direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C6×D13, C78⋊6C6, C13⋊3C62, (C3×C78)⋊3C2, C26⋊3(C3×C6), C39⋊8(C2×C6), (C3×C39)⋊8C22, SmallGroup(468,50)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C3×C6×D13 |
Generators and relations for C3×C6×D13
G = < a,b,c,d | a3=b6=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 276 in 60 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C6, C6, C32, C2×C6, C13, C3×C6, C3×C6, D13, C26, C62, C39, D26, C3×D13, C78, C3×C39, C6×D13, C32×D13, C3×C78, C3×C6×D13
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, D13, C62, D26, C3×D13, C6×D13, C32×D13, C3×C6×D13
►On 234 points
Generators in S234
(1 91 49)(2 79 50)(3 80 51)(4 81 52)(5 82 40)(6 83 41)(7 84 42)(8 85 43)(9 86 44)(10 87 45)(11 88 46)(12 89 47)(13 90 48)(14 100 58)(15 101 59)(16 102 60)(17 103 61)(18 104 62)(19 92 63)(20 93 64)(21 94 65)(22 95 53)(23 96 54)(24 97 55)(25 98 56)(26 99 57)(27 116 78)(28 117 66)(29 105 67)(30 106 68)(31 107 69)(32 108 70)(33 109 71)(34 110 72)(35 111 73)(36 112 74)(37 113 75)(38 114 76)(39 115 77)(118 198 168)(119 199 169)(120 200 157)(121 201 158)(122 202 159)(123 203 160)(124 204 161)(125 205 162)(126 206 163)(127 207 164)(128 208 165)(129 196 166)(130 197 167)(131 213 179)(132 214 180)(133 215 181)(134 216 182)(135 217 170)(136 218 171)(137 219 172)(138 220 173)(139 221 174)(140 209 175)(141 210 176)(142 211 177)(143 212 178)(144 233 184)(145 234 185)(146 222 186)(147 223 187)(148 224 188)(149 225 189)(150 226 190)(151 227 191)(152 228 192)(153 229 193)(154 230 194)(155 231 195)(156 232 183)
(1 136 36 119 25 144)(2 137 37 120 26 145)(3 138 38 121 14 146)(4 139 39 122 15 147)(5 140 27 123 16 148)(6 141 28 124 17 149)(7 142 29 125 18 150)(8 143 30 126 19 151)(9 131 31 127 20 152)(10 132 32 128 21 153)(11 133 33 129 22 154)(12 134 34 130 23 155)(13 135 35 118 24 156)(40 175 78 160 60 188)(41 176 66 161 61 189)(42 177 67 162 62 190)(43 178 68 163 63 191)(44 179 69 164 64 192)(45 180 70 165 65 193)(46 181 71 166 53 194)(47 182 72 167 54 195)(48 170 73 168 55 183)(49 171 74 169 56 184)(50 172 75 157 57 185)(51 173 76 158 58 186)(52 174 77 159 59 187)(79 219 113 200 99 234)(80 220 114 201 100 222)(81 221 115 202 101 223)(82 209 116 203 102 224)(83 210 117 204 103 225)(84 211 105 205 104 226)(85 212 106 206 92 227)(86 213 107 207 93 228)(87 214 108 208 94 229)(88 215 109 196 95 230)(89 216 110 197 96 231)(90 217 111 198 97 232)(91 218 112 199 98 233)
(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)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 31)(28 30)(32 39)(33 38)(34 37)(35 36)(40 44)(41 43)(45 52)(46 51)(47 50)(48 49)(53 58)(54 57)(55 56)(59 65)(60 64)(61 63)(66 68)(69 78)(70 77)(71 76)(72 75)(73 74)(79 89)(80 88)(81 87)(82 86)(83 85)(90 91)(92 103)(93 102)(94 101)(95 100)(96 99)(97 98)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(118 119)(120 130)(121 129)(122 128)(123 127)(124 126)(131 140)(132 139)(133 138)(134 137)(135 136)(141 143)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)(157 167)(158 166)(159 165)(160 164)(161 163)(168 169)(170 171)(172 182)(173 181)(174 180)(175 179)(176 178)(183 184)(185 195)(186 194)(187 193)(188 192)(189 191)(196 201)(197 200)(198 199)(202 208)(203 207)(204 206)(209 213)(210 212)(214 221)(215 220)(216 219)(217 218)(222 230)(223 229)(224 228)(225 227)(231 234)(232 233)
