direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×C33.3C32, C6.3C3≀C3, C33.3(C3×C6), (C3×C6).16He3, C32⋊C9.10C6, C6.2(He3.C3), (C32×C6).3C32, C32.31(C2×He3), C6.1(C3.He3), (C3×3- 1+2).4C6, (C6×3- 1+2).1C3, C3.6(C2×C3≀C3), (C2×C32⋊C9).1C3, C3.5(C2×He3.C3), C3.3(C2×C3.He3), SmallGroup(486,65)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C33.3C32
G = < a,b,c,d,e,f | a2=b3=c3=e3=1, d3=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: 180 in 60 conjugacy classes, 24 normal (all 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, C33.3C32, C2×C33.3C32
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C2×He3, C3≀C3, He3.C3, C3.He3, C2×C3≀C3, C2×He3.C3, C2×C3.He3, C33.3C32, C2×C33.3C32
►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 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 82)(18 83)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 91)(27 92)(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 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 43 36)(2 44 28)(3 45 29)(4 37 30)(5 38 31)(6 39 32)(7 40 33)(8 41 34)(9 42 35)(10 155 23)(11 156 24)(12 157 25)(13 158 26)(14 159 27)(15 160 19)(16 161 20)(17 162 21)(18 154 22)(46 56 66)(47 57 67)(48 58 68)(49 59 69)(50 60 70)(51 61 71)(52 62 72)(53 63 64)(54 55 65)(73 96 83)(74 97 84)(75 98 85)(76 99 86)(77 91 87)(78 92 88)(79 93 89)(80 94 90)(81 95 82)(100 124 117)(101 125 109)(102 126 110)(103 118 111)(104 119 112)(105 120 113)(106 121 114)(107 122 115)(108 123 116)(127 137 147)(128 138 148)(129 139 149)(130 140 150)(131 141 151)(132 142 152)(133 143 153)(134 144 145)(135 136 146)
(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 28 44)(3 45 29)(5 31 38)(6 39 32)(8 34 41)(9 42 35)(10 161 26)(11 17 14)(12 22 160)(13 155 20)(15 25 154)(16 158 23)(18 19 157)(21 27 24)(46 49 52)(47 70 63)(48 61 65)(50 64 57)(51 55 68)(53 67 60)(54 58 71)(56 59 62)(66 69 72)(73 89 99)(74 94 87)(75 81 78)(76 83 93)(77 97 90)(79 86 96)(80 91 84)(82 88 85)(92 98 95)(101 109 125)(102 126 110)(104 112 119)(105 120 113)(107 115 122)(108 123 116)(127 130 133)(128 151 144)(129 142 146)(131 145 138)(132 136 149)(134 148 141)(135 139 152)(137 140 143)(147 150 153)(156 162 159)
(1 95 49 4 98 52 7 92 46)(2 79 60 5 73 63 8 76 57)(3 91 51 6 94 54 9 97 48)(10 139 126 13 142 120 16 136 123)(11 143 121 14 137 124 17 140 118)(12 128 109 15 131 112 18 134 115)(19 151 119 22 145 122 25 148 125)(20 135 108 23 129 102 26 132 105)(21 130 103 24 133 106 27 127 100)(28 89 50 31 83 53 34 86 47)(29 77 71 32 80 65 35 74 68)(30 75 72 33 78 66 36 81 69)(37 85 62 40 88 56 43 82 59)(38 96 64 41 99 67 44 93 70)(39 90 55 42 84 58 45 87 61)(101 160 141 104 154 144 107 157 138)(110 158 152 113 161 146 116 155 149)(111 156 153 114 159 147 117 162 150)
G:=sub<Sym(162)| (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,82)(18,83)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,91)(27,92)(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), (1,4,7)(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), (1,43,36)(2,44,28)(3,45,29)(4,37,30)(5,38,31)(6,39,32)(7,40,33)(8,41,34)(9,42,35)(10,155,23)(11,156,24)(12,157,25)(13,158,26)(14,159,27)(15,160,19)(16,161,20)(17,162,21)(18,154,22)(46,56,66)(47,57,67)(48,58,68)(49,59,69)(50,60,70)(51,61,71)(52,62,72)(53,63,64)(54,55,65)(73,96,83)(74,97,84)(75,98,85)(76,99,86)(77,91,87)(78,92,88)(79,93,89)(80,94,90)(81,95,82)(100,124,117)(101,125,109)(102,126,110)(103,118,111)(104,119,112)(105,120,113)(106,121,114)(107,122,115)(108,123,116)(127,137,147)(128,138,148)(129,139,149)(130,140,150)(131,141,151)(132,142,152)(133,143,153)(134,144,145)(135,136,146), (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), (2,28,44)(3,45,29)(5,31,38)(6,39,32)(8,34,41)(9,42,35)(10,161,26)(11,17,14)(12,22,160)(13,155,20)(15,25,154)(16,158,23)(18,19,157)(21,27,24)(46,49,52)(47,70,63)(48,61,65)(50,64,57)(51,55,68)(53,67,60)(54,58,71)(56,59,62)(66,69,72)(73,89,99)(74,94,87)(75,81,78)(76,83,93)(77,97,90)(79,86,96)(80,91,84)(82,88,85)(92,98,95)(101,109,125)(102,126,110)(104,112,119)(105,120,113)(107,115,122)(108,123,116)(127,130,133)(128,151,144)(129,142,146)(131,145,138)(132,136,149)(134,148,141)(135,139,152)(137,140,143)(147,150,153)(156,162,159), (1,95,49,4,98,52,7,92,46)(2,79,60,5,73,63,8,76,57)(3,91,51,6,94,54,9,97,48)(10,139,126,13,142,120,16,136,123)(11,143,121,14,137,124,17,140,118)(12,128,109,15,131,112,18,134,115)(19,151,119,22,145,122,25,148,125)(20,135,108,23,129,102,26,132,105)(21,130,103,24,133,106,27,127,100)(28,89,50,31,83,53,34,86,47)(29,77,71,32,80,65,35,74,68)(30,75,72,33,78,66,36,81,69)(37,85,62,40,88,56,43,82,59)(38,96,64,41,99,67,44,93,70)(39,90,55,42,84,58,45,87,61)(101,160,141,104,154,144,107,157,138)(110,158,152,113,161,146,116,155,149)(111,156,153,114,159,147,117,162,150)>;
G:=Group( (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,82)(18,83)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,91)(27,92)(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), (1,4,7)(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), (1,43,36)(2,44,28)(3,45,29)(4,37,30)(5,38,31)(6,39,32)(7,40,33)(8,41,34)(9,42,35)(10,155,23)(11,156,24)(12,157,25)(13,158,26)(14,159,27)(15,160,19)(16,161,20)(17,162,21)(18,154,22)(46,56,66)(47,57,67)(48,58,68)(49,59,69)(50,60,70)(51,61,71)(52,62,72)(53,63,64)(54,55,65)(73,96,83)(74,97,84)(75,98,85)(76,99,86)(77,91,87)(78,92,88)(79,93,89)(80,94,90)(81,95,82)(100,124,117)(101,125,109)(102,126,110)(103,118,111)(104,119,112)(105,120,113)(106,121,114)(107,122,115)(108,123,116)(127,137,147)(128,138,148)(129,139,149)(130,140,150)(131,141,151)(132,142,152)(133,143,153)(134,144,145)(135,136,146), (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), (2,28,44)(3,45,29)(5,31,38)(6,39,32)(8,34,41)(9,42,35)(10,161,26)(11,17,14)(12,22,160)(13,155,20)(15,25,154)(16,158,23)(18,19,157)(21,27,24)(46,49,52)(47,70,63)(48,61,65)(50,64,57)(51,55,68)(53,67,60)(54,58,71)(56,59,62)(66,69,72)(73,89,99)(74,94,87)(75,81,78)(76,83,93)(77,97,90)(79,86,96)(80,91,84)(82,88,85)(92,98,95)(101,109,125)(102,126,110)(104,112,119)(105,120,113)(107,115,122)(108,123,116)(127,130,133)(128,151,144)(129,142,146)(131,145,138)(132,136,149)(134,148,141)(135,139,152)(137,140,143)(147,150,153)(156,162,159), (1,95,49,4,98,52,7,92,46)(2,79,60,5,73,63,8,76,57)(3,91,51,6,94,54,9,97,48)(10,139,126,13,142,120,16,136,123)(11,143,121,14,137,124,17,140,118)(12,128,109,15,131,112,18,134,115)(19,151,119,22,145,122,25,148,125)(20,135,108,23,129,102,26,132,105)(21,130,103,24,133,106,27,127,100)(28,89,50,31,83,53,34,86,47)(29,77,71,32,80,65,35,74,68)(30,75,72,33,78,66,36,81,69)(37,85,62,40,88,56,43,82,59)(38,96,64,41,99,67,44,93,70)(39,90,55,42,84,58,45,87,61)(101,160,141,104,154,144,107,157,138)(110,158,152,113,161,146,116,155,149)(111,156,153,114,159,147,117,162,150) );
