direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×C32.27He3, C6.4C3≀C3, C32⋊C9⋊11C6, (C6×He3).2C3, C33.4(C3×C6), (C3×C6).17He3, (C3×He3).15C6, C6.3(He3.C3), C32.32(C2×He3), (C32×C6).4C32, C6.2(He3⋊C3), C3.7(C2×C3≀C3), (C2×C32⋊C9)⋊3C3, C3.6(C2×He3.C3), C3.4(C2×He3⋊C3), SmallGroup(486,66)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32.27He3
G = < a,b,c,d,e,f | a2=b3=c3=d3=e3=1, f3=c-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bc-1, fbf-1=bd-1, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bc-1d-1e >
Subgroups: 288 in 72 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C3, C3, C3, C6, 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, C32⋊C9, C3×He3, C2×C32⋊C9, C2×C32⋊C9, C6×He3, C32.27He3, C2×C32.27He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C2×He3, C3≀C3, He3.C3, He3⋊C3, C2×C3≀C3, C2×He3.C3, C2×He3⋊C3, C32.27He3, C2×C32.27He3
►On 162 points
Generators in S162
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 100)(8 101)(9 102)(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)
(2 31 41)(3 42 32)(5 34 44)(6 45 35)(8 28 38)(9 39 29)(10 159 20)(11 14 17)(12 19 155)(13 162 23)(15 22 158)(16 156 26)(18 25 161)(21 24 27)(46 52 49)(47 67 60)(48 55 65)(50 70 63)(51 58 68)(53 64 57)(54 61 71)(56 62 59)(66 72 69)(73 76 79)(74 89 96)(75 94 84)(77 83 99)(78 97 87)(80 86 93)(81 91 90)(82 85 88)(92 95 98)(101 109 119)(102 120 110)(104 112 122)(105 123 113)(107 115 125)(108 126 116)(127 133 130)(128 148 141)(129 136 146)(131 151 144)(132 139 149)(134 145 138)(135 142 152)(137 143 140)(147 153 150)(154 157 160)
(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)(10 156 23)(11 157 24)(12 158 25)(13 159 26)(14 160 27)(15 161 19)(16 162 20)(17 154 21)(18 155 22)(46 56 69)(47 57 70)(48 58 71)(49 59 72)(50 60 64)(51 61 65)(52 62 66)(53 63 67)(54 55 68)(73 98 85)(74 99 86)(75 91 87)(76 92 88)(77 93 89)(78 94 90)(79 95 82)(80 96 83)(81 97 84)(100 118 117)(101 119 109)(102 120 110)(103 121 111)(104 122 112)(105 123 113)(106 124 114)(107 125 115)(108 126 116)(127 137 150)(128 138 151)(129 139 152)(130 140 153)(131 141 145)(132 142 146)(133 143 147)(134 144 148)(135 136 149)
(1 92 49)(2 80 60)(3 91 51)(4 95 52)(5 74 63)(6 94 54)(7 98 46)(8 77 57)(9 97 48)(10 142 123)(11 140 121)(12 128 109)(13 136 126)(14 143 124)(15 131 112)(16 139 120)(17 137 118)(18 134 115)(19 145 122)(20 129 102)(21 127 100)(22 148 125)(23 132 105)(24 130 103)(25 151 119)(26 135 108)(27 133 106)(28 89 47)(29 81 71)(30 76 72)(31 83 50)(32 75 65)(33 79 66)(34 86 53)(35 78 68)(36 73 69)(37 85 56)(38 93 70)(39 84 58)(40 88 59)(41 96 64)(42 87 61)(43 82 62)(44 99 67)(45 90 55)(101 158 138)(104 161 141)(107 155 144)(110 162 152)(111 157 153)(113 156 146)(114 160 147)(116 159 149)(117 154 150)
(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)