G:=sub<Sym(234)| (1,91,49)(2,79,50)(3,80,51)(4,81,52)(5,82,40)(6,83,41)(7,84,42)(8,85,43)(9,86,44)(10,87,45)(11,88,46)(12,89,47)(13,90,48)(14,100,58)(15,101,59)(16,102,60)(17,103,61)(18,104,62)(19,92,63)(20,93,64)(21,94,65)(22,95,53)(23,96,54)(24,97,55)(25,98,56)(26,99,57)(27,116,78)(28,117,66)(29,105,67)(30,106,68)(31,107,69)(32,108,70)(33,109,71)(34,110,72)(35,111,73)(36,112,74)(37,113,75)(38,114,76)(39,115,77)(118,198,168)(119,199,169)(120,200,157)(121,201,158)(122,202,159)(123,203,160)(124,204,161)(125,205,162)(126,206,163)(127,207,164)(128,208,165)(129,196,166)(130,197,167)(131,213,179)(132,214,180)(133,215,181)(134,216,182)(135,217,170)(136,218,171)(137,219,172)(138,220,173)(139,221,174)(140,209,175)(141,210,176)(142,211,177)(143,212,178)(144,233,184)(145,234,185)(146,222,186)(147,223,187)(148,224,188)(149,225,189)(150,226,190)(151,227,191)(152,228,192)(153,229,193)(154,230,194)(155,231,195)(156,232,183), (1,136,36,119,25,144)(2,137,37,120,26,145)(3,138,38,121,14,146)(4,139,39,122,15,147)(5,140,27,123,16,148)(6,141,28,124,17,149)(7,142,29,125,18,150)(8,143,30,126,19,151)(9,131,31,127,20,152)(10,132,32,128,21,153)(11,133,33,129,22,154)(12,134,34,130,23,155)(13,135,35,118,24,156)(40,175,78,160,60,188)(41,176,66,161,61,189)(42,177,67,162,62,190)(43,178,68,163,63,191)(44,179,69,164,64,192)(45,180,70,165,65,193)(46,181,71,166,53,194)(47,182,72,167,54,195)(48,170,73,168,55,183)(49,171,74,169,56,184)(50,172,75,157,57,185)(51,173,76,158,58,186)(52,174,77,159,59,187)(79,219,113,200,99,234)(80,220,114,201,100,222)(81,221,115,202,101,223)(82,209,116,203,102,224)(83,210,117,204,103,225)(84,211,105,205,104,226)(85,212,106,206,92,227)(86,213,107,207,93,228)(87,214,108,208,94,229)(88,215,109,196,95,230)(89,216,110,197,96,231)(90,217,111,198,97,232)(91,218,112,199,98,233), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,31)(28,30)(32,39)(33,38)(34,37)(35,36)(40,44)(41,43)(45,52)(46,51)(47,50)(48,49)(53,58)(54,57)(55,56)(59,65)(60,64)(61,63)(66,68)(69,78)(70,77)(71,76)(72,75)(73,74)(79,89)(80,88)(81,87)(82,86)(83,85)(90,91)(92,103)(93,102)(94,101)(95,100)(96,99)(97,98)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(118,119)(120,130)(121,129)(122,128)(123,127)(124,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,143)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(157,167)(158,166)(159,165)(160,164)(161,163)(168,169)(170,171)(172,182)(173,181)(174,180)(175,179)(176,178)(183,184)(185,195)(186,194)(187,193)(188,192)(189,191)(196,201)(197,200)(198,199)(202,208)(203,207)(204,206)(209,213)(210,212)(214,221)(215,220)(216,219)(217,218)(222,230)(223,229)(224,228)(225,227)(231,234)(232,233)>;
