direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×He3⋊C9, He3⋊2C18, (C2×He3)⋊C9, C6.7C3≀C3, C32⋊C9⋊13C6, (C32×C9)⋊27C6, (C32×C18)⋊1C3, (C6×He3).3C3, (C3×C6).24He3, C6.7(C32⋊C9), (C3×He3).16C6, C32.1(C3×C18), C33.34(C3×C6), C6.7(He3.C3), C32.22(C2×He3), C6.5(He3⋊C3), (C32×C6).22C32, (C3×C6).13- 1+2, C32.1(C2×3- 1+2), C3.2(C2×C3≀C3), (C3×C6).1(C3×C9), (C2×C32⋊C9)⋊5C3, C3.7(C2×C32⋊C9), C3.2(C2×He3.C3), C3.2(C2×He3⋊C3), SmallGroup(486,77)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×He3⋊C9
G = < a,b,c,d,e | a2=b3=c3=d3=e9=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe-1=bc-1, cd=dc, ce=ec, ede-1=b-1c-1d >
Subgroups: 306 in 94 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, He3, He3, C33, C33, C3×C18, C2×He3, C2×He3, C32×C6, C32×C6, C32⋊C9, C32×C9, C3×He3, C2×C32⋊C9, C32×C18, C6×He3, He3⋊C9, C2×He3⋊C9
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, C3≀C3, He3.C3, He3⋊C3, C2×C32⋊C9, C2×C3≀C3, C2×He3.C3, C2×He3⋊C3, He3⋊C9, C2×He3⋊C9
►On 162 points
Generators in S162
(1 141)(2 142)(3 143)(4 144)(5 136)(6 137)(7 138)(8 139)(9 140)(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 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 82)(36 83)(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)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 109)(63 110)(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 21 78)(3 76 25)(5 24 81)(6 79 19)(8 27 75)(9 73 22)(10 46 85)(11 83 50)(12 15 18)(13 49 88)(14 86 53)(16 52 82)(17 89 47)(20 23 26)(28 31 34)(29 91 127)(30 134 95)(32 94 130)(33 128 98)(35 97 133)(36 131 92)(37 111 68)(38 41 44)(39 67 116)(40 114 71)(42 70 110)(43 117 65)(45 64 113)(48 51 54)(55 149 118)(56 59 62)(57 126 145)(58 152 121)(60 120 148)(61 146 124)(63 123 151)(66 69 72)(74 77 80)(84 87 90)(93 96 99)(100 137 160)(101 104 107)(102 159 142)(103 140 154)(105 162 136)(106 143 157)(108 156 139)(109 112 115)(119 122 125)(129 132 135)(138 141 144)(147 150 153)(155 158 161)
(1 80 26)(2 81 27)(3 73 19)(4 74 20)(5 75 21)(6 76 22)(7 77 23)(8 78 24)(9 79 25)(10 88 52)(11 89 53)(12 90 54)(13 82 46)(14 83 47)(15 84 48)(16 85 49)(17 86 50)(18 87 51)(28 129 96)(29 130 97)(30 131 98)(31 132 99)(32 133 91)(33 134 92)(34 135 93)(35 127 94)(36 128 95)(37 117 71)(38 109 72)(39 110 64)(40 111 65)(41 112 66)(42 113 67)(43 114 68)(44 115 69)(45 116 70)(55 146 121)(56 147 122)(57 148 123)(58 149 124)(59 150 125)(60 151 126)(61 152 118)(62 153 119)(63 145 120)(100 143 154)(101 144 155)(102 136 156)(103 137 157)(104 138 158)(105 139 159)(106 140 160)(107 141 161)(108 142 162)
(1 42 14)(2 65 51)(3 72 13)(4 45 17)(5 68 54)(6 66 16)(7 39 11)(8 71 48)(9 69 10)(12 75 43)(15 78 37)(18 81 40)(19 109 46)(20 70 50)(21 114 90)(22 112 49)(23 64 53)(24 117 84)(25 115 52)(26 67 47)(27 111 87)(28 105 61)(29 157 122)(30 155 60)(31 108 55)(32 160 125)(33 158 63)(34 102 58)(35 154 119)(36 161 57)(38 82 73)(41 85 76)(44 88 79)(56 130 103)(59 133 106)(62 127 100)(74 116 86)(77 110 89)(80 113 83)(91 140 150)(92 138 120)(93 156 124)(94 143 153)(95 141 123)(96 159 118)(97 137 147)(98 144 126)(99 162 121)(101 151 131)(104 145 134)(107 148 128)(129 139 152)(132 142 146)(135 136 149)
