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G = C3⋊S3×C26order 468 = 22·32·13

Direct product of C26 and C3⋊S3

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

Aliases: C3⋊S3×C26, C783S3, C398D6, C6⋊(S3×C13), C32(S3×C26), (C3×C78)⋊5C2, (C3×C6)⋊2C26, C323(C2×C26), (C3×C39)⋊10C22, SmallGroup(468,53)

Series: Derived Chief Lower central Upper central

C1C32 — C3⋊S3×C26
C1C3C32C3×C39C13×C3⋊S3 — C3⋊S3×C26
C32 — C3⋊S3×C26
C1C26

Generators and relations for C3⋊S3×C26
 G = < a,b,c,d | a26=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 156 in 60 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C2 [×2], C3 [×4], C22, S3 [×8], C6 [×4], C32, D6 [×4], C13, C3⋊S3 [×2], C3×C6, C26, C26 [×2], C2×C3⋊S3, C39 [×4], C2×C26, S3×C13 [×8], C78 [×4], C3×C39, S3×C26 [×4], C13×C3⋊S3 [×2], C3×C78, C3⋊S3×C26
Quotients: C1, C2 [×3], C22, S3 [×4], D6 [×4], C13, C3⋊S3, C26 [×3], C2×C3⋊S3, C2×C26, S3×C13 [×4], S3×C26 [×4], C13×C3⋊S3, C3⋊S3×C26

Smallest permutation representation of C3⋊S3×C26
On 234 points
Generators in S234
(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)
(1 136 114)(2 137 115)(3 138 116)(4 139 117)(5 140 118)(6 141 119)(7 142 120)(8 143 121)(9 144 122)(10 145 123)(11 146 124)(12 147 125)(13 148 126)(14 149 127)(15 150 128)(16 151 129)(17 152 130)(18 153 105)(19 154 106)(20 155 107)(21 156 108)(22 131 109)(23 132 110)(24 133 111)(25 134 112)(26 135 113)(27 206 176)(28 207 177)(29 208 178)(30 183 179)(31 184 180)(32 185 181)(33 186 182)(34 187 157)(35 188 158)(36 189 159)(37 190 160)(38 191 161)(39 192 162)(40 193 163)(41 194 164)(42 195 165)(43 196 166)(44 197 167)(45 198 168)(46 199 169)(47 200 170)(48 201 171)(49 202 172)(50 203 173)(51 204 174)(52 205 175)(53 223 95)(54 224 96)(55 225 97)(56 226 98)(57 227 99)(58 228 100)(59 229 101)(60 230 102)(61 231 103)(62 232 104)(63 233 79)(64 234 80)(65 209 81)(66 210 82)(67 211 83)(68 212 84)(69 213 85)(70 214 86)(71 215 87)(72 216 88)(73 217 89)(74 218 90)(75 219 91)(76 220 92)(77 221 93)(78 222 94)
(1 157 220)(2 158 221)(3 159 222)(4 160 223)(5 161 224)(6 162 225)(7 163 226)(8 164 227)(9 165 228)(10 166 229)(11 167 230)(12 168 231)(13 169 232)(14 170 233)(15 171 234)(16 172 209)(17 173 210)(18 174 211)(19 175 212)(20 176 213)(21 177 214)(22 178 215)(23 179 216)(24 180 217)(25 181 218)(26 182 219)(27 85 155)(28 86 156)(29 87 131)(30 88 132)(31 89 133)(32 90 134)(33 91 135)(34 92 136)(35 93 137)(36 94 138)(37 95 139)(38 96 140)(39 97 141)(40 98 142)(41 99 143)(42 100 144)(43 101 145)(44 102 146)(45 103 147)(46 104 148)(47 79 149)(48 80 150)(49 81 151)(50 82 152)(51 83 153)(52 84 154)(53 117 190)(54 118 191)(55 119 192)(56 120 193)(57 121 194)(58 122 195)(59 123 196)(60 124 197)(61 125 198)(62 126 199)(63 127 200)(64 128 201)(65 129 202)(66 130 203)(67 105 204)(68 106 205)(69 107 206)(70 108 207)(71 109 208)(72 110 183)(73 111 184)(74 112 185)(75 113 186)(76 114 187)(77 115 188)(78 116 189)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 53)(51 54)(52 55)(79 187)(80 188)(81 189)(82 190)(83 191)(84 192)(85 193)(86 194)(87 195)(88 196)(89 197)(90 198)(91 199)(92 200)(93 201)(94 202)(95 203)(96 204)(97 205)(98 206)(99 207)(100 208)(101 183)(102 184)(103 185)(104 186)(105 140)(106 141)(107 142)(108 143)(109 144)(110 145)(111 146)(112 147)(113 148)(114 149)(115 150)(116 151)(117 152)(118 153)(119 154)(120 155)(121 156)(122 131)(123 132)(124 133)(125 134)(126 135)(127 136)(128 137)(129 138)(130 139)(157 233)(158 234)(159 209)(160 210)(161 211)(162 212)(163 213)(164 214)(165 215)(166 216)(167 217)(168 218)(169 219)(170 220)(171 221)(172 222)(173 223)(174 224)(175 225)(176 226)(177 227)(178 228)(179 229)(180 230)(181 231)(182 232)

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

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

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

156 conjugacy classes

class 1 2A2B2C3A3B3C3D6A6B6C6D13A···13L26A···26L26M···26AJ39A···39AV78A···78AV
order12223333666613···1326···2626···2639···3978···78
size1199222222221···11···19···92···22···2

156 irreducible representations

dim1111112222
type+++++
imageC1C2C2C13C26C26S3D6S3×C13S3×C26
kernelC3⋊S3×C26C13×C3⋊S3C3×C78C2×C3⋊S3C3⋊S3C3×C6C78C39C6C3
# reps121122412444848

Matrix representation of C3⋊S3×C26 in GL4(𝔽79) generated by

33000
03300
00180
00018
,
78100
78000
0010
0001
,
78100
78000
007878
0010
,
1000
17800
0010
007878
G:=sub<GL(4,GF(79))| [33,0,0,0,0,33,0,0,0,0,18,0,0,0,0,18],[78,78,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[78,78,0,0,1,0,0,0,0,0,78,1,0,0,78,0],[1,1,0,0,0,78,0,0,0,0,1,78,0,0,0,78] >;

C3⋊S3×C26 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_{26}
% in TeX

G:=Group("C3:S3xC26");
// GroupNames label

G:=SmallGroup(468,53);
// by ID

G=gap.SmallGroup(468,53);
# by ID

G:=PCGroup([5,-2,-2,-13,-3,-3,2083,7804]);
// Polycyclic

G:=Group<a,b,c,d|a^26=b^3=c^3=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

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