C33×C9 |
| C1 — C33×C9 |
Generators and relations for C33×C9
G = < a,b,c,d | a3=b3=c3=d9=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 396, all normal (4 characteristic)
C1, C3, C3, C9, C32, C3×C9, C33, C32×C9, C34, C33×C9
Quotients: C1, C3, C9, C32, C3×C9, C33, C32×C9, C34, C33×C9
►Regular action on 243 points
Generators in S243
(1 152 161)(2 153 162)(3 145 154)(4 146 155)(5 147 156)(6 148 157)(7 149 158)(8 150 159)(9 151 160)(10 70 79)(11 71 80)(12 72 81)(13 64 73)(14 65 74)(15 66 75)(16 67 76)(17 68 77)(18 69 78)(19 55 29)(20 56 30)(21 57 31)(22 58 32)(23 59 33)(24 60 34)(25 61 35)(26 62 36)(27 63 28)(37 46 139)(38 47 140)(39 48 141)(40 49 142)(41 50 143)(42 51 144)(43 52 136)(44 53 137)(45 54 138)(82 208 207)(83 209 199)(84 210 200)(85 211 201)(86 212 202)(87 213 203)(88 214 204)(89 215 205)(90 216 206)(91 100 193)(92 101 194)(93 102 195)(94 103 196)(95 104 197)(96 105 198)(97 106 190)(98 107 191)(99 108 192)(109 118 127)(110 119 128)(111 120 129)(112 121 130)(113 122 131)(114 123 132)(115 124 133)(116 125 134)(117 126 135)(163 172 184)(164 173 185)(165 174 186)(166 175 187)(167 176 188)(168 177 189)(169 178 181)(170 179 182)(171 180 183)(217 226 235)(218 227 236)(219 228 237)(220 229 238)(221 230 239)(222 231 240)(223 232 241)(224 233 242)(225 234 243)
(1 205 25)(2 206 26)(3 207 27)(4 199 19)(5 200 20)(6 201 21)(7 202 22)(8 203 23)(9 204 24)(10 134 169)(11 135 170)(12 127 171)(13 128 163)(14 129 164)(15 130 165)(16 131 166)(17 132 167)(18 133 168)(28 154 208)(29 155 209)(30 156 210)(31 157 211)(32 158 212)(33 159 213)(34 160 214)(35 161 215)(36 162 216)(37 91 217)(38 92 218)(39 93 219)(40 94 220)(41 95 221)(42 96 222)(43 97 223)(44 98 224)(45 99 225)(46 100 226)(47 101 227)(48 102 228)(49 103 229)(50 104 230)(51 105 231)(52 106 232)(53 107 233)(54 108 234)(55 146 83)(56 147 84)(57 148 85)(58 149 86)(59 150 87)(60 151 88)(61 152 89)(62 153 90)(63 145 82)(64 110 172)(65 111 173)(66 112 174)(67 113 175)(68 114 176)(69 115 177)(70 116 178)(71 117 179)(72 109 180)(73 119 184)(74 120 185)(75 121 186)(76 122 187)(77 123 188)(78 124 189)(79 125 181)(80 126 182)(81 118 183)(136 190 241)(137 191 242)(138 192 243)(139 193 235)(140 194 236)(141 195 237)(142 196 238)(143 197 239)(144 198 240)
(1 70 233)(2 71 234)(3 72 226)(4 64 227)(5 65 228)(6 66 229)(7 67 230)(8 68 231)(9 69 232)(10 224 161)(11 225 162)(12 217 154)(13 218 155)(14 219 156)(15 220 157)(16 221 158)(17 222 159)(18 223 160)(19 172 101)(20 173 102)(21 174 103)(22 175 104)(23 176 105)(24 177 106)(25 178 107)(26 179 108)(27 180 100)(28 171 91)(29 163 92)(30 164 93)(31 165 94)(32 166 95)(33 167 96)(34 168 97)(35 169 98)(36 170 99)(37 208 127)(38 209 128)(39 210 129)(40 211 130)(41 212 131)(42 213 132)(43 214 133)(44 215 134)(45 216 135)(46 207 109)(47 199 110)(48 200 111)(49 201 112)(50 202 113)(51 203 114)(52 204 115)(53 205 116)(54 206 117)(55 184 194)(56 185 195)(57 186 196)(58 187 197)(59 188 198)(60 189 190)(61 181 191)(62 182 192)(63 183 193)(73 236 146)(74 237 147)(75 238 148)(76 239 149)(77 240 150)(78 241 151)(79 242 152)(80 243 153)(81 235 145)(82 118 139)(83 119 140)(84 120 141)(85 121 142)(86 122 143)(87 123 144)(88 124 136)(89 125 137)(90 126 138)
