Copied to
clipboard

G = C33×C9order 243 = 35

Abelian group of type [3,3,3,9]

direct product, p-group, abelian, monomial

Aliases: C33×C9, SmallGroup(243,61)

Series: Derived Chief Lower central Upper central Jennings

C1 — C33×C9
C1C3C32C33C34 — C33×C9
C1 — C33×C9
C1 — C33×C9
C1C3C3 — 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 [×39], C9 [×27], C32 [×130], C3×C9 [×117], C33 [×40], C32×C9 [×39], C34, C33×C9
Quotients: C1, C3 [×40], C9 [×27], C32 [×130], C3×C9 [×117], C33 [×40], C32×C9 [×39], C34, C33×C9

Smallest permutation representation of 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 67 79)(11 68 80)(12 69 81)(13 70 73)(14 71 74)(15 72 75)(16 64 76)(17 65 77)(18 66 78)(19 181 29)(20 182 30)(21 183 31)(22 184 32)(23 185 33)(24 186 34)(25 187 35)(26 188 36)(27 189 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)(55 178 165)(56 179 166)(57 180 167)(58 172 168)(59 173 169)(60 174 170)(61 175 171)(62 176 163)(63 177 164)(82 208 202)(83 209 203)(84 210 204)(85 211 205)(86 212 206)(87 213 207)(88 214 199)(89 215 200)(90 216 201)(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)(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)
(1 200 25)(2 201 26)(3 202 27)(4 203 19)(5 204 20)(6 205 21)(7 206 22)(8 207 23)(9 199 24)(10 131 61)(11 132 62)(12 133 63)(13 134 55)(14 135 56)(15 127 57)(16 128 58)(17 129 59)(18 130 60)(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)(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)(73 125 165)(74 126 166)(75 118 167)(76 119 168)(77 120 169)(78 121 170)(79 122 171)(80 123 163)(81 124 164)(82 189 145)(83 181 146)(84 182 147)(85 183 148)(86 184 149)(87 185 150)(88 186 151)(89 187 152)(90 188 153)(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)(10 221 158)(11 222 159)(12 223 160)(13 224 161)(14 225 162)(15 217 154)(16 218 155)(17 219 156)(18 220 157)(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 57 91)(29 58 92)(30 59 93)(31 60 94)(32 61 95)(33 62 96)(34 63 97)(35 55 98)(36 56 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)(46 202 109)(47 203 110)(48 204 111)(49 205 112)(50 206 113)(51 207 114)(52 199 115)(53 200 116)(54 201 117)(73 242 152)(74 243 153)(75 235 145)(76 236 146)(77 237 147)(78 238 148)(79 239 149)(80 240 150)(81 241 151)(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)(163 198 185)(164 190 186)(165 191 187)(166 192 188)(167 193 189)(168 194 181)(169 195 182)(170 196 183)(171 197 184)
(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)(10,67,79)(11,68,80)(12,69,81)(13,70,73)(14,71,74)(15,72,75)(16,64,76)(17,65,77)(18,66,78)(19,181,29)(20,182,30)(21,183,31)(22,184,32)(23,185,33)(24,186,34)(25,187,35)(26,188,36)(27,189,28)(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)(55,178,165)(56,179,166)(57,180,167)(58,172,168)(59,173,169)(60,174,170)(61,175,171)(62,176,163)(63,177,164)(82,208,202)(83,209,203)(84,210,204)(85,211,205)(86,212,206)(87,213,207)(88,214,199)(89,215,200)(90,216,201)(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)(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), (1,200,25)(2,201,26)(3,202,27)(4,203,19)(5,204,20)(6,205,21)(7,206,22)(8,207,23)(9,199,24)(10,131,61)(11,132,62)(12,133,63)(13,134,55)(14,135,56)(15,127,57)(16,128,58)(17,129,59)(18,130,60)(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)(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)(73,125,165)(74,126,166)(75,118,167)(76,119,168)(77,120,169)(78,121,170)(79,122,171)(80,123,163)(81,124,164)(82,189,145)(83,181,146)(84,182,147)(85,183,148)(86,184,149)(87,185,150)(88,186,151)(89,187,152)(90,188,153)(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)(10,221,158)(11,222,159)(12,223,160)(13,224,161)(14,225,162)(15,217,154)(16,218,155)(17,219,156)(18,220,157)(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)(28,57,91)(29,58,92)(30,59,93)(31,60,94)(32,61,95)(33,62,96)(34,63,97)(35,55,98)(36,56,99)(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)(46,202,109)(47,203,110)(48,204,111)(49,205,112)(50,206,113)(51,207,114)(52,199,115)(53,200,116)(54,201,117)(73,242,152)(74,243,153)(75,235,145)(76,236,146)(77,237,147)(78,238,148)(79,239,149)(80,240,150)(81,241,151)(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)(163,198,185)(164,190,186)(165,191,187)(166,192,188)(167,193,189)(168,194,181)(169,195,182)(170,196,183)(171,197,184), (