G=PermutationGroup([[(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,82),(18,83),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,91),(27,92),(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)], [(1,4,7),(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)], [(1,43,36),(2,44,28),(3,45,29),(4,37,30),(5,38,31),(6,39,32),(7,40,33),(8,41,34),(9,42,35),(10,155,23),(11,156,24),(12,157,25),(13,158,26),(14,159,27),(15,160,19),(16,161,20),(17,162,21),(18,154,22),(46,56,66),(47,57,67),(48,58,68),(49,59,69),(50,60,70),(51,61,71),(52,62,72),(53,63,64),(54,55,65),(73,96,83),(74,97,84),(75,98,85),(76,99,86),(77,91,87),(78,92,88),(79,93,89),(80,94,90),(81,95,82),(100,124,117),(101,125,109),(102,126,110),(103,118,111),(104,119,112),(105,120,113),(106,121,114),(107,122,115),(108,123,116),(127,137,147),(128,138,148),(129,139,149),(130,140,150),(131,141,151),(132,142,152),(133,143,153),(134,144,145),(135,136,146)], [(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)], [(2,28,44),(3,45,29),(5,31,38),(6,39,32),(8,34,41),(9,42,35),(10,161,26),(11,17,14),(12,22,160),(13,155,20),(15,25,154),(16,158,23),(18,19,157),(21,27,24),(46,49,52),(47,70,63),(48,61,65),(50,64,57),(51,55,68),(53,67,60),(54,58,71),(56,59,62),(66,69,72),(73,89,99),(74,94,87),(75,81,78),(76,83,93),(77,97,90),(79,86,96),(80,91,84),(82,88,85),(92,98,95),(101,109,125),(102,126,110),(104,112,119),(105,120,113),(107,115,122),(108,123,116),(127,130,133),(128,151,144),(129,142,146),(131,145,138),(132,136,149),(134,148,141),(135,139,152),(137,140,143),(147,150,153),(156,162,159)], [(1,95,49,4,98,52,7,92,46),(2,79,60,5,73,63,8,76,57),(3,91,51,6,94,54,9,97,48),(10,139,126,13,142,120,16,136,123),(11,143,121,14,137,124,17,140,118),(12,128,109,15,131,112,18,134,115),(19,151,119,22,145,122,25,148,125),(20,135,108,23,129,102,26,132,105),(21,130,103,24,133,106,27,127,100),(28,89,50,31,83,53,34,86,47),(29,77,71,32,80,65,35,74,68),(30,75,72,33,78,66,36,81,69),(37,85,62,40,88,56,43,82,59),(38,96,64,41,99,67,44,93,70),(39,90,55,42,84,58,45,87,61),(101,160,141,104,154,144,107,157,138),(110,158,152,113,161,146,116,155,149),(111,156,153,114,159,147,117,162,150)]])
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 | 3 | 3 |
| type | + | + | ||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | He3 | C2×He3 | C3≀C3 | He3.C3 | C3.He3 | C2×C3≀C3 | C2×He3.C3 | C2×C3.He3 |
| kernel | C2×C33.3C32 | C33.3C32 | C2×C32⋊C9 | C6×3- 1+2 | C32⋊C9 | C3×3- 1+2 | C3×C6 | C32 | C6 | C6 | C6 | C3 | C3 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 2 | 2 | 6 | 12 | 6 | 6 | 12 | 6 |
Matrix representation of C2×C33.3C32 ►in GL7(𝔽19)
| 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 |
G:=sub<GL(7,GF(19))| [18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,5,0,0,0,0,17,0,0] >;
C2×C33.3C32 in GAP, Magma, Sage, TeX
C_2\times C_3^3._3C_3^2
% in TeX
G:=Group("C2xC3^3.3C3^2"); // GroupNames label
G:=SmallGroup(486,65);
// by ID
G=gap.SmallGroup(486,65);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,176,873,735]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=e^3=1,d^3=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