(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)| (1,141)(2,142)(3,143)(4,144)(5,136)(6,137)(7,138)(8,139)(9,140)(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)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,82)(36,83)(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)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,109)(63,110)(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)(2,21,78)(3,76,25)(5,24,81)(6,79,19)(8,27,75)(9,73,22)(10,46,85)(11,83,50)(12,15,18)(13,49,88)(14,86,53)(16,52,82)(17,89,47)(20,23,26)(28,31,34)(29,91,127)(30,134,95)(32,94,130)(33,128,98)(35,97,133)(36,131,92)(37,111,68)(38,41,44)(39,67,116)(40,114,71)(42,70,110)(43,117,65)(45,64,113)(48,51,54)(55,149,118)(56,59,62)(57,126,145)(58,152,121)(60,120,148)(61,146,124)(63,123,151)(66,69,72)(74,77,80)(84,87,90)(93,96,99)(100,137,160)(101,104,107)(102,159,142)(103,140,154)(105,162,136)(106,143,157)(108,156,139)(109,112,115)(119,122,125)(129,132,135)(138,141,144)(147,150,153)(155,158,161), (1,80,26)(2,81,27)(3,73,19)(4,74,20)(5,75,21)(6,76,22)(7,77,23)(8,78,24)(9,79,25)(10,88,52)(11,89,53)(12,90,54)(13,82,46)(14,83,47)(15,84,48)(16,85,49)(17,86,50)(18,87,51)(28,129,96)(29,130,97)(30,131,98)(31,132,99)(32,133,91)(33,134,92)(34,135,93)(35,127,94)(36,128,95)(37,117,71)(38,109,72)(39,110,64)(40,111,65)(41,112,66)(42,113,67)(43,114,68)(44,115,69)(45,116,70)(55,146,121)(56,147,122)(57,148,123)(58,149,124)(59,150,125)(60,151,126)(61,152,118)(62,153,119)(63,145,120)(100,143,154)(101,144,155)(102,136,156)(103,137,157)(104,138,158)(105,139,159)(106,140,160)(107,141,161)(108,142,162), (1,42,14)(2,65,51)(3,72,13)(4,45,17)(5,68,54)(6,66,16)(7,39,11)(8,71,48)(9,69,10)(12,75,43)(15,78,37)(18,81,40)(19,109,46)(20,70,50)(21,114,90)(22,112,49)(23,64,53)(24,117,84)(25,115,52)(26,67,47)(27,111,87)(28,105,61)(29,157,122)(30,155,60)(31,108,55)(32,160,125)(33,158,63)(34,102,58)(35,154,119)(36,161,57)(38,82,73)(41,85,76)(44,88,79)(56,130,103)(59,133,106)(62,127,100)(74,116,86)(77,110,89)(80,113,83)(91,140,150)(92,138,120)(93,156,124)(94,143,153)(95,141,123)(96,159,118)(97,137,147)(98,144,126)(99,162,121)(101,151,131)(104,145,134)(107,148,128)(129,139,152)(132,142,146)(135,136,149), (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( (1,141)(2,142)(3,143)(4,144)(5,136)(6,137)(7,138)(8,139)(9,140)(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)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,82)(36,83)(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)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,109)(63,110)(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)(2,21,78)(3,76,25)(5,24,81)(6,79,19)(8,27,75)(9,73,22)(10,46,85)(11,83,50)(