direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C3×C6×He3, C34⋊13C6, C6.1C34, (C3×C6)⋊C33, (C33×C6)⋊3C3, C33⋊12(C3×C6), C3.1(C33×C6), (C32×C6)⋊5C32, C32⋊2(C32×C6), SmallGroup(486,251)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C6×He3
G = < a,b,c,d,e | a3=b6=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 1764 in 900 conjugacy classes, 468 normal (8 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C32, C32, C3×C6, C3×C6, He3, C33, C33, C33, C2×He3, C32×C6, C32×C6, C32×C6, C3×He3, C34, C6×He3, C33×C6, C32×He3, C3×C6×He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C34, C6×He3, C33×C6, C32×He3, C3×C6×He3
►On 162 points
Generators in S162
(1 105 135)(2 106 136)(3 107 137)(4 108 138)(5 103 133)(6 104 134)(7 158 16)(8 159 17)(9 160 18)(10 161 13)(11 162 14)(12 157 15)(19 59 45)(20 60 46)(21 55 47)(22 56 48)(23 57 43)(24 58 44)(25 67 120)(26 68 115)(27 69 116)(28 70 117)(29 71 118)(30 72 119)(31 73 65)(32 74 66)(33 75 61)(34 76 62)(35 77 63)(36 78 64)(37 130 84)(38 131 79)(39 132 80)(40 127 81)(41 128 82)(42 129 83)(49 124 85)(50 125 86)(51 126 87)(52 121 88)(53 122 89)(54 123 90)(91 142 102)(92 143 97)(93 144 98)(94 139 99)(95 140 100)(96 141 101)(109 149 156)(110 150 151)(111 145 152)(112 146 153)(113 147 154)(114 148 155)
(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 54 115)(2 49 116)(3 50 117)(4 51 118)(5 52 119)(6 53 120)(7 75 95)(8 76 96)(9 77 91)(10 78 92)(11 73 93)(12 74 94)(13 36 97)(14 31 98)(15 32 99)(16 33 100)(17 34 101)(18 35 102)(19 37 111)(20 38 112)(21 39 113)(22 40 114)(23 41 109)(24 42 110)(25 104 122)(26 105 123)(27 106 124)(28 107 125)(29 108 126)(30 103 121)(43 82 156)(44 83 151)(45 84 152)(46 79 153)(47 80 154)(48 81 155)(55 132 147)(56 127 148)(57 128 149)(58 129 150)(59 130 145)(60 131 146)(61 140 158)(62 141 159)(63 142 160)(64 143 161)(65 144 162)(66 139 157)(67 134 89)(68 135 90)(69 136 85)(70 137 86)(71 138 87)(72 133 88)
(1 48 96)(2 43 91)(3 44 92)(4 45 93)(5 46 94)(6 47 95)(7 53 80)(8 54 81)(9 49 82)(10 50 83)(11 51 84)(12 52 79)(13 86 129)(14 87 130)(15 88 131)(16 89 132)(17 90 127)(18 85 128)(19 144 108)(20 139 103)(21 140 104)(22 141 105)(23 142 106)(24 143 107)(25 113 61)(26 114 62)(27 109 63)(28 110 64)(29 111 65)(30 112 66)(31 71 145)(32 72 146)(33 67 147)(34 68 148)(35 69 149)(36 70 150)(37 162 126)(38 157 121)(39 158 122)(40 159 123)(41 160 124)(42 161 125)(55 100 134)(56 101 135)(57 102 136)(58 97 137)(59 98 138)(60 99 133)(73 118 152)(74 119 153)(75 120 154)(76 115 155)(77 116 156)(78 117 151)
(1 20 97)(2 21 98)(3 22 99)(4 23 100)(5 24 101)(6 19 102)(7 37 85)(8 38 86)(9 39 87)(10 40 88)(11 41 89)(12 42 90)(13 81 121)(14 82 122)(15 83 123)(16 84 124)(17 79 125)(18 80 126)(25 71 116)(26 72 117)(27 67 118)(28 68 119)(29 69 120)(30 70 115)(31 77 61)(32 78 62)(33 73 63)(34 74 64)(35 75 65)(36 76 66)(43 140 138)(44 141 133)(45 142 134)(46 143 135)(47 144 136)(48 139 137)(49 158 130)(50 159 131)(51 160 132)(52 161 127)(53 162 128)(54 157 129)(55 93 106)(56 94 107)(57 95 108)(58 96 103)(59 91 104)(60 92 105)(109 147 152)(110 148 153)(111 149 154)(112 150 155)(113 145 156)(114 146 151)