G:=Group( (1,91,49)(2,79,50)(3,80,51)(4,81,52)(5,82,40)(6,83,41)(7,84,42)(8,85,43)(9,86,44)(10,87,45)(11,88,46)(12,89,47)(13,90,48)(14,100,58)(15,101,59)(16,102,60)(17,103,61)(18,104,62)(19,92,63)(20,93,64)(21,94,65)(22,95,53)(23,96,54)(24,97,55)(25,98,56)(26,99,57)(27,116,78)(28,117,66)(29,105,67)(30,106,68)(31,107,69)(32,108,70)(33,109,71)(34,110,72)(35,111,73)(36,112,74)(37,113,75)(38,114,76)(39,115,77)(118,198,168)(119,199,169)(120,200,157)(121,201,158)(122,202,159)(123,203,160)(124,204,161)(125,205,162)(126,206,163)(127,207,164)(128,208,165)(129,196,166)(130,197,167)(131,213,179)(132,214,180)(133,215,181)(134,216,182)(135,217,170)(136,218,171)(137,219,172)(138,220,173)(139,221,174)(140,209,175)(141,210,176)(142,211,177)(143,212,178)(144,233,184)(145,234,185)(146,222,186)(147,223,187)(148,224,188)(149,225,189)(150,226,190)(151,227,191)(152,228,192)(153,229,193)(154,230,194)(155,231,195)(156,232,183), (1,136,36,119,25,144)(2,137,37,120,26,145)(3,138,38,121,14,146)(4,139,39,122,15,147)(5,140,27,123,16,148)(6,141,28,124,17,149)(7,142,29,125,18,150)(8,143,30,126,19,151)(9,131,31,127,20,152)(10,132,32,128,21,153)(11,133,33,129,22,154)(12,134,34,130,23,155)(13,135,35,118,24,156)(40,175,78,160,60,188)(41,176,66,161,61,189)(42,177,67,162,62,190)(43,178,68,163,63,191)(44,179,69,164,64,192)(45,180,70,165,65,193)(46,181,71,166,53,194)(47,182,72,167,54,195)(48,170,73,168,55,183)(49,171,74,169,56,184)(50,172,75,157,57,185)(51,173,76,158,58,186)(52,174,77,159,59,187)(79,219,113,200,99,234)(80,220,114,201,100,222)(81,221,115,202,101,223)(82,209,116,203,102,224)(83,210,117,204,103,225)(84,211,105,205,104,226)(85,212,106,206,92,227)(86,213,107,207,93,228)(87,214,108,208,94,229)(88,215,109,196,95,230)(89,216,110,197,96,231)(90,217,111,198,97,232)(91,218,112,199,98,233), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,31)(28,30)(32,39)(33,38)(34,37)(35,36)(40,44)(41,43)(45,52)(46,51)(47,50)(48,49)(53,58)(54,57)(55,56)(59,65)(60,64)(61,63)(66,68)(69,78)(70,77)(71,76)(72,75)(73,74)(79,89)(80,88)(81,87)(82,86)(83,85)(90,91)(92,103)(93,102)(94,101)(95,100)(96,99)(97,98)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(118,119)(120,130)(121,129)(122,128)(123,127)(124,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,143)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(157,167)(158,166)(159,165)(160,164)(161,163)(168,169)(170,171)(172,182)(173,181)(174,180)(175,179)(176,178)(183,184)(185,195)(186,194)(187,193)(188,192)(189,191)(196,201)(197,200)(198,199)(202,208)(203,207)(204,206)(209,213)(210,212)(214,221)(215,220)(216,219)(217,218)(222,230)(223,229)(224,228)(225,227)(231,234)(232,233) );
G=PermutationGroup([[(1,91,49),(2,79,50),(3,80,51),(4,81,52),(5,82,40),(6,83,41),(7,84,42),(8,85,43),(9,86,44),(10,87,45),(11,88,46),(12,89,47),(13,90,48),(14,100,58),(15,101,59),(16,102,60),(17,103,61),(18,104,62),(19,92,63),(20,93,64),(21,94,65),(22,95,53),(23,96,54),(24,97,55),(25,98,56),(26,99,57),(27,116,78),(28,117,66),(29,105,67),(30,106,68),(31,107,69),(32,108,70),(33,109,71),(34,110,72),(35,111,73),(36,112,74),(37,113,75),(38,114,76),(39,115,77),(118,198,168),(119,199,169),(120,200,157),(121,201,158),(122,202,159),(123,203,160),(124,204,161),(125,205,162),(126,206,163),(127,207,164),(128,208,165),(129,196,166),(130,197,167),(131,213,179),(132,214,180),(133,215,181),(134,216,182),(135,217,170),(136,218,171),(137,219,172),(138,220,173),(139,221,174),(140,209,175),(141,210,176),(142,211,177),(143,212,178),(144,233,184),(145,234,185),(146,222,186),(147,223,187),(148,224,188),(149,225,189),(150,226,190),(151,227,191),(152,228,192),(153,229,193),(154,230,194),(155,231,195),(156,232,183)], [(1,136,36,119,25,144),(2,137,37