direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×C32.19He3, (C3×C9)⋊9C18, (C3×C18)⋊2C9, (C3×C6).22He3, C32⋊C9.14C6, C6.4(C32⋊C9), (C32×C9).15C6, C33.32(C3×C6), C32.8(C3×C18), (C32×C18).3C3, C6.6(He3.C3), C32.20(C2×He3), (C32×C6).20C32, (C3×C6).53- 1+2, C32.5(C2×3- 1+2), (C3×C6).8(C3×C9), C3.4(C2×C32⋊C9), (C2×C32⋊C9).5C3, C3.1(C2×He3.C3), SmallGroup(486,74)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32.19He3
G = < a,b,c,d,e,f | a2=b3=c3=e3=1, d3=c, 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 (14 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, C32.19He3, C2×C32.19He3
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, C2×C32⋊C9, C2×He3.C3, C32.19He3, C2×C32.19He3
►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 88)(11 89)(12 90)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 97)(20 98)(21 99)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 108)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(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 44 32)(2 45 33)(3 37 34)(4 38 35)(5 39 36)(6 40 28)(7 41 29)(8 42 30)(9 43 31)(10 26 162)(11 27 154)(12 19 155)(13 20 156)(14 21 157)(15 22 158)(16 23 159)(17 24 160)(18 25 161)(46 56 67)(47 57 68)(48 58 69)(49 59 70)(50 60 71)(51 61 72)(52 62 64)(53 63 65)(54 55 66)(73 89 96)(74 90 97)(75 82 98)(76 83 99)(77 84 91)(78 85 92)(79 86 93)(80 87 94)(81 88 95)(100 112 122)(101 113 123)(102 114 124)(103 115 125)(104 116 126)(105 117 118)(106 109 119)(107 110 120)(108 111 121)(127 137 148)(128 138 149)(129 139 150)(130 140 151)(131 141 152)(132 142 153)(133 143 145)(134 144 146)(135 136 147)
(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 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 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(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 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)(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)(154 160 157)(155 161 158)(156 162 159)
(1 92 70 44 78 49 32 85 59)(2 99 71 45 76 50 33 83 60)(3 97 72 37 74 51 34 90 61)(4 95 64 38 81 52 35 88 62)(5 93 65 39 79 53 36 86 63)(6 91 66 40 77 54 28 84 55)(7 98 67 41 75 46 29 82 56)(8 96 68 42 73 47 30 89 57)(9 94 69 43 80 48 31 87 58)(10 143 109 26 145 119 162 133 106)(11 138 113 27 149 123 154 128 101)(12 142 117 19 153 118 155 132 105)(13 137 112 20 148 122 156 127 100)(14 141 116 21 152 126 157 131 104)(15 136 111 22 147 121 158 135 108)(16 140 115 23 151 125 159 130 103)(17 144 110 24 146 120 160 134 107)(18 139 114 25 150 124 161 129 102)
G:=sub<Sym(162)| (1,115)(2,116)(3,117)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,88)(11,89)(12,90)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,97)(20,98)(21,99)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,108)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(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), (1,44,32)(2,45,33)(3,37,34)(4,38,35)(5,39,36)(6,40,28)(7,41,29)(8,42,30)(9,43,31)(10,26,162)(11,27,154)(12,19,155)(13,20,156)(14,21,157)(15,22,158)(16,23,159)(17,24,160)(18,25,161)(46,56,67)(47,57,68)(48,58,69)(49,59,70)(50,60,71)(51,61,72)(52,62,64)(53,63,65)(54,55,66)(73,89,96)(74,90,97)(75,82,98)(76,83,99)(77,84,91)(78,85,92)(79,86,93)(80,87,94)(81,88,95)(100,112,122)(101,113,123)(102,114,124)(103,115,125)(104,116,126)(105,117,118)(106,109,119)(107,110,120)(108,111,121)(127,137,148)(128,138,149)(129,139,150)(130,140,151)(131,141,152)(132,142,153)(133,143,145)(134,144,146)(135,136,147), (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), (1,2,3,4,5,6,7,8,9)(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), (10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(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)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(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)(154,160,157)(155,161,158)(156,162,159), (1,92,70,44,78,49,32,85,59)(2,99,71,45,76,50,33,83,60)(3,97,72,37,74,51,34,90,61)(4,95,64,38,81,52,35,88,62)(5,93,65,39,79,53,36,86,63)(6,91,66,40,77,54,28,84,55)(7,98,67,41,75,46,29,82,56)(8,96,68,42,73,47,30,89,57)(9,94,69,43,80,48,31,87,58)(10,143,109,26,145,119,162,133,106)(11,138,113,27,149,123,154,128,101)(12,142,117,19,153,118,155,132,105)(13,137,112,20,148,122,156,127,100)(14,141,116,21,152,126,157,131,104)(15,136,111,22,147,121,158,135,108)(16,140,115,23,151,125,159,130,103)(17,144,110,24,146,120,160,134,107)(18,139,114,25,150,124,161,129,102)>;
G:=Group( (1,115)(2,116)(3,117)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,88)(11,89)(12,90)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,97)(20,98)(21,99)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,108)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(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), (1,44,32)(2,45,33)(3,37,34)(4,38,35)(5,39,36)(6,40,28)(7,41,29)(8,42,30)(9,43,31)(10,26,162)(11,27,154)(12,19,155)(13,20,156)(14,21,157)(15,22,158)(16,23,159)(17,24,160)(18,25,161)(46,56,67)(47,57,68)(48,58,69)(49,59,70)(50,60,71)(51,61,72)(52,62,64)(53,63,65)(54,55,66)(73,89,96)(74,90,97)(75,82,98)(76,83,99)(77,84,91)(78,85,92)(79,86,93)(80,87,94)(81,88,95)(100,112,122)(101,113,123)(102,114,124)(103,115,125)(104,116,126)(105,117,118)(106,109,119)(107,110,120)(108,111,121)(127,137,148)(128,138,149)(129,139,150)(130,140,151)(131,141,152)(132,142,153)(133,143,145)(134,144,146)(135,136,147), (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), (1,2,3,4,5,6,7,8,9)(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), (10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(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)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(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)(154,160,157)(155,161,158)(156,162,159), (1,92,70,44,78,49,32,85,59)(2,99,71,45,76,50,33,83,60)(3,97,72,37,74,51,34,90,61)(4,95,64,38,81,52,35,88,62)(5,93,65,39,79,53,36,86,63)(6,91,66,40,77,54,28,84,55)(7,98,67,41,75,46,29,82,56)(8,96,68,42,73,47,30,89,57)(9,94,69,43,80,48,31,87,58)(10,143,109,26,145,119,162,133,106)(11,138,113,27,149,123,154,128,101)(12,142,117,19,153,118,155,132,105)(13,137,112,20,148,122,156,127,100)(14,141,116,21,152,126,157,131,104)(15,136,111,22,147,121,158,135,108)(16,140,115,23,151,125,159,130,103)(17,144,110,24,146,120,160,134,107)(18,139,114,25,150,124,161,129,102) );
G=PermutationGroup([[(1,115),(2,116),(3,117),(4,109),(5,110),(6,111),(7,112),(8,113),(9,114),(10,88),(11,89),(12,90),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,97),(20,98),(21,99),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,108),(29,100),(30,101),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(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)], [(1,44,32),(2,45,33),(3,37,34),(4,38,35),(5,39,36),(6,40,28),(7,41,29),(8,42,30),(9,43,31),(10,26,162),(11,27,154),(12,19,155),(13,20,156),(14,21,157),(15,22,158),(16,23,159),(17,24,160),(18,25,161),(46,56,67),(47,57,68),(48,58,69),(49,59,70),(50,60,71),(51,61,72),(52,62,64),(53,63,65),(54,55,66),(73,89,96),(74,90,97),(75,82,98),(76,83,99),(77,84,91),(78,85,92),(79,86,93),(80,87,94),(81,88,95),(100,112,122),(101,113,123),(102,114,124),(103,115,125),(104,116,126),(105,117,118),(106,109,119),(107,110,120),(108,111,121),(127,137,148),(128,138,149),(129,139,150),(130,140,151),(131,141,152),(132,142,153),(133,143,145),(134,144,146),(135,136,147)], [(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)], [(1,2,3,4,5,6,7,8,9),(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)], [(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(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),(73,79,76),(74,80,77),(75,81,78),(82,88,85),(83,89,86),(84,90,87),(91,97,94),(92,98,95),(93,99,96),(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),(154,160,157),(155,161,158),(156,162,159)], [(1,92,70,44,78,49,32,85,59),(2,99,71,45,76,50,33,83,60),(3,97,72,37,74,51,34,90,61),(4,95,64,38,81,52,35,88,62),(5,93,65,39,79,53,36,86,63),(6,91,66,40,77,54,28,84,55),(7,98,67,41,75,46,29,82,56),(8,96,68,42,73,47,30,89,57),(9,94,69,43,80,48,31,87,58),(10,143,109,26,145,119,162,133,106),(11,138,113,27,149,123,154,128,101),(12,142,117,19,153,118,155,132,105),(13,137,112,20,148,122,156,127,100),(14,141,116,21,152,126,157,131,104),(15,136,111,22,147,121,158,135,108),(16,140,115,23,151,125,159,130,103),(17,144,110,24,146,120,160,134,107),(18,139,114,25,150,124,161,129,102)]])
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 |
| type | + | + | ||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 | He3 | 3- 1+2 | C2×He3 | C2×3- 1+2 | He3.C3 | C2×He3.C3 |
| kernel | C2×C32.19He3 | C32.19He3 | C2×C32⋊C9 | C32×C18 | C32⋊C9 | C32×C9 | C3×C18 | C3×C9 | C3×C6 | C3×C6 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 18 | 18 | 2 | 4 | 2 | 4 | 18 | 18 |
Matrix representation of C2×C32.19He3 ►in GL4(𝔽19) generated by
| 1 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 7 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 11 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 8 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 9 | 7 | 0 |
| 0 | 16 | 0 | 11 |
| 6 | 0 | 0 | 0 |
| 0 | 9 | 6 | 0 |
| 0 | 9 | 10 | 1 |
| 0 | 13 | 17 | 0 |
G:=sub<GL(4,GF(19))| [1,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[7,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[11,0,0,0,0,4,0,8,0,0,4,0,0,0,0,9],[1,0,0,0,0,1,9,16,0,0,7,0,0,0,0,11],[6,0,0,0,0,9,9,13,0,6,10,17,0,0,1,0] >;
C2×C32.19He3 in GAP, Magma, Sage, TeX
C_2\times C_3^2._{19}{\rm He}_3 % in TeX
G:=Group("C2xC3^2.19He3"); // GroupNames label
G:=SmallGroup(486,74);
// by ID
G=gap.SmallGroup(486,74);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,500,2169]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=e^3=1,d^3=c,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