G:=sub<Sym(162)| (1,105,135)(2,106,136)(3,107,137)(4,108,138)(5,103,133)(6,104,134)(7,158,16)(8,159,17)(9,160,18)(10,161,13)(11,162,14)(12,157,15)(19,59,45)(20,60,46)(21,55,47)(22,56,48)(23,57,43)(24,58,44)(25,67,120)(26,68,115)(27,69,116)(28,70,117)(29,71,118)(30,72,119)(31,73,65)(32,74,66)(33,75,61)(34,76,62)(35,77,63)(36,78,64)(37,130,84)(38,131,79)(39,132,80)(40,127,81)(41,128,82)(42,129,83)(49,124,85)(50,125,86)(51,126,87)(52,121,88)(53,122,89)(54,123,90)(91,142,102)(92,143,97)(93,144,98)(94,139,99)(95,140,100)(96,141,101)(109,149,156)(110,150,151)(111,145,152)(112,146,153)(113,147,154)(114,148,155), (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), (1,54,115)(2,49,116)(3,50,117)(4,51,118)(5,52,119)(6,53,120)(7,75,95)(8,76,96)(9,77,91)(10,78,92)(11,73,93)(12,74,94)(13,36,97)(14,31,98)(15,32,99)(16,33,100)(17,34,101)(18,35,102)(19,37,111)(20,38,112)(21,39,113)(22,40,114)(23,41,109)(24,42,110)(25,104,122)(26,105,123)(27,106,124)(28,107,125)(29,108,126)(30,103,121)(43,82,156)(44,83,151)(45,84,152)(46,79,153)(47,80,154)(48,81,155)(55,132,147)(56,127,148)(57,128,149)(58,129,150)(59,130,145)(60,131,146)(61,140,158)(62,141,159)(63,142,160)(64,143,161)(65,144,162)(66,139,157)(67,134,89)(68,135,90)(69,136,85)(70,137,86)(71,138,87)(72,133,88), (1,48,96)(2,43,91)(3,44,92)(4,45,93)(5,46,94)(6,47,95)(7,53,80)(8,54,81)(9,49,82)(10,50,83)(11,51,84)(12,52,79)(13,86,129)(14,87,130)(15,88,131)(16,89,132)(17,90,127)(18,85,128)(19,144,108)(20,139,103)(21,140,104)(22,141,105)(23,142,106)(24,143,107)(25,113,61)(26,114,62)(27,109,63)(28,110,64)(29,111,65)(30,112,66)(31,71,145)(32,72,146)(33,67,147)(34,68,148)(35,69,149)(36,70,150)(37,162,126)(38,157,121)(39,158,122)(40,159,123)(41,160,124)(42,161,125)(55,100,134)(56,101,135)(57,102,136)(58,97,137)(59,98,138)(60,99,133)(73,118,152)(74,119,153)(75,120,154)(76,115,155)(77,116,156)(78,117,151), (1,20,97)(2,21,98)(3,22,99)(4,23,100)(5,24,101)(6,19,102)(7,37,85)(8,38,86)(9,39,87)(10,40,88)(11,41,89)(12,42,90)(13,81,121)(14,82,122)(15,83,123)(16,84,124)(17,79,125)(18,80,126)(25,71,116)(26,72,117)(27,67,118)(28,68,119)(29,69,120)(30,70,115)(31,77,61)(32,78,62)(33,73,63)(34,74,64)(35,75,65)(36,76,66)(43,140,138)(44,141,133)(45,142,134)(46,143,135)(47,144,136)(48,139,137)(49,158,130)(50,159,131)(51,160,132)(52,161,127)(53,162,128)(54,157,129)(55,93,106)(56,94,107)(57,95,108)(58,96,103)(59,91,104)(60,92,105)(109,147,152)(110,148,153)(111,149,154)(112,150,155)(113,145,156)(114,146,151)>;