G:=sub<Sym(162)| (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,100)(8,101)(9,102)(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), (2,31,41)(3,42,32)(5,34,44)(6,45,35)(8,28,38)(9,39,29)(10,159,20)(11,14,17)(12,19,155)(13,162,23)(15,22,158)(16,156,26)(18,25,161)(21,24,27)(46,52,49)(47,67,60)(48,55,65)(50,70,63)(51,58,68)(53,64,57)(54,61,71)(56,62,59)(66,72,69)(73,76,79)(74,89,96)(75,94,84)(77,83,99)(78,97,87)(80,86,93)(81,91,90)(82,85,88)(92,95,98)(101,109,119)(102,120,110)(104,112,122)(105,123,113)(107,115,125)(108,126,116)(127,133,130)(128,148,141)(129,136,146)(131,151,144)(132,139,149)(134,145,138)(135,142,152)(137,143,140)(147,153,150)(154,157,160), (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)(10,156,23)(11,157,24)(12,158,25)(13,159,26)(14,160,27)(15,161,19)(16,162,20)(17,154,21)(18,155,22)(46,56,69)(47,57,70)(48,58,71)(49,59,72)(50,60,64)(51,61,65)(52,62,66)(53,63,67)(54,55,68)(73,98,85)(74,99,86)(75,91,87)(76,92,88)(77,93,89)(78,94,90)(79,95,82)(80,96,83)(81,97,84)(100,118,117)(101,119,109)(102,120,110)(103,121,111)(104,122,112)(105,123,113)(106,124,114)(107,125,115)(108,126,116)(127,137,150)(128,138,151)(129,139,152)(130,140,153)(131,141,145)(132,142,146)(133,143,147)(134,144,148)(135,136,149), (1,92,49)(2,80,60)(3,91,51)(4,95,52)(5,74,63)(6,94,54)(7,98,46)(8,77,57)(9,97,48)(10,142,123)(11,140,121)(12,128,109)(13,136,126)(14,143,124)(15,131,112)(16,139,120)(17,137,118)(18,134,115)(19,145,122)(20,129,102)(21,127,100)(22,148,125)(23,132,105)(24,130,103)(25,151,119)(26,135,108)(27,133,106)(28,89,47)(29,81,71)(30,76,72)(31,83,50)(32,75,65)(33,79,66)(34,86,53)(35,78,68)(36,73,69)(37,85,56)(38,93,70)(39,84,58)(40,88,59)(41,96,64)(42,87,61)(43,82,62)(44,99,67)(45,90,55)(101,158,138)(104,161,141)(107,155,144)(110,162,152)(111,157,153)(113,156,146)(114,160,147)(116,159,149)(117,154,150), (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)>;
G:=Group( (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,100)(8,101)(9,102)(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), (2,31,41)(3,42,32)(5,34,44)(6,45,35)(8,28,38)(9,39,29)(10,159,20)(11,14,17)(12,19,155)(13,162,23)(15,22,158)(16,156,26)(18,25,161)(21,24,27)(46,52,49)(47,67,60)(48,55,65)(50,70,63)(51,58,68)(53,64,57)(54,61,71)(56,62,59)(66,72,69)(73,76,79)(74,89,96)(75,94,84)(77,83,99)(78,97,87)(80,86,93)(81,91,90)(82,85,88)(92,95,98)(101,109,119)(102,120,110)(104,112,122)(105,123,113)(107,115,125)(108,126,116)(127,133,130)(128,148,141)(129,136,146)(131,151,144)(132,139,149)(134,145,138)(135,142,152)(137,143,140)(147,153,150)(154,157,160), (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)(10,156,23)(11,157,24)(12,158,25)(13,159,26)(14,160,27)(15,161,19)(16,162,20)(17,154,21)(18,155,22)(46,56,69)(47,57,70)(48,58,71)(49,59,72)(50,60,64)(51,61,65)(52,62,66)(53,63,67)(54,55,68)(73,98,85)(74,99,86)(75,91,87)(76,92,88)(77,93,89)(78,94,90)(79,95,82)(80,96,83)(81,97,84)(100,118,117)(101,119,109)(102,120,110)(103,121,111)(104,122,112)(105,123,113)(106,124,114)(107,125,115)(108,126,116)(127,137,150)(128,138,151)(129,139,152)(130,140,153)(131,141,145)(132,142,146)(133,143,147)(134,144,148)(135,136,149), (1,92,49)(2,80,60)(3,91,51)(4,95,52)(5,74,63)(6,94,54)(7,98,46)(8,77,57)(9,97,48)(10,142,123)(11,140,121)(12,128,109)(13,136,126)(14,143,124)(15,131,112)(16,139,120)(17,137,118)(18,134,115)(19,145,122)(20,129,102)(21,127,100)(22,148,125)(23,132,105)(24,130,103)(25,151,119)(26,135,108)(27,133,106)(28,89,47)(29,81,71)(30,76,72)(31,83,50)(32,75,65)(33,79,66)(34,86,53)(35,78,68)(36,73,69)(37,85,56)(38,93,70)(39,84,58)(40,88,59)(41,96,64)(42,87,61)(43,82,62)(44,99,67)(45,90,55)(101,158,138)(104,161,141)(107,155,144)(110,162,152)(111,157,153)(113,156,146)(114,160,147)(116,159,149)(117,154,150), (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) );