12,15,18)(13,49,88)(14,86,53)(16,52,82)(17,89,47)(20,23,26)(28,31,34)(29,91,127)(30,134,95)(32,94,130)(33,128,98)(35,97,133)(36,131,92)(37,111,68)(38,41,44)(39,67,116)(40,114,71)(42,70,110)(43,117,65)(45,64,113)(48,51,54)(55,149,118)(56,59,62)(57,126,145)(58,152,121)(60,120,148)(61,146,124)(63,123,151)(66,69,72)(74,77,80)(84,87,90)(93,96,99)(100,137,160)(101,104,107)(102,159,142)(103,140,154)(105,162,136)(106,143,157)(108,156,139)(109,112,115)(119,122,125)(129,132,135)(138,141,144)(147,150,153)(155,158,161), (1,80,26)(2,81,27)(3,73,19)(4,74,20)(5,75,21)(6,76,22)(7,77,23)(8,78,24)(9,79,25)(10,88,52)(11,89,53)(12,90,54)(13,82,46)(14,83,47)(15,84,48)(16,85,49)(17,86,50)(18,87,51)(28,129,96)(29,130,97)(30,131,98)(31,132,99)(32,133,91)(33,134,92)(34,135,93)(35,127,94)(36,128,95)(37,117,71)(38,109,72)(39,110,64)(40,111,65)(41,112,66)(42,113,67)(43,114,68)(44,115,69)(45,116,70)(55,146,121)(56,147,122)(57,148,123)(58,149,124)(59,150,125)(60,151,126)(61,152,118)(62,153,119)(63,145,120)(100,143,154)(101,144,155)(102,136,156)(103,137,157)(104,138,158)(105,139,159)(106,140,160)(107,141,161)(108,142,162), (1,42,14)(2,65,51)(3,72,13)(4,45,17)(5,68,54)(6,66,16)(7,39,11)(8,71,48)(9,69,10)(12,75,43)(15,78,37)(18,81,40)(19,109,46)(20,70,50)(21,114,90)(22,112,49)(23,64,53)(24,117,84)(25,115,52)(26,67,47)(27,111,87)(28,105,61)(29,157,122)(30,155,60)(31,108,55)(32,160,125)(33,158,63)(34,102,58)(35,154,119)(36,161,57)(38,82,73)(41,85,76)(44,88,79)(56,130,103)(59,133,106)(62,127,100)(74,116,86)(77,110,89)(80,113,83)(91,140,150)(92,138,120)(93,156,124)(94,143,153)(95,141,123)(96,159,118)(97,137,147)(98,144,126)(99,162,121)(101,151,131)(104,145,134)(107,148,128)(129,139,152)(132,142,146)(135,136,149), (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([[(1,141),(2,142),(3,143),(4,144),(5,136),(6,137),(7,138),(8,139),(9,140),(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),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,82),(36,83),(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),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(61,117),(62,109),(63,110),(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),(2,21,78),(3,76,25),(5,24,81),(6,79,19),(8,27,75),(9,73,22),(10,46,85),(11,83,50),(12,15,18),(13,49,88),(14,86,53),(16,52,82),(17,89,47),(20,23,26),(28,31,34),(29,91,127),(30,134,95),(32,94,130),(33,128,98),(35,97,133),(36,131,92),(37,111,68),(38,41,44),(39,67,116),(40,114,71),(42,70,110),(43,117,65),(45,64,113),(48,51,54),(55,149,118),(56,59,62),(57,126,145),(58,152,121),(60,120,148),(61,146,124),(63,123,151),(66,69,72),(74,77,80),(84,87,90),(93,96,99),(100,137,160),(101,104,107),(102,159,142),(103,140,154),(105,162,136),(106,143,157),(108,156,139),(109,112,115),(119,122,125),(129,132,135),(138,141,144),(147,150,153),(155,158,161)], [(1,80,26),(2,81,27),(3,73,19),(4,74,20),(5,75,21),(6,76,22),(7,77,23),(8,78,24),(9,79,25),(10,88,52),(11,89,53),(12,90,54),