G:=Group( (1,105,135)(2,106,136)(3,107,137)(4,108,138)(5,103,133)(6,104,134)(7,158,16)(8,159,17)(9,160,18)(10,161,13)(11,162,14)(12,157,15)(19,59,45)(20,60,46)(21,55,47)(22,56,48)(23,57,43)(24,58,44)(25,67,120)(26,68,115)(27,69,116)(28,70,117)(29,71,118)(30,72,119)(31,73,65)(32,74,66)(33,75,61)(34,76,62)(35,77,63)(36,78,64)(37,130,84)(38,131,79)(39,132,80)(40,127,81)(41,128,82)(42,129,83)(49,124,85)(50,125,86)(51,126,87)(52,121,88)(53,122,89)(54,123,90)(91,142,102)(92,143,97)(93,144,98)(94,139,99)(95,140,100)(96,141,101)(109,149,156)(110,150,151)(111,145,152)(112,146,153)(113,147,154)(114,148,155), (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), (1,54,115)(2,49,116)(3,50,117)(4,51,118)(5,52,119)(6,53,120)(7,75,95)(8,76,96)(9,77,91)(10,78,92)(11,73,93)(12,74,94)(13,36,97)(14,31,98)(15,32,99)(16,33,100)(17,34,101)(18,35,102)(19,37,111)(20,38,112)(21,39,113)(22,40,114)(23,41,109)(24,42,110)(25,104,122)(26,105,123)(27,106,124)(28,107,125)(29,108,126)(30,103,121)(43,82,156)(44,83,151)(45,84,152)(46,79,153)(47,80,154)(48,81,155)(55,132,147)(56,127,148)(57,128,149)(58,129,150)(59,130,145)(60,131,146)(61,140,158)(62,141,159)(63,142,160)(64,143,161)(65,144,162)(66,139,157)(67,134,89)(68,135,90)(69,136,85)(70,137,86)(71,138,87)(72,133,88), (1,48,96)(2,43,91)(3,44,92)(4,45,93)(5,46,94)(6,47,95)(7,53,80)(8,54,81)(9,49,82)(10,50,83)(11,51,84)(12,52,79)(13,86,129)(14,87,130)(15,88,131)(16,89,132)(17,90,127)(18,85,128)(19,144,108)(20,139,103)(21,140,104)(22,141,105)(23,142,106)(24,143,107)(25,113,61)(26,114,62)(27,109,63)(28,110,64)(29,111,65)(30,112,66)(31,71,145)(32,72,146)(33,67,147)(34,68,148)(35,69,149)(36,70,150)(37,162,126)(38,157,121)(39,158,122)(40,159,123)(41,160,124)(42,161,125)(55,100,134)(56,101,135)(57,102,136)(58,97,137)(59,98,138)(60,99,133)(73,118,152)(74,119,153)(75,120,154)(76,115,155)(77,116,156)(78,117,151), (1,20,97)(2,21,98)(3,22,99)(4,23,100)(5,24,101)(6,19,102)(7,37,85)(8,38,86)(9,39,87)(10,40,88)(11,41,89)(12,42,90)(13,81,121)(14,82,122)(15,83,123)(16,84,124)(17,79,125)(18,80,126)(25,71,116)(26,72,117)(27,67,118)(28,68,119)(29,69,120)(30,70,115)(31,77,61)(32,78,62)(33,73,63)(34,74,64)(35,75,65)(36,76,66)(43,140,138)(44,141,133)(45,142,134)(46,143,135)(47,144,136)(48,139,137)(49,158,130)(50,159,131)(51,160,132)(52,161,127)(53,162,128)(54,157,129)(55,93,106)(56,94,107)(57,95,108)(58,96,103)(59,91,104)(60,92,105)(109,147,152)(110,148,153)(111,149,154)(112,150,155)(113,145,156)(114,146,151) );
G=PermutationGroup([[(1,105,135),(2,106,136),(3,107,137),(4,108,138),(5,103,133),(6,104,134),(7,158,16),(8,159,17),(9,160,18),(10,161,13),(11,162,14),(12,157,15),(19,59,45),(20,60,46),(21,55,47),(22,56,48),(23,57,43),(24,58,44),(25,67,120),(26,68,115),(27,69,116),(28,70,117),(29,71,118),(30,72,119),(31,73,65),(32,74,66),(33,75,61),(34,76,62),(35,77,63),(36,78,64),(37,130,84),(38,131,79),(39,132,80),(40,127,81),(41,128,82),(42,129,83),(49,124,85),(50,125,86),(51,126,87),(52,121,88),(53,122,89),(54,123,90),(91,142,102),(92,143,97),(93,144,98),(94,139,99),(95,140,100),(96,141,101),(109,149,156),(110,150,151),(111,145,152),(112,146,153),(113,147,154),(114,148,155)], [(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)], [(1,54,115),(2,49,116),(3,50,117),(4,51,118),(5,52,119),(6,53,120),(7,75,95),(8,76,96),(9,77,91),(10,78,92),(11,73,93),(12,74,94),(13,36,97),(14,31,98),(15,32