G=PermutationGroup([[(1,103),(2,104),(3,105),(4,106),(5,107),(6,108),(7,100),(8,101),(9,102),(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)], [(2,31,41),(3,42,32),(5,34,44),(6,45,35),(8,28,38),(9,39,29),(10,159,20),(11,14,17),(12,19,155),(13,162,23),(15,22,158),(16,156,26),(18,25,161),(21,24,27),(46,52,49),(47,67,60),(48,55,65),(50,70,63),(51,58,68),(53,64,57),(54,61,71),(56,62,59),(66,72,69),(73,76,79),(74,89,96),(75,94,84),(77,83,99),(78,97,87),(80,86,93),(81,91,90),(82,85,88),(92,95,98),(101,109,119),(102,120,110),(104,112,122),(105,123,113),(107,115,125),(108,126,116),(127,133,130),(128,148,141),(129,136,146),(131,151,144),(132,139,149),(134,145,138),(135,142,152),(137,143,140),(147,153,150),(154,157,160)], [(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),(10,156,23),(11,157,24),(12,158,25),(13,159,26),(14,160,27),(15,161,19),(16,162,20),(17,154,21),(18,155,22),(46,56,69),(47,57,70),(48,58,71),(49,59,72),(50,60,64),(51,61,65),(52,62,66),(53,63,67),(54,55,68),(73,98,85),(74,99,86),(75,91,87),(76,92,88),(77,93,89),(78,94,90),(79,95,82),(80,96,83),(81,97,84),(100,118,117),(101,119,109),(102,120,110),(103,121,111),(104,122,112),(105,123,113),(106,124,114),(107,125,115),(108,126,116),(127,137,150),(128,138,151),(129,139,152),(130,140,153),(131,141,145),(132,142,146),(133,143,147),(134,144,148),(135,136,149)], [(1,92,49),(2,80,60),(3,91,51),(4,95,52),(5,74,63),(6,94,54),(7,98,46),(8,77,57),(9,97,48),(10,142,123),(11,140,121),(12,128,109),(13,136,126),(14,143,124),(15,131,112),(16,139,120),(17,137,118),(18,134,115),(19,145,122),(20,129,102),(21,127,100),(22,148,125),(23,132,105),(24,130,103),(25,151,119),(26,135,108),(27,133,106),(28,89,47),(29,81,71),(30,76,72),(31,83,50),(32,75,65),(33,79,66),(34,86,53),(35,78,68),(36,73,69),(37,85,56),(38,93,70),(39,84,58),(40,88,59),(41,96,64),(42,87,61),(43,82,62),(44,99,67),(45,90,55),(101,158,138),(104,161,141),(107,155,144),(110,162,152),(111,157,153),(113,156,146),(114,160,147),(116,159,149),(117,154,150)], [(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)]])
70 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3P | 6A | ··· | 6H | 6I | ··· | 6P | 9A | ··· | 9R | 18A | ··· | 18R |
| 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 | He3⋊C3 | C2×C3≀C3 | C2×He3.C3 | C2×He3⋊C3 |
| kernel | C2×C32.27He3 | C32.27He3 | C2×C32⋊C9 | C6×He3 | C32⋊C9 | C3×He3 | 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×C32.27He3 ►in GL6(𝔽19)
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 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 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 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 | 0 | 0 | 7 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;
C2×C32.27He3 in GAP, Magma, Sage, TeX
C_2\times C_3^2._{27}{\rm He}_3 % in TeX
G:=Group("C2xC3^2.27He3"); // GroupNames label
G:=SmallGroup(486,66);
// by ID
G=gap.SmallGroup(486,66);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,338,873,735]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^3=1,f^3=c^-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,e*b*e^-1=b*c^-1,f*b*f^-1=b*d^-1,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*c^-1*d^-1*e>;
// generators/relations