C2xC3^2.28He3 - GroupNames

G = C₂×C₃².₂₈He₃ order 486 = 2·3⁵

Direct product of C₂ and C₃².₂₈He₃

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

Aliases: C₂×C₃².₂₈He₃, C₆.₅C₃≀C₃, (C₃×C₆).₁₈He₃, C₃³.₅(C₃×C₆), C₃²⋊C₉.₁₁C₆, (C₃²×C₆).₅C₃², C₃².₃₃(C₂×He₃), C₆.₂(C₃.He₃), (C₆×3_-¹⁺²).₂C₃, (C₃×3_-¹⁺²).₅C₆, C₃.₈(C₂×C₃≀C₃), (C₂×C₃²⋊C₉).₂C₃, C₃.₄(C₂×C₃.He₃), SmallGroup(486,67)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₂×C₃².₂₈He₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²⋊C₉ — C₃².₂₈He₃ — C₂×C₃².₂₈He₃

Lower central

C₁ — C₃² — C₃³ — C₂×C₃².₂₈He₃

Upper central

C₁ — C₃×C₆ — C₃²×C₆ — C₂×C₃².₂₈He₃

Generators and relations for C₂×C₃².₂₈He₃
G = < a,b,c,d,e,f | a²=b³=c³=e³=1, d³=b^-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: 198 in 70 conjugacy classes, 24 normal (8 characteristic)
C₁, C₂, C₃, C₃, C₆, C₆, C₉, C₃², C₃², C₁₈, C₃×C₆, C₃×C₆, C₃×C₉, 3_-¹⁺², C₃³, C₃×C₁₈, C₂×3_-¹⁺², C₃²×C₆, C₃²⋊C₉, C₃×3_-¹⁺², C₂×C₃²⋊C₉, C₆×3_-¹⁺², C₃².₂₈He₃, C₂×C₃².₂₈He₃
Quotients: C₁, C₂, C₃, C₆, C₃², C₃×C₆, He₃, C₂×He₃, C₃≀C₃, C₃.He₃, C₂×C₃≀C₃, C₂×C₃.He₃, C₃².₂₈He₃, C₂×C₃².₂₈He₃

Smallest permutation representation of C₂×C₃².₂₈He₃
►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 87)(11 88)(12 89)(13 90)(14 82)(15 83)(16 84)(17 85)(18 86)(19 96)(20 97)(21 98)(22 99)(23 91)(24 92)(25 93)(26 94)(27 95)(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 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 40 30)(2 41 31)(3 42 32)(4 43 33)(5 44 34)(6 45 35)(7 37 36)(8 38 28)(9 39 29)(10 159 23)(11 160 24)(12 161 25)(13 162 26)(14 154 27)(15 155 19)(16 156 20)(17 157 21)(18 158 22)(46 59 72)(47 60 64)(48 61 65)(49 62 66)(50 63 67)(51 55 68)(52 56 69)(53 57 70)(54 58 71)(73 95 82)(74 96 83)(75 97 84)(76 98 85)(77 99 86)(78 91 87)(79 92 88)(80 93 89)(81 94 90)(100 121 111)(101 122 112)(102 123 113)(103 124 114)(104 125 115)(105 126 116)(106 118 117)(107 119 109)(108 120 110)(127 140 153)(128 141 145)(129 142 146)(130 143 147)(131 144 148)(132 136 149)(133 137 150)(134 138 151)(135 139 152)

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

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

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

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

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

70 conjugacy classes

class	1	2	3A	···	3H	3I	3J	6A	···	6H	6I	6J	9A	···	9X	18A	···	18X
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	3	3	3	3	3	3
type	+	+
image	C₁	C₂	C₃	C₃	C₆	C₆	He₃	C₂×He₃	C₃≀C₃	C₃.He₃	C₂×C₃≀C₃	C₂×C₃.He₃
kernel	C₂×C₃².₂₈He₃	C₃².₂₈He₃	C₂×C₃²⋊C₉	C₆×3_-¹⁺²	C₃²⋊C₉	C₃×3_-¹⁺²	C₃×C₆	C₃²	C₆	C₆	C₃	C₃
# reps	1	1	4	4	4	4	2	2	12	12	12	12

Matrix representation of C₂×C₃².₂₈He₃ ►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

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

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

0	1	0	0	0	0
0	0	11	0	0	0
11	0	0	0	0	0
0	0	0	17	0	0
0	0	0	0	17	0
0	0	0	8	13	5

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	7	0
0	0	0	6	2	11

0	11	0	0	0	0
0	0	11	0	0	0
7	0	0	0	0	0
0	0	0	0	1	0
0	0	0	13	14	9
0	0	0	8	0	5

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],[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],[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],[0,0,11,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,0,17,0,8,0,0,0,0,17,13,0,0,0,0,0,5],[11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,6,0,0,0,0,7,2,0,0,0,0,0,11],[0,0,7,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,13,8,0,0,0,1,14,0,0,0,0,0,9,5] >;

C₂×C₃².₂₈He₃ in GAP, Magma, Sage, TeX

C_2\times C_3^2._{28}{\rm He}_3

% in TeX

G:=Group("C2xC3^2.28He3");

// GroupNames label

G:=SmallGroup(486,67);

// by ID

G=gap.SmallGroup(486,67);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,338,873,735]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=e^3=1,d^3=b^-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×C32.28He3 order 486 = 2·35

Direct product of C2 and C32.28He3

G = C₂×C₃².₂₈He₃ order 486 = 2·3⁵

Direct product of C₂ and C₃².₂₈He₃