C2xC3^2.20He3 - GroupNames

G = C₂×C₃².₂₀He₃ order 486 = 2·3⁵

Direct product of C₂ and C₃².₂₀He₃

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

Aliases: C₂×C₃².₂₀He₃, (C₃×C₁₈)⋊₃C₉, (C₃×C₉)⋊₁₀C₁₈, (C₃×C₆).₂₃He₃, C₃²⋊C₉.₁₅C₆, C₆.₅(C₃²⋊C₉), C₃³.₃₃(C₃×C₆), (C₃²×C₉).₁₆C₆, (C₃²×C₁₈).₄C₃, C₃².₉(C₃×C₁₈), C₃².₂₁(C₂×He₃), C₆.₄(He₃⋊C₃), C₆.₄(C₃.He₃), (C₃²×C₆).₂₁C₃², (C₃×C₆).₆3_-¹⁺², C₃².₆(C₂×3_-¹⁺²), (C₃×C₆).₉(C₃×C₉), C₃.₅(C₂×C₃²⋊C₉), (C₂×C₃²⋊C₉).₆C₃, C₃.₁(C₂×He₃⋊C₃), C₃.₁(C₂×C₃.He₃), SmallGroup(486,75)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂×C₃².₂₀He₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₉ — C₃².₂₀He₃ — C₂×C₃².₂₀He₃

Lower central

C₁ — C₃ — C₃² — C₂×C₃².₂₀He₃

Upper central

C₁ — C₃×C₆ — C₃²×C₆ — C₂×C₃².₂₀He₃

Generators and relations for C₂×C₃².₂₀He₃
G = < a,b,c,d,e,f | a²=b³=c³=e³=1, d³=c^-1, f³=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf^-1=cde^-1, fef^-1=c^-1e >

Subgroups: 180 in 72 conjugacy classes, 36 normal (16 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, C₃²×C₉, C₂×C₃²⋊C₉, C₃²×C₁₈, C₃².₂₀He₃, C₂×C₃².₂₀He₃
Quotients: C₁, C₂, C₃, C₆, C₉, C₃², C₁₈, C₃×C₆, C₃×C₉, He₃, 3_-¹⁺², C₃×C₁₈, C₂×He₃, C₂×3_-¹⁺², C₃²⋊C₉, He₃⋊C₃, C₃.He₃, C₂×C₃²⋊C₉, C₂×He₃⋊C₃, C₂×C₃.He₃, C₃².₂₀He₃, C₂×C₃².₂₀He₃

Smallest permutation representation of C₂×C₃².₂₀He₃
►On 162 points

Generators in S₁₆₂

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

(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(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)(100 106 103)(101 107 104)(102 108 105)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)(136 142 139)(137 143 140)(138 144 141)(145 151 148)(146 152 149)(147 153 150)(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)

(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(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)(127 133 130)(128 134 131)(129 135 132)(136 142 139)(137 143 140)(138 144 141)(145 151 148)(146 152 149)(147 153 150)(154 157 160)(155 158 161)(156 159 162)

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

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

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

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

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	3	3	3	3
type	+	+
image	C₁	C₂	C₃	C₃	C₆	C₆	C₉	C₁₈	He₃	3_-¹⁺²	C₂×He₃	C₂×3_-¹⁺²	He₃⋊C₃	C₃.He₃	C₂×He₃⋊C₃	C₂×C₃.He₃
kernel	C₂×C₃².₂₀He₃	C₃².₂₀He₃	C₂×C₃²⋊C₉	C₃²×C₁₈	C₃²⋊C₉	C₃²×C₉	C₃×C₁₈	C₃×C₉	C₃×C₆	C₃×C₆	C₃²	C₃²	C₆	C₆	C₃	C₃
# reps	1	1	6	2	6	2	18	18	2	4	2	4	6	12	6	12

Matrix representation of C₂×C₃².₂₀He₃ ►in GL₅(𝔽₁₉)

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

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

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

1	0	0	0	0
0	1	0	0	0
0	0	6	0	0
0	0	0	6	0
0	0	1	0	9

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	7	11	0
0	0	2	0	7

16	0	0	0	0
0	1	0	0	0
0	0	7	10	0
0	0	10	12	1
0	0	7	3	0

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[1,0,0,0,0,0,1,0,0,0,0,0,6,0,1,0,0,0,6,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,1,7,2,0,0,0,11,0,0,0,0,0,7],[16,0,0,0,0,0,1,0,0,0,0,0,7,10,7,0,0,10,12,3,0,0,0,1,0] >;

C₂×C₃².₂₀He₃ in GAP, Magma, Sage, TeX

C_2\times C_3^2._{20}{\rm He}_3

% in TeX

G:=Group("C2xC3^2.20He3");

// GroupNames label

G:=SmallGroup(486,75);

// by ID

G=gap.SmallGroup(486,75);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,986,2169]);

// Polycyclic

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

// generators/relations

G = C2×C32.20He3 order 486 = 2·35

Direct product of C2 and C32.20He3

G = C₂×C₃².₂₀He₃ order 486 = 2·3⁵

Direct product of C₂ and C₃².₂₀He₃