,120,26,145),(3,138,38,121,14,146),(4,139,39,122,15,147),(5,140,27,123,16,148),(6,141,28,124,17,149),(7,142,29,125,18,150),(8,143,30,126,19,151),(9,131,31,127,20,152),(10,132,32,128,21,153),(11,133,33,129,22,154),(12,134,34,130,23,155),(13,135,35,118,24,156),(40,175,78,160,60,188),(41,176,66,161,61,189),(42,177,67,162,62,190),(43,178,68,163,63,191),(44,179,69,164,64,192),(45,180,70,165,65,193),(46,181,71,166,53,194),(47,182,72,167,54,195),(48,170,73,168,55,183),(49,171,74,169,56,184),(50,172,75,157,57,185),(51,173,76,158,58,186),(52,174,77,159,59,187),(79,219,113,200,99,234),(80,220,114,201,100,222),(81,221,115,202,101,223),(82,209,116,203,102,224),(83,210,117,204,103,225),(84,211,105,205,104,226),(85,212,106,206,92,227),(86,213,107,207,93,228),(87,214,108,208,94,229),(88,215,109,196,95,230),(89,216,110,197,96,231),(90,217,111,198,97,232),(91,218,112,199,98,233)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,22),(15,21),(16,20),(17,19),(23,26),(24,25),(27,31),(28,30),(32,39),(33,38),(34,37),(35,36),(40,44),(41,43),(45,52),(46,51),(47,50),(48,49),(53,58),(54,57),(55,56),(59,65),(60,64),(61,63),(66,68),(69,78),(70,77),(71,76),(72,75),(73,74),(79,89),(80,88),(81,87),(82,86),(83,85),(90,91),(92,103),(93,102),(94,101),(95,100),(96,99),(97,98),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112),(118,119),(120,130),(121,129),(122,128),(123,127),(124,126),(131,140),(132,139),(133,138),(134,137),(135,136),(141,143),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151),(157,167),(158,166),(159,165),(160,164),(161,163),(168,169),(170,171),(172,182),(173,181),(174,180),(175,179),(176,178),(183,184),(185,195),(186,194),(187,193),(188,192),(189,191),(196,201),(197,200),(198,199),(202,208),(203,207),(204,206),(209,213),(210,212),(214,221),(215,220),(216,219),(217,218),(222,230),(223,229),(224,228),(225,227),(231,234),(232,233)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 6A | ··· | 6H | 6I | ··· | 6X | 13A | ··· | 13F | 26A | ··· | 26F | 39A | ··· | 39AV | 78A | ··· | 78AV |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 13 | 13 | 1 | ··· | 1 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D13 | D26 | C3×D13 | C6×D13 |
| kernel | C3×C6×D13 | C32×D13 | C3×C78 | C6×D13 | C3×D13 | C78 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 2 | 1 | 8 | 16 | 8 | 6 | 6 | 48 | 48 |
Matrix representation of C3×C6×D13 ►in GL4(𝔽79) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 55 | 0 |
| 0 | 0 | 0 | 55 |
| 23 | 0 | 0 | 0 |
| 0 | 78 | 0 | 0 |
| 0 | 0 | 55 | 0 |
| 0 | 0 | 0 | 55 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 1 |
| 0 | 0 | 78 | 0 |
| 78 | 0 | 0 | 0 |
| 0 | 78 | 0 | 0 |
| 0 | 0 | 50 | 72 |
| 0 | 0 | 41 | 29 |
G:=sub<GL(4,GF(79))| [1,0,0,0,0,1,0,0,0,0,55,0,0,0,0,55],[23,0,0,0,0,78,0,0,0,0,55,0,0,0,0,55],[1,0,0,0,0,1,0,0,0,0,18,78,0,0,1,0],[78,0,0,0,0,78,0,0,0,0,50,41,0,0,72,29] >;
C3×C6×D13 in GAP, Magma, Sage, TeX
C_3\times C_6\times D_{13} % in TeX
G:=Group("C3xC6xD13"); // GroupNames label
G:=SmallGroup(468,50);
// by ID
G=gap.SmallGroup(468,50);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-13,10804]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^13=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations