C2xC3^3.C3^2 - GroupNames

G = C₂×C₃³.C₃² order 486 = 2·3⁵

Direct product of C₂ and C₃³.C₃²

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

Aliases: C₂×C₃³.C₃², C₆.₂C₃≀C₃, C₃²⋊C₉⋊₁₀C₆, (C₆×He₃).₁C₃, C₃³.₂(C₃×C₆), (C₃×C₆).₁₅He₃, (C₃×He₃).₁₄C₆, C₆.₁(He₃.C₃), C₃².₃₀(C₂×He₃), (C₃²×C₆).₂C₃², (C₃×3_-¹⁺²)⋊₈C₆, (C₆×3_-¹⁺²)⋊₁C₃, C₃.₅(C₂×C₃≀C₃), (C₂×C₃²⋊C₉)⋊₂C₃, C₃.₄(C₂×He₃.C₃), SmallGroup(486,64)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₂×C₃³.C₃²

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃×He₃ — C₃³.C₃² — C₂×C₃³.C₃²

Lower central

C₁ — C₃² — C₃³ — C₂×C₃³.C₃²

Upper central

C₁ — C₃×C₆ — C₃²×C₆ — C₂×C₃³.C₃²

Generators and relations for C₂×C₃³.C₃²
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, bd=db, be=eb, bf=fb, ede^-1=cd=dc, ce=ec, cf=fc, fdf^-1=bde^-1, fef^-1=b^-1e >

Subgroups: 324 in 92 conjugacy classes, 24 normal (16 characteristic)
C₁, C₂, C₃, C₃, C₃, C₆, C₆, C₆, C₉, C₃², C₃², C₁₈, C₃×C₆, C₃×C₆, C₃×C₉, He₃, 3_-¹⁺², C₃³, C₃³, C₃×C₁₈, C₂×He₃, C₂×3_-¹⁺², C₃²×C₆, C₃²×C₆, C₃²⋊C₉, C₃×He₃, C₃×3_-¹⁺², C₂×C₃²⋊C₉, C₆×He₃, C₆×3_-¹⁺², C₃³.C₃², C₂×C₃³.C₃²
Quotients: C₁, C₂, C₃, C₆, C₃², C₃×C₆, He₃, C₂×He₃, C₃≀C₃, He₃.C₃, C₂×C₃≀C₃, C₂×He₃.C₃, C₃³.C₃², C₂×C₃³.C₃²

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

Generators in S₁₆₂

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

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

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

70 conjugacy classes

class	1	2	3A	···	3H	3I	···	3P	6A	···	6H	6I	···	6P	9A	···	9R	18A	···	18R
order	1	2	3	···	3	3	···	3	6	···	6	6	···	6	9	···	9	18	···	18
size	1	1	1	···	1	9	···	9	1	···	1	9	···	9	9	···	9	9	···	9

70 irreducible representations

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

Matrix representation of C₂×C₃³.C₃² ►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	7	0	0
0	0	0	0	7	0
0	0	0	0	0	7

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

0	0	1	0	0	0
1	0	0	0	0	0
0	11	0	0	0	0
0	0	0	0	17	0
0	0	0	0	0	17
0	0	0	16	0	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],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11],[0,1,0,0,0,0,0,0,11,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,17,0,0,0,0,0,0,17,0] >;

C₂×C₃³.C₃² in GAP, Magma, Sage, TeX

C_2\times C_3^3.C_3^2

% in TeX

G:=Group("C2xC3^3.C3^2");

// GroupNames label

G:=SmallGroup(486,64);

// by ID

G=gap.SmallGroup(486,64);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,873,735]);

// 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,b*d=d*b,b*e=e*b,b*f=f*b,e*d*e^-1=c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=b*d*e^-1,f*e*f^-1=b^-1*e>;

// generators/relations

G = C2×C33.C32 order 486 = 2·35

Direct product of C2 and C33.C32

G = C₂×C₃³.C₃² order 486 = 2·3⁵

Direct product of C₂ and C₃³.C₃²