C2xC3^3.31C3^2 - GroupNames

G = C₂×C₃³.₃₁C₃² order 486 = 2·3⁵

Direct product of C₂ and C₃³.₃₁C₃²

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

Aliases: C₂×C₃³.₃₁C₃², C₉⋊C₉⋊₇C₆, C₃²⋊C₉.₁₈C₆, C₆.₇(C₉○He₃), (C₃×C₆).₂₅C₃³, (C₃²×C₉).₂₀C₆, (C₃×C₁₈).₈C₃², C₃³.₄₂(C₃×C₆), (C₃²×C₁₈).₈C₃, (C₃²×C₆).₃₀C₃², C₃².₂₉(C₃²×C₆), C₆.₇(C₃×3_-¹⁺²), (C₃×C₆).₈3_-¹⁺², C₃.₇(C₆×3_-¹⁺²), C₃².₈(C₂×3_-¹⁺²), (C₂×C₉⋊C₉)⋊₄C₃, (C₃×C₉).₂₅(C₃×C₆), C₃.₇(C₂×C₉○He₃), (C₂×C₃²⋊C₉).₉C₃, SmallGroup(486,201)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂×C₃³.₃₁C₃²

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₃²×C₉ — C₃³.₃₁C₃² — C₂×C₃³.₃₁C₃²

Lower central

C₁ — C₃² — C₂×C₃³.₃₁C₃²

Upper central

C₁ — C₃×C₆ — C₂×C₃³.₃₁C₃²

Generators and relations for C₂×C₃³.₃₁C₃²
G = < a,b,c,d,e,f | a²=b³=c³=e³=1, d³=c, 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: 198 in 114 conjugacy classes, 72 normal (12 characteristic)
C₁, C₂, C₃, C₃, C₃, C₆, C₆, C₆, C₉, C₃², C₃², C₃², C₁₈, C₃×C₆, C₃×C₆, C₃×C₆, C₃×C₉, C₃×C₉, C₃³, C₃×C₁₈, C₃×C₁₈, C₃²×C₆, C₃²⋊C₉, C₉⋊C₉, C₃²×C₉, C₂×C₃²⋊C₉, C₂×C₉⋊C₉, C₃²×C₁₈, C₃³.₃₁C₃², C₂×C₃³.₃₁C₃²
Quotients: C₁, C₂, C₃, C₆, C₃², C₃×C₆, 3_-¹⁺², C₃³, C₂×3_-¹⁺², C₃²×C₆, C₃×3_-¹⁺², C₉○He₃, C₆×3_-¹⁺², C₂×C₉○He₃, C₃³.₃₁C₃², C₂×C₃³.₃₁C₃²

Smallest permutation representation of C₂×C₃³.₃₁C₃²
►On 162 points

Generators in S₁₆₂

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

(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 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)

(2 54 81)(3 73 46)(5 48 75)(6 76 49)(8 51 78)(9 79 52)(10 23 114)(11 115 24)(13 26 117)(14 109 27)(16 20 111)(17 112 21)(28 121 151)(30 153 123)(31 124 145)(33 147 126)(34 118 148)(36 150 120)(37 67 85)(39 87 69)(40 70 88)(42 90 72)(43 64 82)(45 84 66)(55 105 92)(57 94 107)(58 108 95)(60 97 101)(61 102 98)(63 91 104)(127 136 154)(129 156 138)(130 139 157)(132 159 141)(133 142 160)(135 162 144)

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

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

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

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

102 conjugacy classes

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

102 irreducible representations

dim	1	1	1	1	1	1	1	1	3	3	3	3
type	+	+
image	C₁	C₂	C₃	C₃	C₃	C₆	C₆	C₆	3_-¹⁺²	C₂×3_-¹⁺²	C₉○He₃	C₂×C₉○He₃
kernel	C₂×C₃³.₃₁C₃²	C₃³.₃₁C₃²	C₂×C₃²⋊C₉	C₂×C₉⋊C₉	C₃²×C₁₈	C₃²⋊C₉	C₉⋊C₉	C₃²×C₉	C₃×C₆	C₃²	C₆	C₃
# reps	1	1	6	18	2	6	18	2	6	6	18	18

Matrix representation of C₂×C₃³.₃₁C₃² ►in GL₆(𝔽₁₉)

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

7	0	0	0	0	0
0	7	0	0	0	0
0	0	7	0	0	0
0	0	0	11	0	0
0	0	0	0	11	0
0	0	0	0	0	11

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

0	1	0	0	0	0
0	0	1	0	0	0
1	0	0	0	0	0
0	0	0	7	10	0
0	0	0	13	12	1
0	0	0	5	11	0

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

9	0	0	0	0	0
0	9	0	0	0	0
0	0	9	0	0	0
0	0	0	16	0	0
0	0	0	17	5	0
0	0	0	16	0	17

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,13,5,0,0,0,10,12,11,0,0,0,0,1,0],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,7,1,0,0,0,0,11,0,0,0,0,0,0,7],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,16,17,16,0,0,0,0,5,0,0,0,0,0,0,17] >;

C₂×C₃³.₃₁C₃² in GAP, Magma, Sage, TeX

C_2\times C_3^3._{31}C_3^2

% in TeX

G:=Group("C2xC3^3.31C3^2");

// GroupNames label

G:=SmallGroup(486,201);

// by ID

G=gap.SmallGroup(486,201);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,548,2169,93]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=e^3=1,d^3=c,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×C33.31C32 order 486 = 2·35

Direct product of C2 and C33.31C32

G = C₂×C₃³.₃₁C₃² order 486 = 2·3⁵

Direct product of C₂ and C₃³.₃₁C₃²