(13,82,46),(14,83,47),(15,84,48),(16,85,49),(17,86,50),(18,87,51),(28,129,96),(29,130,97),(30,131,98),(31,132,99),(32,133,91),(33,134,92),(34,135,93),(35,127,94),(36,128,95),(37,117,71),(38,109,72),(39,110,64),(40,111,65),(41,112,66),(42,113,67),(43,114,68),(44,115,69),(45,116,70),(55,146,121),(56,147,122),(57,148,123),(58,149,124),(59,150,125),(60,151,126),(61,152,118),(62,153,119),(63,145,120),(100,143,154),(101,144,155),(102,136,156),(103,137,157),(104,138,158),(105,139,159),(106,140,160),(107,141,161),(108,142,162)], [(1,42,14),(2,65,51),(3,72,13),(4,45,17),(5,68,54),(6,66,16),(7,39,11),(8,71,48),(9,69,10),(12,75,43),(15,78,37),(18,81,40),(19,109,46),(20,70,50),(21,114,90),(22,112,49),(23,64,53),(24,117,84),(25,115,52),(26,67,47),(27,111,87),(28,105,61),(29,157,122),(30,155,60),(31,108,55),(32,160,125),(33,158,63),(34,102,58),(35,154,119),(36,161,57),(38,82,73),(41,85,76),(44,88,79),(56,130,103),(59,133,106),(62,127,100),(74,116,86),(77,110,89),(80,113,83),(91,140,150),(92,138,120),(93,156,124),(94,143,153),(95,141,123),(96,159,118),(97,137,147),(98,144,126),(99,162,121),(101,151,131),(104,145,134),(107,148,128),(129,139,152),(132,142,146),(135,136,149)], [(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)]])
102 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3N | 3O | ··· | 3T | 6A | ··· | 6H | 6I | ··· | 6N | 6O | ··· | 6T | 9A | ··· | 9R | 9S | ··· | 9AD | 18A | ··· | 18R | 18S | ··· | 18AD |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 3 | ··· | 3 | 9 | ··· | 9 | 3 | ··· | 3 | 9 | ··· | 9 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | C9 | C18 | He3 | 3- 1+2 | C2×He3 | C2×3- 1+2 | C3≀C3 | He3.C3 | He3⋊C3 | C2×C3≀C3 | C2×He3.C3 | C2×He3⋊C3 |
| kernel | C2×He3⋊C9 | He3⋊C9 | C2×C32⋊C9 | C32×C18 | C6×He3 | C32⋊C9 | C32×C9 | C3×He3 | C2×He3 | He3 | C3×C6 | C3×C6 | C32 | C32 | C6 | C6 | C6 | C3 | C3 | C3 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 2 | 18 | 18 | 2 | 4 | 2 | 4 | 6 | 6 | 6 | 6 | 6 | 6 |
Matrix representation of C2×He3⋊C9 ►in GL4(𝔽19) generated by
| 18 | 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 | 1 | 0 |
| 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 7 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 11 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 7 | 0 | 0 |
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,7,0,0,0,0,1,0,0,0,0,11],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[7,0,0,0,0,0,0,11,0,11,0,0,0,0,11,0],[16,0,0,0,0,0,0,7,0,11,0,0,0,0,7,0] >;
C2×He3⋊C9 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes C_9 % in TeX
G:=Group("C2xHe3:C9"); // GroupNames label
G:=SmallGroup(486,77);
// by ID
G=gap.SmallGroup(486,77);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,2169,735]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^9=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=b*c^-1,c*d=d*c,c*e=e*c,e*d*e^-1=b^-1*c^-1*d>;
// generators/relations