(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)| (1,152,161)(2,153,162)(3,145,154)(4,146,155)(5,147,156)(6,148,157)(7,149,158)(8,150,159)(9,151,160)(10,70,79)(11,71,80)(12,72,81)(13,64,73)(14,65,74)(15,66,75)(16,67,76)(17,68,77)(18,69,78)(19,55,29)(20,56,30)(21,57,31)(22,58,32)(23,59,33)(24,60,34)(25,61,35)(26,62,36)(27,63,28)(37,46,139)(38,47,140)(39,48,141)(40,49,142)(41,50,143)(42,51,144)(43,52,136)(44,53,137)(45,54,138)(82,208,207)(83,209,199)(84,210,200)(85,211,201)(86,212,202)(87,213,203)(88,214,204)(89,215,205)(90,216,206)(91,100,193)(92,101,194)(93,102,195)(94,103,196)(95,104,197)(96,105,198)(97,106,190)(98,107,191)(99,108,192)(109,118,127)(110,119,128)(111,120,129)(112,121,130)(113,122,131)(114,123,132)(115,124,133)(116,125,134)(117,126,135)(163,172,184)(164,173,185)(165,174,186)(166,175,187)(167,176,188)(168,177,189)(169,178,181)(170,179,182)(171,180,183)(217,226,235)(218,227,236)(219,228,237)(220,229,238)(221,230,239)(222,231,240)(223,232,241)(224,233,242)(225,234,243), (1,205,25)(2,206,26)(3,207,27)(4,199,19)(5,200,20)(6,201,21)(7,202,22)(8,203,23)(9,204,24)(10,134,169)(11,135,170)(12,127,171)(13,128,163)(14,129,164)(15,130,165)(16,131,166)(17,132,167)(18,133,168)(28,154,208)(29,155,209)(30,156,210)(31,157,211)(32,158,212)(33,159,213)(34,160,214)(35,161,215)(36,162,216)(37,91,217)(38,92,218)(39,93,219)(40,94,220)(41,95,221)(42,96,222)(43,97,223)(44,98,224)(45,99,225)(46,100,226)(47,101,227)(48,102,228)(49,103,229)(50,104,230)(51,105,231)(52,106,232)(53,107,233)(54,108,234)(55,146,83)(56,147,84)(57,148,85)(58,149,86)(59,150,87)(60,151,88)(61,152,89)(62,153,90)(63,145,82)(64,110,172)(65,111,173)(66,112,174)(67,113,175)(68,114,176)(69,115,177)(70,116,178)(71,117,179)(72,109,180)(73,119,184)(74,120,185)(75,121,186)(76,122,187)(77,123,188)(78,124,189)(79,125,181)(80,126,182)(81,118,183)(136,190,241)(137,191,242)(138,192,243)(139,193,235)(140,194,236)(141,195,237)(142,196,238)(143,197,239)(144,198,240), (1,70,233)(2,71,234)(3,72,226)(4,64,227)(5,65,228)(6,66,229)(7,67,230)(8,68,231)(9,69,232)(10,224,161)(11,225,162)(12,217,154)(13,218,155)(14,219,156)(15,220,157)(16,221,158)(17,222,159)(18,223,160)(19,172,101)(20,173,102)(21,174,103)(22,175,104)(23,176,105)(24,177,106)(25,178,107)(26,179,108)(27,180,100)(28,171,91)(29,163,92)(30,164,93)(31,165,94)(32,166,95)(33,167,96)(34,168,97)(35,169,98)(36,170,99)(37,208,127)(38,209,128)(39,210,129)(40,211,130)(41,212,131)(42,213,132)(43,214,133)(44,215,134)(45,216,135)(46,207,109)(47,199,110)(48,200,111)(49,201,112)(50,202,113)(51,203,114)(52,204,115)(53,205,116)(54,206,117)(55,184,194)(56,185,195)(57,186,196)(58,187,197)(59,188,198)(60,189,190)(61,181,191)(62,182,192)(63,183,193)(73,236,146)(74,237,147)(75,238,148)(76,239,149)(77,240,150)(78,241,151)(79,242,152)(80,243,153)(81,235,145)(82,118,139)(83,119,140)(84,120,141)(85,121,142)(86,122,143)(87,123,144)(88,124,136)(89,125,137)(90,126,138), (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( (1,152,161)(2,153,162)(3,145,154)(4,146,155)(5,147,156)(6,148,157)(7,149,158)(8,150,159)(9,151,160)(10,70,79)(11,71,80)(12,72,81)(13,64,73)(14,65,74)(15,66,75)(16,67,76)(17,68,77)(18,69,78)(19,55,29)(20,56,30)(21,57,31)(22,58,32)(23,59,33)(24,60,34)(25,61,35)(26,62,36)(27,63,28)(37,46,139)(38,47,140)(39,48,141)(40,49,142)(41,50,143)(42,51,144)(43,52,136)(44,53,137)(45,54,138)(82,208,207)(83,209,199)(84,210,200)(85,211,201)(86,212,202)(87,213,203)(88,214,204)(89,215,205)(90,216,206)(91,100,193)(92,101,194)(93,102,195)(94,103,196)(95,104,197)(96,105,198)(97,106,190)(98,107,191)(99,108,192)(109,118,127)(110,119,128)(111,120,129)(112,121,130)(113,122,131)(114,123,132)(115,124,133)(116,125,134)(117,126,135)(163,172,184)(164,173,185)(165,174,186)(166,175,187)(167,176,188)(168,177,189)(169,178,181)(170,179,182)(171,180,183)(217,226,235)(218,227,236)(219,228,237)(220,229,238)(221,230,239)(222,231,240)(223,232,241)(224,233,242)(225,234,243), (1,205,25)(2,206,26)(3,207,27)(4,199,19)(5,200,20)(6,201,21)(7,202,22)(8,203,23)(9,204,24)(10,134,169)(11,135,170)(12,127,171)(13,128,163)(14,129,164)(15,130,165)(16,131,166)(17,132,167)(18,133,168)(28,154,208)(29,155,209)(30,156,210)(31,157,211)(32,158,212)(33,159,213)(34,160,214)(35,161,215)(36,162,216)(37,91,217)(38,92,218)(39,93,219)(40,94,220)(41,95,221)(42,96,222)(43,97,223)(44,98,224)(45,99,225)(46,100,226)(47,101,227)(48,102,228)(49,103,229)(50,104,230)(51,105,231)(52,106,232)(53,107,233)(54,108,234)(55,146,83)(56,147,84)(57,148,85)(58,149,86)(59,150,87)(60,151,88)(61,152,89)(62,153,90)(63,145,82)(64,110,172)(65,111,173)(66,112,174)(67,113,175)(68,114,176)(69,115,177)(70,116,178)(71,117,179)(72,109,180)(73,119,184)(74,120,185)(75,121,186)(76,122,187)(77,123,188)(78,124,189)(79,125,181)(80,126,182)(81,118,183)(136,190,241)(137,191,242)(138,192,243)(139,193,235)(140,194,236)(141,195,237)(142,196,238)(143,197,239)(144,198,240), (1,70,233)(2,71,234)(3,72,226)(4,64,227)(5,65,228)(6,66,229)(7,67,230)(8,68,231)(9,69,232)(10,224,161)(11,225,162)(12,217,154)(13,218,155)(14,219,156)(15,220,157)(16,221,158)(17,222,159)(18,223,160)(19,172,101)(20,173,102)(21,174,103)(22,175,104)(23,176,105)(24,177,106)(25,178,107)(26,179,108)(27,180,100)(28,171,91)(29,163,92)(30,164,93)(31,165,94)(32,166,95)(33,167,96)(34,168,97)(35,169,98)(36,170,99)(37,208,127)(38,209,128)(39,210,129)(40,211,130)(41,212,131)(42,213,132)(43,214,133)(44,215,134)(45,216,135)(46,207,109)(47,199,110)(48,200,111)(49,201,112)(50,202,113)(51,203,114)(52,204,115)(53,205,116)(54,206,117)(55,184,194)(56,185,195)(57,186,196)(58,187,197)(59,188,198)(60,189,190)(61,181,191)(62,182,192)(63,183,193)(73,236,146)(74,237,147)(75,238,148)(76,239,149)(77,240,150)(78,241,151)(79,242,152)(80,243,153)(81,235,145)(82,118,139)(83,119,140)(84,120,141)(85,121,142)(86,122,143)(87,123,144)(88,124,136)(89,125,137)(90,126,138), (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([[(1,152,161),(2,153,162),(3,145,154),(4,146,155),(5,147,156),(6,148,157),(7,149,158),(8,150,159),(9,151,160),(10,70,79),(11,71,80),(12,72,81),(13,64,73),(14,65,74),(15,66,75),(16,67,76),(17,68,77),(18,69,78),(19,55,29),(20,56,30),(21,57,31),(22,58,32),(23,59,33),(24,60,34),(25,61,35),(26,62,36),(27,63,28),(37,46,139),(38,47,140),(39,48,141),(40,49,142),(41,50,143),(42,51,144),(43,52,136),(44,53,137),(45,54,138),(82,208,207),(83,209,199),(84,210,200),(85,211,201),(86,212,202),(87,213,203),(88,214,204),(89,215,205),(90,216,206),(91,100,193),(92,101,194),(93,102,195),(94,103,196),(95,104,197),(96,105,198),(97,106,190),(98,107,191),(99,108,192),(109,118,127),(110,119,128),(111,120,129),(112,121,130),(113,122,131),(114,123,132),(115,124,133),(116,125,134),(117,126,135),(163,172,184),(164,173,185),(165,174,186),(166,175,187),(167,176,188),(168,177,189),(169,178,181),(170,179,182),(171,180,183),(217,226,235),(218,227,236),(219,228,237),(220,229,238),(221,230,239),(222,231,240),(223,232,241),(224,233,242),(225,234,243)], [(1,205,25),(2,206,26),(3,207,27),(4,199,19),(5,200,20),(6,201,21),(7,202,22),(8,203,23),(9,204,24),(10,134,169),(11,135,170),(12,127,171),(13,128,163),(14,129,164),(15,130,165),(16,131,166),(17,132,167),(18,133,168),(28,154,208),(29,155,209),(30,156,210),(31,157,211),(32,158,212),(33,159,213),(34,160,214),(35,161,215),(36,162,216),(37,91,217),(38,92,218),(39,93,219),(40,94,220),(41,95,221),(42,96,222),(43,97,223),(44,98,224),(45,99,225),(46,100,226),(47,101,227),(48,102,228),(49,103,229),(50,104,230),(51,105,231),(52,106,232),(53,107,233),(54,108,234),(55,146,83),(56,147,84),(57,148,85),(58,149,86),(59,150,87),(60,151,88),(61,152,89),(62,153,90),(63,145,82),(64,110,172),(65,111,173),(66,112,174),(67,113,175),(68,114,176),(69,115,177),(70,116,178),(71,117,179),(72,109,180),(73,119,184),(74,120,185),(75,121,186),(76,122,187),(77,123,188),(78,124,189),(79,125,181),(80,126,182),(81,118,183),(136,190,241),(137,191,242),(138,192,243),(139,193,235),(140,194,236),(141,195,237),(142,196,238),(143,197,239),(144,198,240)], [(1,70,233),(2,71,234),(3,72,226),(4,64,227),(5,65,228),(6,66,229),(7,67,230),(8,68,231),(9,69,232),(10,224,161),(11,225,162),(12,217,154),(13,218,155),(14,219,156),(15,220,157),(16,221,158),(17,222,159),(18,223,160),(19,172,101),(20,173,102),(21,174,103),(22,175,104),(23,176,105),(24,177,106),(25,178,107),(26,179,108),(27,180,100),(28,171,91),(29,163,92),(30,164,93),(31,165,94),(32,166,95),(33,167,96),(34,168,97),(35,169,98),(36,170,99),(37,208,127),(38,209,128),(39,210,129),(40,211,130),(41,212,131),(42,213,132),(43,214,133),(44,215,134),(45,216,135),(46,207,109),(47,199,110),(48,200,111),(49,201,112),(50,202,113),(51,203,114),(52,204,115),(53,205,116),(54,206,117),(55,184,194),(56,185,195),(57,186,196),(58,187,197),(59,188,198),(60,189,190),(61,181,191),(62,182,192),(63,183,193),(73,236,146),(74,237,147),(75,238,148),(76,239,149),(77,240,150),(78,241,151),(79,242,152),(80,243,153),(81,235,145),(82,118,139),(83,119,140),(84,120,141),(85,121,142),(86,122,143),(87,123,144),(88,124,136),(89,125,137),(90,126,138)], [(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)]])
C33×C9 is a maximal subgroup of
C33⋊9D9
243 conjugacy classes
| class | 1 | 3A | ··· | 3CB | 9A | ··· | 9FF |
| 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 | C33×C9 | C32×C9 | C34 | C33 |
| # reps | 1 | 78 | 2 | 162 |
Matrix representation of C33×C9 ►in GL4(𝔽19) generated by
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 7 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 11 |
G:=sub<GL(4,GF(19))| [11,0,0,0,0,11,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,1,0,0,0,0,1,0,0,0,0,11],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[7,0,0,0,0,4,0,0,0,0,6,0,0,0,0,11] >;
C33×C9 in GAP, Magma, Sage, TeX
C_3^3\times C_9
% in TeX
G:=Group("C3^3xC9"); // GroupNames label
G:=SmallGroup(243,61);
// by ID
G=gap.SmallGroup(243,61);
# by ID
G:=PCGroup([5,-3,3,3,3,-3,405]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^9=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations