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## G = C3×C6×D13order 468 = 22·32·13

### Direct product of C3×C6 and D13

Aliases: C3×C6×D13, C786C6, C133C62, (C3×C78)⋊3C2, C263(C3×C6), C398(C2×C6), (C3×C39)⋊8C22, SmallGroup(468,50)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C3×C6×D13
 Chief series C1 — C13 — C39 — C3×C39 — C32×D13 — C3×C6×D13
 Lower central C13 — C3×C6×D13
 Upper central C1 — C3×C6

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 [×2], C3 [×4], C22, C6 [×4], C6 [×8], C32, C2×C6 [×4], C13, C3×C6, C3×C6 [×2], D13 [×2], C26, C62, C39 [×4], D26, C3×D13 [×8], C78 [×4], C3×C39, C6×D13 [×4], C32×D13 [×2], C3×C78, C3×C6×D13
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], C32, C2×C6 [×4], C3×C6 [×3], D13, C62, D26, C3×D13 [×4], C6×D13 [×4], C32×D13, C3×C6×D13

Smallest permutation representation of C3×C6×D13
On 234 points
Generators in S234
(1 86 44)(2 87 45)(3 88 46)(4 89 47)(5 90 48)(6 91 49)(7 79 50)(8 80 51)(9 81 52)(10 82 40)(11 83 41)(12 84 42)(13 85 43)(14 93 58)(15 94 59)(16 95 60)(17 96 61)(18 97 62)(19 98 63)(20 99 64)(21 100 65)(22 101 53)(23 102 54)(24 103 55)(25 104 56)(26 92 57)(27 114 76)(28 115 77)(29 116 78)(30 117 66)(31 105 67)(32 106 68)(33 107 69)(34 108 70)(35 109 71)(36 110 72)(37 111 73)(38 112 74)(39 113 75)(118 203 159)(119 204 160)(120 205 161)(121 206 162)(122 207 163)(123 208 164)(124 196 165)(125 197 166)(126 198 167)(127 199 168)(128 200 169)(129 201 157)(130 202 158)(131 216 182)(132 217 170)(133 218 171)(134 219 172)(135 220 173)(136 221 174)(137 209 175)(138 210 176)(139 211 177)(140 212 178)(141 213 179)(142 214 180)(143 215 181)(144 226 186)(145 227 187)(146 228 188)(147 229 189)(148 230 190)(149 231 191)(150 232 192)(151 233 193)(152 234 194)(153 222 195)(154 223 183)(155 224 184)(156 225 185)
(1 143 34 118 21 148)(2 131 35 119 22 149)(3 132 36 120 23 150)(4 133 37 121 24 151)(5 134 38 122 25 152)(6 135 39 123 26 153)(7 136 27 124 14 154)(8 137 28 125 15 155)(9 138 29 126 16 156)(10 139 30 127 17 144)(11 140 31 128 18 145)(12 141 32 129 19 146)(13 142 33 130 20 147)(40 177 66 168 61 186)(41 178 67 169 62 187)(42 179 68 157 63 188)(43 180 69 158 64 189)(44 181 70 159 65 190)(45 182 71 160 53 191)(46 170 72 161 54 192)(47 171 73 162 55 193)(48 172 74 163 56 194)(49 173 75 164 57 195)(50 174 76 165 58 183)(51 175 77 166 59 184)(52 176 78 167 60 185)(79 221 114 196 93 223)(80 209 115 197 94 224)(81 210 116 198 95 225)(82 211 117 199 96 226)(83 212 105 200 97 227)(84 213 106 201 98 228)(85 214 107 202 99 229)(86 215 108 203 100 230)(87 216 109 204 101 231)(88 217 110 205 102 232)(89 218 111 206 103 233)(90 219 112 207 104 234)(91 220 113 208 92 222)
(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)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(40 47)(41 46)(42 45)(43 44)(48 52)(49 51)(53 63)(54 62)(55 61)(56 60)(57 59)(64 65)(66 73)(67 72)(68 71)(69 70)(74 78)(75 77)(80 91)(81 90)(82 89)(83 88)(84 87)(85 86)(92 94)(95 104)(96 103)(97 102)(98 101)(99 100)(105 110)(106 109)(107 108)(111 117)(112 116)(113 115)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)(131 141)(132 140)(133 139)(134 138)(135 137)(142 143)(144 151)(145 150)(146 149)(147 148)(152 156)(153 155)(157 160)(158 159)(161 169)(162 168)(163 167)(164 166)(170 178)(171 177)(172 176)(173 175)(179 182)(180 181)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)(197 208)(198 207)(199 206)(200 205)(201 204)(202 203)(209 220)(210 219)(211 218)(212 217)(213 216)(214 215)(222 224)(225 234)(226 233)(227 232)(228 231)(229 230)

G:=sub<Sym(234)| (1,86,44)(2,87,45)(3,88,46)(4,89,47)(5,90,48)(6,91,49)(7,79,50)(8,80,51)(9,81,52)(10,82,40)(11,83,41)(12,84,42)(13,85,43)(14,93,58)(15,94,59)(16,95,60)(17,96,61)(18,97,62)(19,98,63)(20,99,64)(21,100,65)(22,101,53)(23,102,54)(24,103,55)(25,104,56)(26,92,57)(27,114,76)(28,115,77)(29,116,78)(30,117,66)(31,105,67)(32,106,68)(33,107,69)(34,108,70)(35,109,71)(36,110,72)(37,111,73)(38,112,74)(39,113,75)(118,203,159)(119,204,160)(120,205,161)(121,206,162)(122,207,163)(123,208,164)(124,196,165)(125,197,166)(126,198,167)(127,199,168)(128,200,169)(129,201,157)(130,202,158)(131,216,182)(132,217,170)(133,218,171)(134,219,172)(135,220,173)(136,221,174)(137,209,175)(138,210,176)(139,211,177)(140,212,178)(141,213,179)(142,214,180)(143,215,181)(144,226,186)(145,227,187)(146,228,188)(147,229,189)(148,230,190)(149,231,191)(150,232,192)(151,233,193)(152,234,194)(153,222,195)(154,223,183)(155,224,184)(156,225,185), (1,143,34,118,21,148)(2,131,35,119,22,149)(3,132,36,120,23,150)(4,133,37,121,24,151)(5,134,38,122,25,152)(6,135,39,123,26,153)(7,136,27,124,14,154)(8,137,28,125,15,155)(9,138,29,126,16,156)(10,139,30,127,17,144)(11,140,31,128,18,145)(12,141,32,129,19,146)(13,142,33,130,20,147)(40,177,66,168,61,186)(41,178,67,169,62,187)(42,179,68,157,63,188)(43,180,69,158,64,189)(44,181,70,159,65,190)(45,182,71,160,53,191)(46,170,72,161,54,192)(47,171,73,162,55,193)(48,172,74,163,56,194)(49,173,75,164,57,195)(50,174,76,165,58,183)(51,175,77,166,59,184)(52,176,78,167,60,185)(79,221,114,196,93,223)(80,209,115,197,94,224)(81,210,116,198,95,225)(82,211,117,199,96,226)(83,212,105,200,97,227)(84,213,106,201,98,228)(85,214,107,202,99,229)(86,215,108,203,100,230)(87,216,109,204,101,231)(88,217,110,205,102,232)(89,218,111,206,103,233)(90,219,112,207,104,234)(91,220,113,208,92,222), (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)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(40,47)(41,46)(42,45)(43,44)(48,52)(49,51)(53,63)(54,62)(55,61)(56,60)(57,59)(64,65)(66,73)(67,72)(68,71)(69,70)(74,78)(75,77)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(92,94)(95,104)(96,103)(97,102)(98,101)(99,100)(105,110)(106,109)(107,108)(111,117)(112,116)(113,115)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(131,141)(132,140)(133,139)(134,138)(135,137)(142,143)(144,151)(145,150)(146,149)(147,148)(152,156)(153,155)(157,160)(158,159)(161,169)(162,168)(163,167)(164,166)(170,178)(171,177)(172,176)(173,175)(179,182)(180,181)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215)(222,224)(225,234)(226,233)(227,232)(228,231)(229,230)>;