,99),(16,33,100),(17,34,101),(18,35,102),(19,37,111),(20,38,112),(21,39,113),(22,40,114),(23,41,109),(24,42,110),(25,104,122),(26,105,123),(27,106,124),(28,107,125),(29,108,126),(30,103,121),(43,82,156),(44,83,151),(45,84,152),(46,79,153),(47,80,154),(48,81,155),(55,132,147),(56,127,148),(57,128,149),(58,129,150),(59,130,145),(60,131,146),(61,140,158),(62,141,159),(63,142,160),(64,143,161),(65,144,162),(66,139,157),(67,134,89),(68,135,90),(69,136,85),(70,137,86),(71,138,87),(72,133,88)], [(1,48,96),(2,43,91),(3,44,92),(4,45,93),(5,46,94),(6,47,95),(7,53,80),(8,54,81),(9,49,82),(10,50,83),(11,51,84),(12,52,79),(13,86,129),(14,87,130),(15,88,131),(16,89,132),(17,90,127),(18,85,128),(19,144,108),(20,139,103),(21,140,104),(22,141,105),(23,142,106),(24,143,107),(25,113,61),(26,114,62),(27,109,63),(28,110,64),(29,111,65),(30,112,66),(31,71,145),(32,72,146),(33,67,147),(34,68,148),(35,69,149),(36,70,150),(37,162,126),(38,157,121),(39,158,122),(40,159,123),(41,160,124),(42,161,125),(55,100,134),(56,101,135),(57,102,136),(58,97,137),(59,98,138),(60,99,133),(73,118,152),(74,119,153),(75,120,154),(76,115,155),(77,116,156),(78,117,151)], [(1,20,97),(2,21,98),(3,22,99),(4,23,100),(5,24,101),(6,19,102),(7,37,85),(8,38,86),(9,39,87),(10,40,88),(11,41,89),(12,42,90),(13,81,121),(14,82,122),(15,83,123),(16,84,124),(17,79,125),(18,80,126),(25,71,116),(26,72,117),(27,67,118),(28,68,119),(29,69,120),(30,70,115),(31,77,61),(32,78,62),(33,73,63),(34,74,64),(35,75,65),(36,76,66),(43,140,138),(44,141,133),(45,142,134),(46,143,135),(47,144,136),(48,139,137),(49,158,130),(50,159,131),(51,160,132),(52,161,127),(53,162,128),(54,157,129),(55,93,106),(56,94,107),(57,95,108),(58,96,103),(59,91,104),(60,92,105),(109,147,152),(110,148,153),(111,149,154),(112,150,155),(113,145,156),(114,146,151)]])
198 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3Z | 3AA | ··· | 3CT | 6A | ··· | 6Z | 6AA | ··· | 6CT |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
198 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | He3 | C2×He3 |
| kernel | C3×C6×He3 | C32×He3 | C6×He3 | C33×C6 | C3×He3 | C34 | C3×C6 | C32 |
| # reps | 1 | 1 | 72 | 8 | 72 | 8 | 18 | 18 |
Matrix representation of C3×C6×He3 ►in GL5(𝔽7)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 5 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 5 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 1 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(7))| [4,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[3,0,0,0,0,0,5,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,1,0,0,0,0,0,5,0,1,0,0,5,0,5,0,0,5,1,2],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,2,2,0,0,0,1,0,4] >;
C3×C6×He3 in GAP, Magma, Sage, TeX
C_3\times C_6\times {\rm He}_3 % in TeX
G:=Group("C3xC6xHe3"); // GroupNames label
G:=SmallGroup(486,251);
// by ID
G=gap.SmallGroup(486,251);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1520]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations