C2xC3^2.24He3 - GroupNames

G = C₂×C₃².₂₄He₃ order 486 = 2·3⁵

Direct product of C₂ and C₃².₂₄He₃

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

Aliases: C₂×C₃².₂₄He₃, C₆.₁C₃≀C₃, C₃²⋊C₉⋊₉C₆, (C₆×He₃)⋊₁C₃, (C₃×He₃)⋊₁₃C₆, C₃³.₁(C₃×C₆), (C₃×C₆).₁₄He₃, C₃².₂₉(C₂×He₃), (C₃²×C₆).₁C₃², C₆.₁(He₃⋊C₃), C₃.₄(C₂×C₃≀C₃), (C₂×C₃²⋊C₉)⋊₁C₃, C₃.₃(C₂×He₃⋊C₃), SmallGroup(486,63)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₂×C₃².₂₄He₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃×He₃ — C₃².₂₄He₃ — C₂×C₃².₂₄He₃

Lower central

C₁ — C₃² — C₃³ — C₂×C₃².₂₄He₃

Upper central

C₁ — C₃×C₆ — C₃²×C₆ — C₂×C₃².₂₄He₃

Generators and relations for C₂×C₃².₂₄He₃
G = < a,b,c,d,e,f | a²=b³=c³=d³=e³=f³=1, 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: 414 in 94 conjugacy classes, 24 normal (8 characteristic)
C₁, C₂, C₃, C₃, C₆, C₆, C₉, C₃², C₃², C₁₈, C₃×C₆, C₃×C₆, C₃×C₉, He₃, C₃³, C₃³, C₃×C₁₈, C₂×He₃, C₃²×C₆, C₃²×C₆, C₃²⋊C₉, C₃×He₃, C₂×C₃²⋊C₉, C₆×He₃, C₃².₂₄He₃, C₂×C₃².₂₄He₃
Quotients: C₁, C₂, C₃, C₆, C₃², C₃×C₆, He₃, C₂×He₃, C₃≀C₃, He₃⋊C₃, C₂×C₃≀C₃, C₂×He₃⋊C₃, C₃².₂₄He₃, C₂×C₃².₂₄He₃

Smallest permutation representation of C₂×C₃².₂₄He₃
►On 162 points

Generators in S₁₆₂

(1 88)(2 89)(3 90)(4 84)(5 82)(6 83)(7 87)(8 85)(9 86)(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)(28 109)(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 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)

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

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

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

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

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

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

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

70 conjugacy classes

class	1	2	3A	···	3H	3I	···	3V	6A	···	6H	6I	···	6V	9A	···	9L	18A	···	18L
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	He₃⋊C₃	C₂×C₃≀C₃	C₂×He₃⋊C₃
kernel	C₂×C₃².₂₄He₃	C₃².₂₄He₃	C₂×C₃²⋊C₉	C₆×He₃	C₃²⋊C₉	C₃×He₃	C₃×C₆	C₃²	C₆	C₆	C₃	C₃
# reps	1	1	4	4	4	4	2	2	12	12	12	12

Matrix representation of C₂×C₃².₂₄He₃ ►in GL₆(𝔽₁₉)

18	0	0	0	0	0
0	18	0	0	0	0
0	0	18	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	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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	7	0	0	0
0	0	0	1	6	0
0	0	0	0	18	1
0	0	0	0	18	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	1	7	0
0	0	0	8	0	11

0	1	0	0	0	0
0	0	1	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	8	0	11

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,6,18,18,0,0,0,0,1,0],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,1,8,0,0,0,0,7,0,0,0,0,0,0,11],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,8,0,0,0,0,1,0,0,0,0,0,0,11] >;

C₂×C₃².₂₄He₃ in GAP, Magma, Sage, TeX

C_2\times C_3^2._{24}{\rm He}_3

% in TeX

G:=Group("C2xC3^2.24He3");

// GroupNames label

G:=SmallGroup(486,63);

// by ID

G=gap.SmallGroup(486,63);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^3=f^3=1,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×C32.24He3 order 486 = 2·35

Direct product of C2 and C32.24He3

G = C₂×C₃².₂₄He₃ order 486 = 2·3⁵

Direct product of C₂ and C₃².₂₄He₃