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G = C2×C3⋊D39order 468 = 22·32·13

Direct product of C2 and C3⋊D39

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2×C3⋊D39, C6⋊D39, C781S3, C32D78, C396D6, C326D26, C26⋊(C3⋊S3), (C3×C78)⋊1C2, (C3×C6)⋊2D13, (C3×C39)⋊6C22, C132(C2×C3⋊S3), SmallGroup(468,54)

Series: Derived Chief Lower central Upper central

C1C3×C39 — C2×C3⋊D39
C1C13C39C3×C39C3⋊D39 — C2×C3⋊D39
C3×C39 — C2×C3⋊D39
C1C2

Generators and relations for C2×C3⋊D39
 G = < a,b,c,d | a2=b3=c39=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 948 in 60 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2 [×2], C3 [×4], C22, S3 [×8], C6 [×4], C32, D6 [×4], C13, C3⋊S3 [×2], C3×C6, D13 [×2], C26, C2×C3⋊S3, C39 [×4], D26, D39 [×8], C78 [×4], C3×C39, D78 [×4], C3⋊D39 [×2], C3×C78, C2×C3⋊D39
Quotients: C1, C2 [×3], C22, S3 [×4], D6 [×4], C3⋊S3, D13, C2×C3⋊S3, D26, D39 [×4], D78 [×4], C3⋊D39, C2×C3⋊D39

Smallest permutation representation of C2×C3⋊D39
On 234 points
Generators in S234
(1 156)(2 118)(3 119)(4 120)(5 121)(6 122)(7 123)(8 124)(9 125)(10 126)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 135)(20 136)(21 137)(22 138)(23 139)(24 140)(25 141)(26 142)(27 143)(28 144)(29 145)(30 146)(31 147)(32 148)(33 149)(34 150)(35 151)(36 152)(37 153)(38 154)(39 155)(40 172)(41 173)(42 174)(43 175)(44 176)(45 177)(46 178)(47 179)(48 180)(49 181)(50 182)(51 183)(52 184)(53 185)(54 186)(55 187)(56 188)(57 189)(58 190)(59 191)(60 192)(61 193)(62 194)(63 195)(64 157)(65 158)(66 159)(67 160)(68 161)(69 162)(70 163)(71 164)(72 165)(73 166)(74 167)(75 168)(76 169)(77 170)(78 171)(79 230)(80 231)(81 232)(82 233)(83 234)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)(113 225)(114 226)(115 227)(116 228)(117 229)
(1 81 59)(2 82 60)(3 83 61)(4 84 62)(5 85 63)(6 86 64)(7 87 65)(8 88 66)(9 89 67)(10 90 68)(11 91 69)(12 92 70)(13 93 71)(14 94 72)(15 95 73)(16 96 74)(17 97 75)(18 98 76)(19 99 77)(20 100 78)(21 101 40)(22 102 41)(23 103 42)(24 104 43)(25 105 44)(26 106 45)(27 107 46)(28 108 47)(29 109 48)(30 110 49)(31 111 50)(32 112 51)(33 113 52)(34 114 53)(35 115 54)(36 116 55)(37 117 56)(38 79 57)(39 80 58)(118 233 192)(119 234 193)(120 196 194)(121 197 195)(122 198 157)(123 199 158)(124 200 159)(125 201 160)(126 202 161)(127 203 162)(128 204 163)(129 205 164)(130 206 165)(131 207 166)(132 208 167)(133 209 168)(134 210 169)(135 211 170)(136 212 171)(137 213 172)(138 214 173)(139 215 174)(140 216 175)(141 217 176)(142 218 177)(143 219 178)(144 220 179)(145 221 180)(146 222 181)(147 223 182)(148 224 183)(149 225 184)(150 226 185)(151 227 186)(152 228 187)(153 229 188)(154 230 189)(155 231 190)(156 232 191)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 99)(41 98)(42 97)(43 96)(44 95)(45 94)(46 93)(47 92)(48 91)(49 90)(50 89)(51 88)(52 87)(53 86)(54 85)(55 84)(56 83)(57 82)(58 81)(59 80)(60 79)(61 117)(62 116)(63 115)(64 114)(65 113)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(118 154)(119 153)(120 152)(121 151)(122 150)(123 149)(124 148)(125 147)(126 146)(127 145)(128 144)(129 143)(130 142)(131 141)(132 140)(133 139)(134 138)(135 137)(155 156)(157 226)(158 225)(159 224)(160 223)(161 222)(162 221)(163 220)(164 219)(165 218)(166 217)(167 216)(168 215)(169 214)(170 213)(171 212)(172 211)(173 210)(174 209)(175 208)(176 207)(177 206)(178 205)(179 204)(180 203)(181 202)(182 201)(183 200)(184 199)(185 198)(186 197)(187 196)(188 234)(189 233)(190 232)(191 231)(192 230)(193 229)(194 228)(195 227)

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

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

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

120 conjugacy classes

class 1 2A2B2C3A3B3C3D6A6B6C6D13A···13F26A···26F39A···39AV78A···78AV
order12223333666613···1326···2639···3978···78
size11117117222222222···22···22···22···2

120 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D6D13D26D39D78
kernelC2×C3⋊D39C3⋊D39C3×C78C78C39C3×C6C32C6C3
# reps12144664848

Matrix representation of C2×C3⋊D39 in GL4(𝔽79) generated by

1000
0100
00780
00078
,
44600
783400
006273
00616
,
69700
123100
00107
007211
,
44800
547500
004927
003430
G:=sub<GL(4,GF(79))| [1,0,0,0,0,1,0,0,0,0,78,0,0,0,0,78],[44,78,0,0,6,34,0,0,0,0,62,6,0,0,73,16],[69,12,0,0,7,31,0,0,0,0,10,72,0,0,7,11],[4,54,0,0,48,75,0,0,0,0,49,34,0,0,27,30] >;

C2×C3⋊D39 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes D_{39}
% in TeX

G:=Group("C2xC3:D39");
// GroupNames label

G:=SmallGroup(468,54);
// by ID

G=gap.SmallGroup(468,54);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-13,122,483,10804]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^39=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