G:=Group( (1,86,44)(2,87,45)(3,88,46)(4,89,47)(5,90,48)(6,91,49)(7,79,50)(8,80,51)(9,81,52)(10,82,40)(11,83,41)(12,84,42)(13,85,43)(14,93,58)(15,94,59)(16,95,60)(17,96,61)(18,97,62)(19,98,63)(20,99,64)(21,100,65)(22,101,53)(23,102,54)(24,103,55)(25,104,56)(26,92,57)(27,114,76)(28,115,77)(29,116,78)(30,117,66)(31,105,67)(32,106,68)(33,107,69)(34,108,70)(35,109,71)(36,110,72)(37,111,73)(38,112,74)(39,113,75)(118,203,159)(119,204,160)(120,205,161)(121,206,162)(122,207,163)(123,208,164)(124,196,165)(125,197,166)(126,198,167)(127,199,168)(128,200,169)(129,201,157)(130,202,158)(131,216,182)(132,217,170)(133,218,171)(134,219,172)(135,220,173)(136,221,174)(137,209,175)(138,210,176)(139,211,177)(140,212,178)(141,213,179)(142,214,180)(143,215,181)(144,226,186)(145,227,187)(146,228,188)(147,229,189)(148,230,190)(149,231,191)(150,232,192)(151,233,193)(152,234,194)(153,222,195)(154,223,183)(155,224,184)(156,225,185), (1,143,34,118,21,148)(2,131,35,119,22,149)(3,132,36,120,23,150)(4,133,37,121,24,151)(5,134,38,122,25,152)(6,135,39,123,26,153)(7,136,27,124,14,154)(8,137,28,125,15,155)(9,138,29,126,16,156)(10,139,30,127,17,144)(11,140,31,128,18,145)(12,141,32,129,19,146)(13,142,33,130,20,147)(40,177,66,168,61,186)(41,178,67,169,62,187)(42,179,68,157,63,188)(43,180,69,158,64,189)(44,181,70,159,65,190)(45,182,71,160,53,191)(46,170,72,161,54,192)(47,171,73,162,55,193)(48,172,74,163,56,194)(49,173,75,164,57,195)(50,174,76,165,58,183)(51,175,77,166,59,184)(52,176,78,167,60,185)(79,221,114,196,93,223)(80,209,115,197,94,224)(81,210,116,198,95,225)(82,211,117,199,96,226)(83,212,105,200,97,227)(84,213,106,201,98,228)(85,214,107,202,99,229)(86,215,108,203,100,230)(87,216,109,204,101,231)(88,217,110,205,102,232)(89,218,111,206,103,233)(90,219,112,207,104,234)(91,220,113,208,92,222), (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)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(40,47)(41,46)(42,45)(43,44)(48,52)(49,51)(53,63)(54,62)(55,61)(56,60)(57,59)(64,65)(66,73)(67,72)(68,71)(69,70)(74,78)(75,77)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(92,94)(95,104)(96,103)(97,102)(98,101)(99,100)(105,110)(106,109)(107,108)(111,117)(112,116)(113,115)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(131,141)(132,140)(133,139)(134,138)(135,137)(142,143)(144,151)(145,150)(146,149)(147,148)(152,156)(153,155)(157,160)(158,159)(161,169)(162,168)(163,167)(164,166)(170,178)(171,177)(172,176)(173,175)(179,182)(180,181)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215)(222,224)(225,234)(226,233)(227,232)(228,231)(229,230) );

G=PermutationGroup([(1,86,44),(2,87,45),(3,88,46),(4,89,47),(5,90,48),(6,91,49),(7,79,50),(8,80,51),(9,81,52),(10,82,40),(11,83,41),(12,84,42),(13,85,43),(14,93,58),(15,94,59),(16,95,60),(17,96,61),(18,97,62),(19,98,63),(20,99,64),(21,100,65),(22,101,53),(23,102,54),(24,103,55),(25,104,56),(26,92,57),(27,114,76),(28,115,77),(29,116,78),(30,117,66),(31,105,67),(32,106,68),(33,107,69),(34,108,70),(35,109,71),(36,110,72),(37,111,73),(38,112,74),(39,113,75),(118,203,159),(119,204,160),(120,205,161),(121,206,162),(122,207,163),(123,208,164),(124,196,165),(125,197,166),(126,198,167),(127,199,168),(128,200,169),(129,201,157),(130,202,158),(131,216,182),(132,217,170),(133,218,171),(134,219,172),(135,220,173),(136,221,174),(137,209,175),(138,210,176),(139,211,177),(140,212,178),(141,213,179),(142,214,180),(143,215,181),(144,226,186),(145,227,187),(146,228,188),(147,229,189),(148,230,190),(149,231,191),(150,232,192),(151,233,193),(152,234,194),(153,222,195),(154,223,183),(155,224,184),(156,225,185)], [(1,143,34,118,21,148),(2,131,35,119,22,149),(3,132,36,120,23,150),(4,133,37,121,24,151),(5,134,38,122,25,152),(6,135,39,123,26,153),(7,136,27,124,14,154),(8,137,28,125,15,155),(9,138,29,126,16,156),(10,139,30,127,17,144),(11,140,31,128,18,145),(12,141,32,129,19,146),(13,142,33,130,20,147),(40,177,66,168,61,186),(41,178,67,169,62,187),(42,179,68,157,63,188),(43,180,69,158,64,189),(44,181,70,159,65,190),(45,182,71,160,53,191),(46,170,72,161,54,192),(47,171,73,162,55,193),(48,172,74,163,56,194),(49,173,75,164,57,195),(50,174,76,165,58,183),(51,175,77,166,59,184),(52,176,78,167,60,185),(79,221,114,196,93,223),(80,209,115,197,94,224),(81,210,116,198,95,225),(82,211,117,199,96,226),(83,212,105,200,97,227),(84,213,106,201,98,228),(85,214,107,202,99,229),(86,215,108,203,100,230),(87,216,109,204,101,231),(88,217,110,205,102,232),(89,218,111,206,103,233),(90,219,112,207,104,234),(91,220,113,208,92,222)], [(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),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(40,47),(41,46),(42,45),(43,44),(48,52),(49,51),(53,63),(54,62),(55,61),(56,60),(57,59),(64,65),(66,73),(67,72),(68,71),(69,70),(74,78),(75,77),(80,91),(81,90),(82,89),(83,88),(84,87),(85,86),(92,94),(95,104),(96,103),(97,102),(98,101),(99,100),(105,110),(106,109),(107,108),(111,117),(112,116),(113,115),(118,130),(119,129),(120,128),(121,127),(122,126),(123,125),(131,141),(132,140),(133,139),(134,138),(135,137),(142,143),(144,151),(145,150),(146,149),(147,148),(152,156),(153,155),(157,160),(158,159),(161,169),(162,168),(163,167),(164,166),(170,178),(171,177),(172,176),(173,175),(179,182),(180,181),(184,195),(185,194),(186,193),(187,192),(188,191),(189,190),(197,208),(198,207),(199,206),(200,205),(201,204),(202,203),(209,220),(210,219),(211,218),(212,217),(213,216),(214,215),(222,224),(225,234),(226,233),(227,232),(228,231),(229,230)])

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

׿
×
𝔽