C2xC3^2.23C3^3 - GroupNames

G = C₂×C₃².₂₃C₃³ order 486 = 2·3⁵

Direct product of C₂ and C₃².₂₃C₃³

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C₂×C₃².₂₃C₃³, C₃.₆(C₆×He₃), C₆.₆(C₃×He₃), (C₃×C₆).₈He₃, C₃²⋊C₉⋊₁₆C₆, (C₃²×C₁₈)⋊₄C₃, (C₃²×C₉)⋊₃₀C₆, (C₆×He₃).₅C₃, C₃³.₉(C₃×C₆), (C₃×He₃).₁₈C₆, C₆.₅(C₉○He₃), (C₃×C₁₈).₆C₃², (C₃×C₆).₂₃C₃³, C₃².₈(C₂×He₃), (C₃²×C₆).₈C₃², C₃².₂₇(C₃²×C₆), (C₆×3_-¹⁺²)⋊₄C₃, (C₃×3_-¹⁺²)⋊₁₁C₆, (C₂×C₃²⋊C₉)⋊₈C₃, (C₃×C₉).₂₄(C₃×C₆), C₃.₅(C₂×C₉○He₃), SmallGroup(486,199)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂×C₃².₂₃C₃³

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₃²×C₉ — C₃².₂₃C₃³ — C₂×C₃².₂₃C₃³

Lower central

C₁ — C₃² — C₂×C₃².₂₃C₃³

Upper central

C₁ — C₃×C₆ — C₂×C₃².₂₃C₃³

Generators and relations for C₂×C₃².₂₃C₃³
G = < a,b,c,d,e,f | a²=b³=c³=d³=e³=1, f³=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ede^-1=bd=db, be=eb, bf=fb, fdf^-1=cd=dc, ce=ec, cf=fc, ef=fe >

Subgroups: 360 in 156 conjugacy classes, 72 normal (14 characteristic)
C₁, C₂, C₃, C₃, C₃, C₆, C₆, C₆, C₉, C₃², C₃², C₃², C₁₈, C₃×C₆, C₃×C₆, C₃×C₆, C₃×C₉, C₃×C₉, He₃, 3_-¹⁺², C₃³, C₃³, C₃×C₁₈, C₃×C₁₈, C₂×He₃, C₂×3_-¹⁺², C₃²×C₆, C₃²×C₆, C₃²⋊C₉, C₃²×C₉, C₃×He₃, C₃×3_-¹⁺², C₂×C₃²⋊C₉, C₃²×C₁₈, C₆×He₃, C₆×3_-¹⁺², C₃².₂₃C₃³, C₂×C₃².₂₃C₃³
Quotients: C₁, C₂, C₃, C₆, C₃², C₃×C₆, He₃, C₃³, C₂×He₃, C₃²×C₆, C₃×He₃, C₉○He₃, C₆×He₃, C₂×C₉○He₃, C₃².₂₃C₃³, C₂×C₃².₂₃C₃³

Smallest permutation representation of C₂×C₃².₂₃C₃³
►On 162 points

Generators in S₁₆₂

(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 82)(18 83)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 91)(27 92)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)

(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 112 115)(110 113 116)(111 114 117)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)(145 148 151)(146 149 152)(147 150 153)(154 157 160)(155 158 161)(156 159 162)

(1 43 29)(2 44 30)(3 45 31)(4 37 32)(5 38 33)(6 39 34)(7 40 35)(8 41 36)(9 42 28)(10 158 23)(11 159 24)(12 160 25)(13 161 26)(14 162 27)(15 154 19)(16 155 20)(17 156 21)(18 157 22)(46 61 64)(47 62 65)(48 63 66)(49 55 67)(50 56 68)(51 57 69)(52 58 70)(53 59 71)(54 60 72)(73 93 89)(74 94 90)(75 95 82)(76 96 83)(77 97 84)(78 98 85)(79 99 86)(80 91 87)(81 92 88)(100 124 110)(101 125 111)(102 126 112)(103 118 113)(104 119 114)(105 120 115)(106 121 116)(107 122 117)(108 123 109)(127 142 145)(128 143 146)(129 144 147)(130 136 148)(131 137 149)(132 138 150)(133 139 151)(134 140 152)(135 141 153)

(1 85 69)(2 79 58)(3 91 53)(4 88 72)(5 73 61)(6 94 47)(7 82 66)(8 76 55)(9 97 50)(10 137 123)(11 150 100)(12 133 111)(13 140 126)(14 153 103)(15 127 114)(16 143 120)(17 147 106)(18 130 117)(19 145 119)(20 128 105)(21 144 116)(22 148 122)(23 131 108)(24 138 110)(25 151 125)(26 134 102)(27 141 113)(28 77 68)(29 98 57)(30 86 52)(31 80 71)(32 92 60)(33 89 46)(34 74 65)(35 95 63)(36 83 49)(37 81 54)(38 93 64)(39 90 62)(40 75 48)(41 96 67)(42 84 56)(43 78 51)(44 99 70)(45 87 59)(101 160 139)(104 154 142)(107 157 136)(109 158 149)(112 161 152)(115 155 146)(118 162 135)(121 156 129)(124 159 132)

(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)(145 148 151)(146 149 152)(147 150 153)(154 160 157)(155 161 158)(156 162 159)

(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)

G:=sub<Sym(162)| (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,82)(18,83)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,91)(27,92)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135)(136,139,142)(137,140,143)(138,141,144)(145,148,151)(146,149,152)(147,150,153)(154,157,160)(155,158,161)(156,159,162), (1,43,29)(2,44,30)(3,45,31)(4,37,32)(5,38,33)(6,39,34)(7,40,35)(8,41,36)(9,42,28)(10,158,23)(11,159,24)(12,160,25)(13,161,26)(14,162,27)(15,154,19)(16,155,20)(17,156,21)(18,157,22)(46,61,64)(47,62,65)(48,63,66)(49,55,67)(50,56,68)(51,57,69)(52,58,70)(53,59,71)(54,60,72)(73,93,89)(74,94,90)(75,95,82)(76,96,83)(77,97,84)(78,98,85)(79,99,86)(80,91,87)(81,92,88)(100,124,110)(101,125,111)(102,126,112)(103,118,113)(104,119,114)(105,120,115)(106,121,116)(107,122,117)(108,123,109)(127,142,145)(128,143,146)(129,144,147)(130,136,148)(131,137,149)(132,138,150)(133,139,151)(134,140,152)(135,141,153), (1,85,69)(2,79,58)(3,91,53)(4,88,72)(5,73,61)(6,94,47)(7,82,66)(8,76,55)(9,97,50)(10,137,123)(11,150,100)(12,133,111)(13,140,126)(14,153,103)(15,127,114)(16,143,120)(17,147,106)(18,130,117)(19,145,119)(20,128,105)(21,144,116)(22,148,122)(23,131,108)(24,138,110)(25,151,125)(26,134,102)(27,141,113)(28,77,68)(29,98,57)(30,86,52)(31,80,71)(32,92,60)(33,89,46)(34,74,65)(35,95,63)(36,83,49)(37,81,54)(38,93,64)(39,90,62)(40,75,48)(41,96,67)(42,84,56)(43,78,51)(44,99,70)(45,87,59)(101,160,139)(104,154,142)(107,157,136)(109,158,149)(112,161,152)(115,155,146)(118,162,135)(121,156,129)(124,159,132), (10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(127,130,133)(128,131,134)(129,132,135)(136,139,142)(137,140,143)(138,141,144)(145,148,151)(146,149,152)(147,150,153)(154,160,157)(155,161,158)(156,162,159), (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)>;

G:=Group( (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,82)(18,83)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,91)(27,92)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,161)(81,162), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135)(136,139,142)(137,140,143)(138,141,144)(145,148,151)(146,149,152)(147,150,153)(154,157,160)(155,158,161)(156,159,162), (1,43,29)(2,44,30)(3,45,31)(4,37,32)(5,38,33)(6,39,34)(7,40,35)(8,41,36)(9,42,28)(10,158,23)(11,159,24)(12,160,25)(13,161,26)(14,162,27)(15,154,19)(16,155,20)(17,156,21)(18,157,22)(46,61,64)(47,62,65)(48,63,66)(49,55,67)(50,56,68)(51,57,69)(52,58,70)(53,59,71)(54,60,72)(73,93,89)(74,94,90)(75,95,82)(76,96,83)(77,97,84)(78,98,85)(79,99,86)(80,91,87)(81,92,88)(100,124,110)(101,125,111)(102,126,112)(103,118,113)(104,119,114)(105,120,115)(106,121,116)(107,122,117)(108,123,109)(127,142,145)(128,143,146)(129,144,147)(130,136,148)(131,137,149)(132,138,150)(133,139,151)(134,140,152)(135,141,153), (1,85,69)(2,79,58)(3,91,53)(4,88,72)(5,73,61)(6,94,47)(7,82,66)(8,76,55)(9,97,50)(10,137,123)(11,150,100)(12,133,111)(13,140,126)(14,153,103)(15,127,114)(16,143,120)(17,147,106)(18,130,117)(19,145,119)(20,128,105)(21,144,116)(22,148,122)(23,131,108)(24,138,110)(25,151,125)(26,134,102)(27,141,113)(28,77,68)(29,98,57)(30,86,52)(31,80,71)(32,92,60)(33,89,46)(34,74,65)(35,95,63)(36,83,49)(37,81,54)(38,93,64)(39,90,62)(40,75,48)(41,96,67)(42,84,56)(43,78,51)(44,99,70)(45,87,59)(101,160,139)(104,154,142)(107,157,136)(109,158,149)(112,161,152)(115,155,146)(118,162,135)(121,156,129)(124,159,132), (10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(127,130,133)(128,131,134)(129,132,135)(136,139,142)(137,140,143)(138,141,144)(145,148,151)(146,149,152)(147,150,153)(154,160,157)(155,161,158)(156,162,159), (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) );

G=PermutationGroup([[(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,82),(18,83),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,91),(27,92),(28,109),(29,110),(30,111),(31,112),(32,113),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,129),(49,130),(50,131),(51,132),(52,133),(53,134),(54,135),(55,136),(56,137),(57,138),(58,139),(59,140),(60,141),(61,142),(62,143),(63,144),(64,145),(65,146),(66,147),(67,148),(68,149),(69,150),(70,151),(71,152),(72,153),(73,154),(74,155),(75,156),(76,157),(77,158),(78,159),(79,160),(80,161),(81,162)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90),(91,94,97),(92,95,98),(93,96,99),(100,103,106),(101,104,107),(102,105,108),(109,112,115),(110,113,116),(111,114,117),(118,121,124),(119,122,125),(120,123,126),(127,130,133),(128,131,134),(129,132,135),(136,139,142),(137,140,143),(138,141,144),(145,148,151),(146,149,152),(147,150,153),(154,157,160),(155,158,161),(156,159,162)], [(1,43,29),(2,44,30),(3,45,31),(4,37,32),(5,38,33),(6,39,34),(7,40,35),(8,41,36),(9,42,28),(10,158,23),(11,159,24),(12,160,25),(13,161,26),(14,162,27),(15,154,19),(16,155,20),(17,156,21),(18,157,22),(46,61,64),(47,62,65),(48,63,66),(49,55,67),(50,56,68),(51,57,69),(52,58,70),(53,59,71),(54,60,72),(73,93,89),(74,94,90),(75,95,82),(76,96,83),(77,97,84),(78,98,85),(79,99,86),(80,91,87),(81,92,88),(100,124,110),(101,125,111),(102,126,112),(103,118,113),(104,119,114),(105,120,115),(106,121,116),(107,122,117),(108,123,109),(127,142,145),(128,143,146),(129,144,147),(130,136,148),(131,137,149),(132,138,150),(133,139,151),(134,140,152),(135,141,153)], [(1,85,69),(2,79,58),(3,91,53),(4,88,72),(5,73,61),(6,94,47),(7,82,66),(8,76,55),(9,97,50),(10,137,123),(11,150,100),(12,133,111),(13,140,126),(14,153,103),(15,127,114),(16,143,120),(17,147,106),(18,130,117),(19,145,119),(20,128,105),(21,144,116),(22,148,122),(23,131,108),(24,138,110),(25,151,125),(26,134,102),(27,141,113),(28,77,68),(29,98,57),(30,86,52),(31,80,71),(32,92,60),(33,89,46),(34,74,65),(35,95,63),(36,83,49),(37,81,54),(38,93,64),(39,90,62),(40,75,48),(41,96,67),(42,84,56),(43,78,51),(44,99,70),(45,87,59),(101,160,139),(104,154,142),(107,157,136),(109,158,149),(112,161,152),(115,155,146),(118,162,135),(121,156,129),(124,159,132)], [(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,79,76),(74,80,77),(75,81,78),(82,88,85),(83,89,86),(84,90,87),(91,97,94),(92,98,95),(93,99,96),(127,130,133),(128,131,134),(129,132,135),(136,139,142),(137,140,143),(138,141,144),(145,148,151),(146,149,152),(147,150,153),(154,160,157),(155,161,158),(156,162,159)], [(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)]])

102 conjugacy classes

class	1	2	3A	···	3H	3I	···	3N	3O	···	3T	6A	···	6H	6I	···	6N	6O	···	6T	9A	···	9R	9S	···	9AD	18A	···	18R	18S	···	18AD
order	1	2	3	···	3	3	···	3	3	···	3	6	···	6	6	···	6	6	···	6	9	···	9	9	···	9	18	···	18	18	···	18
size	1	1	1	···	1	3	···	3	9	···	9	1	···	1	3	···	3	9	···	9	3	···	3	9	···	9	3	···	3	9	···	9

102 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	3	3	3	3
type	+	+
image	C₁	C₂	C₃	C₃	C₃	C₃	C₆	C₆	C₆	C₆	He₃	C₂×He₃	C₉○He₃	C₂×C₉○He₃
kernel	C₂×C₃².₂₃C₃³	C₃².₂₃C₃³	C₂×C₃²⋊C₉	C₃²×C₁₈	C₆×He₃	C₆×3_-¹⁺²	C₃²⋊C₉	C₃²×C₉	C₃×He₃	C₃×3_-¹⁺²	C₃×C₆	C₃²	C₆	C₃
# reps	1	1	18	2	2	4	18	2	2	4	6	6	18	18

Matrix representation of C₂×C₃².₂₃C₃³ ►in GL₆(𝔽₁₉)

18	0	0	0	0	0
0	18	0	0	0	0
0	0	18	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

11	0	0	0	0	0
0	11	0	0	0	0
0	0	11	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	7	0	0
0	0	0	0	7	0
0	0	0	0	0	7

0	1	0	0	0	0
0	0	1	0	0	0
1	0	0	0	0	0
0	0	0	18	5	11
0	0	0	15	16	14
0	0	0	0	10	4

1	0	0	0	0	0
0	11	0	0	0	0
0	0	7	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

17	0	0	0	0	0
0	17	0	0	0	0
0	0	17	0	0	0
0	0	0	4	13	0
0	0	0	0	15	1
0	0	0	1	3	0

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,15,0,0,0,0,5,16,10,0,0,0,11,14,4],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,4,0,1,0,0,0,13,15,3,0,0,0,0,1,0] >;

C₂×C₃².₂₃C₃³ in GAP, Magma, Sage, TeX

C_2\times C_3^2._{23}C_3^3

% in TeX

G:=Group("C2xC3^2.23C3^3");

// GroupNames label

G:=SmallGroup(486,199);

// by ID

G=gap.SmallGroup(486,199);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,2169,93]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^3=1,f^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,b*f=f*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,e*f=f*e>;

// generators/relations

G = C2×C32.23C33 order 486 = 2·35

Direct product of C2 and C32.23C33

G = C₂×C₃².₂₃C₃³ order 486 = 2·3⁵

Direct product of C₂ and C₃².₂₃C₃³