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)(10,67,79)(11,68,80)(12,69,81)(13,70,73)(14,71,74)(15,72,75)(16,64,76)(17,65,77)(18,66,78)(19,181,29)(20,182,30)(21,183,31)(22,184,32)(23,185,33)(24,186,34)(25,187,35)(26,188,36)(27,189,28)(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)(55,178,165)(56,179,166)(57,180,167)(58,172,168)(59,173,169)(60,174,170)(61,175,171)(62,176,163)(63,177,164)(82,208,202)(83,209,203)(84,210,204)(85,211,205)(86,212,206)(87,213,207)(88,214,199)(89,215,200)(90,216,201)(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)(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), (1,200,25)(2,201,26)(3,202,27)(4,203,19)(5,204,20)(6,205,21)(7,206,22)(8,207,23)(9,199,24)(10,131,61)(11,132,62)(12,133,63)(13,134,55)(14,135,56)(15,127,57)(16,128,58)(17,129,59)(18,130,60)(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)(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)(73,125,165)(74,126,166)(75,118,167)(76,119,168)(77,120,169)(78,121,170)(79,122,171)(80,123,163)(81,124,164)(82,189,145)(83,181,146)(84,182,147)(85,183,148)(86,184,149)(87,185,150)(88,186,151)(89,187,152)(90,188,153)(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)(10,221,158)(11,222,159)(12,223,160)(13,224,161)(14,225,162)(15,217,154)(16,218,155)(17,219,156)(18,220,157)(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)(28,57,91)(29,58,92)(30,59,93)(31,60,94)(32,61,95)(33,62,96)(34,63,97)(35,55,98)(36,56,99)(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)(46,202,109)(47,203,110)(48,204,111)(49,205,112)(50,206,113)(51,207,114)(52,199,115)(53,200,116)(54,201,117)(73,242,152)(74,243,153)(75,235,145)(76,236,146)(77,237,147)(78,238,148)(79,239,149)(80,240,150)(81,241,151)(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)(163,198,185)(164,190,186)(165,191,187)(166,192,188)(167,193,189)(168,194,181)(169,195,182)(170,196,183)(171,197,184), (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),(10,67,79),(11,68,80),(12,69,81),(13,70,73),(14,71,74),(15,72,75),(16,64,76),(17,65,77),(18,66,78),(19,181,29),(20,182,30),(21,183,31),(22,184,32),(23,185,33),(24,186,34),(25,187,35),(26,188,36),(27,189,28),(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),(55,178,165),(56,179,166),(57,180,167),(58,172,168),(59,173,169),(60,174,170),(61,175,171),(62,176,163),(63,177,164),(82,208,202),(83,209,203),(84,210,204),(85,211,205),(86,212,206),(87,213,207),(88,214,199),(89,215,200),(90,216,201),(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),(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)], [(1,200,25),(2,201,26),(3,202,27),(4,203,19),(5,204,20),(6,205,21),(7,206,22),(8,207,23),(9,199,24),(10,131,61),(11,132,62),(12,133,63),(13,134,55),(14,135,56),(15,127,57),(16,128,58),(17,129,59),(18,130,60),(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),(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),(73,125,165),(74,126,166),(75,118,167),(76,119,168),(77,120,169),(78,121,170),(79,122,171),(80,123,163),(81,124,164),(82,189,145),(83,181,146),(84,182,147),(85,183,148),(86,184,149),(87,185,150),(88,186,151),(89,187,152),(90,188,153),(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),(10,221,158),(11,222,159),(12,223,160),(13,224,161),(14,225,162),(15,217,154),(16,218,155),(17,219,156),(18,220,157),(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),(28,57,91),(29,58,92),(30,59,93),(31,60,94),(32,61,95),(33,62,96),(34,63,97),(35,55,98),(36,56,99),(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),(46,202,109),(47,203,110),(48,204,111),(49,205,112),(50,206,113),(51,207,114),(52,199,115),(53,200,116),(54,201,117),(73,242,152),(74,243,153),(75,235,145),(76,236,146),(77,237,147),(78,238,148),(79,239,149),(80,240,150),(81,241,151),(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),(163,198,185),(164,190,186),(165,191,187),(166,192,188),(167,193,189),(168,194,181),(169,195,182),(170,196,183),(171,197,184)], [(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   C339D9

243 conjugacy classes

class 1 3A···3CB9A···9FF
order13···39···9
size11···11···1

243 irreducible representations

dim1111
type+
imageC1C3C3C9
kernelC33×C9C32×C9C34C33
# reps1782162

Matrix representation of C33×C9 in GL4(𝔽19) generated by

11000
01100
0010
0001
,
7000
0100
0010
00011
,
1000
01100
00110
00011
,
7000
0400
0060
00011
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

׿
×
𝔽