direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C6×He3⋊C3, (C6×He3)⋊4C3, He3⋊3(C3×C6), C6.9(C3×He3), C3.9(C6×He3), (C3×He3)⋊16C6, (C32×C18)⋊6C3, (C3×C18)⋊7C32, (C32×C9)⋊32C6, (C3×C6).3C33, (C3×C6).29He3, C33.44(C3×C6), (C2×He3)⋊2C32, C32.3(C32×C6), C32.27(C2×He3), (C32×C6).32C32, (C3×C9)⋊18(C3×C6), SmallGroup(486,212)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×He3⋊C3
G = < a,b,c,d,e | a6=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe-1=bc-1, cd=dc, ce=ec, ede-1=bcd >
Subgroups: 576 in 168 conjugacy classes, 72 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, He3, He3, C33, C33, C3×C18, C3×C18, C2×He3, C2×He3, C32×C6, C32×C6, He3⋊C3, C32×C9, C3×He3, C2×He3⋊C3, C32×C18, C6×He3, C3×He3⋊C3, C6×He3⋊C3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, He3⋊C3, C3×He3, C2×He3⋊C3, C6×He3, C3×He3⋊C3, C6×He3⋊C3
►On 162 points
Generators in S162
(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 22 46)(2 23 47)(3 24 48)(4 19 43)(5 20 44)(6 21 45)(7 49 86)(8 50 87)(9 51 88)(10 52 89)(11 53 90)(12 54 85)(13 93 136)(14 94 137)(15 95 138)(16 96 133)(17 91 134)(18 92 135)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 83 126)(38 84 121)(39 79 122)(40 80 123)(41 81 124)(42 82 125)(55 140 99)(56 141 100)(57 142 101)(58 143 102)(59 144 97)(60 139 98)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 119 162)(74 120 157)(75 115 158)(76 116 159)(77 117 160)(78 118 161)(103 107 105)(104 108 106)(109 113 111)(110 114 112)(127 131 129)(128 132 130)(145 149 147)(146 150 148)(151 155 153)(152 156 154)
(1 44 24)(2 45 19)(3 46 20)(4 47 21)(5 48 22)(6 43 23)(7 51 90)(8 52 85)(9 53 86)(10 54 87)(11 49 88)(12 50 89)(13 134 95)(14 135 96)(15 136 91)(16 137 92)(17 138 93)(18 133 94)(25 149 62)(26 150 63)(27 145 64)(28 146 65)(29 147 66)(30 148 61)(31 104 71)(32 105 72)(33 106 67)(34 107 68)(35 108 69)(36 103 70)(37 79 124)(38 80 125)(39 81 126)(40 82 121)(41 83 122)(42 84 123)(55 97 142)(56 98 143)(57 99 144)(58 100 139)(59 101 140)(60 102 141)(73 115 160)(74 116 161)(75 117 162)(76 118 157)(77 119 158)(78 120 159)(109 130 154)(110 131 155)(111 132 156)(112 127 151)(113 128 152)(114 129 153)
(1 8 149)(2 9 150)(3 10 145)(4 11 146)(5 12 147)(6 7 148)(13 160 128)(14 161 129)(15 162 130)(16 157 131)(17 158 132)(18 159 127)(19 86 26)(20 87 27)(21 88 28)(22 89 29)(23 90 30)(24 85 25)(31 98 37)(32 99 38)(33 100 39)(34 101 40)(35 102 41)(36 97 42)(43 51 61)(44 52 62)(45 53 63)(46 54 64)(47 49 65)(48 50 66)(55 123 70)(56 124 71)(57 125 72)(58 126 67)(59 121 68)(60 122 69)(73 152 134)(74 153 135)(75 154 136)(76 155 137)(77 156 138)(78 151 133)(79 104 143)(80 105 144)(81 106 139)(82 107 140)(83 108 141)(84 103 142)(91 117 109)(92 118 110)(93 119 111)(94 120 112)(95 115 113)(96 116 114)
(1 79 127)(2 80 128)(3 81 129)(4 82 130)(5 83 131)(6 84 132)(7 105 136)(8 106 137)(9 107 138)(10 108 133)(11 103 134)(12 104 135)(13 88 36)(14 89 31)(15 90 32)(16 85 33)(17 86 34)(18 87 35)(19 38 113)(20 39 114)(21 40 109)(22 41 110)(23 42 111)(24 37 112)(25 141 161)(26 142 162)(27 143 157)(28 144 158)(29 139 159)(30 140 160)(43 123 156)(44 124 151)(45 125 152)(46 126 153)(47 121 154)(48 122 155)(49 70 95)(50 71 96)(51 72 91)(52 67 92)(53 68 93)(54 69 94)(55 75 150)(56 76 145)(57 77 146)(58 78 147)(59 73 148)(60 74 149)(61 101 115)(62 102 116)(63 97 117)(64 98 118)(65 99 119)(66 100 120)
G:=sub<Sym(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), (1,22,46)(2,23,47)(3,24,48)(4,19,43)(5,20,44)(6,21,45)(7,49,86)(8,50,87)(9,51,88)(10,52,89)(11,53,90)(12,54,85)(13,93,136)(14,94,137)(15,95,138)(16,96,133)(17,91,134)(18,92,135)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,83,126)(38,84,121)(39,79,122)(40,80,123)(41,81,124)(42,82,125)(55,140,99)(56,141,100)(57,142,101)(58,143,102)(59,144,97)(60,139,98)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,119,162)(74,120,157)(75,115,158)(76,116,159)(77,117,160)(78,118,161)(103,107,105)(104,108,106)(109,113,111)(110,114,112)(127,131,129)(128,132,130)(145,149,147)(146,150,148)(151,155,153)(152,156,154), (1,44,24)(2,45,19)(3,46,20)(4,47,21)(5,48,22)(6,43,23)(7,51,90)(8,52,85)(9,53,86)(10,54,87)(11,49,88)(12,50,89)(13,134,95)(14,135,96)(15,136,91)(16,137,92)(17,138,93)(18,133,94)(25,149,62)(26,150,63)(27,145,64)(28,146,65)(29,147,66)(30,148,61)(31,104,71)(32,105,72)(33,106,67)(34,107,68)(35,108,69)(36,103,70)(37,79,124)(38,80,125)(39,81,126)(40,82,121)(41,83,122)(42,84,123)(55,97,142)(56,98,143)(57,99,144)(58,100,139)(59,101,140)(60,102,141)(73,115,160)(74,116,161)(75,117,162)(76,118,157)(77,119,158)(78,120,159)(109,130,154)(110,131,155)(111,132,156)(112,127,151)(113,128,152)(114,129,153), (1,8,149)(2,9,150)(3,10,145)(4,11,146)(5,12,147)(6,7,148)(13,160,128)(14,161,129)(15,162,130)(16,157,131)(17,158,132)(18,159,127)(19,86,26)(20,87,27)(21,88,28)(22,89,29)(23,90,30)(24,85,25)(31,98,37)(32,99,38)(33,100,39)(34,101,40)(35,102,41)(36,97,42)(43,51,61)(44,52,62)(45,53,63)(46,54,64)(47,49,65)(48,50,66)(55,123,70)(56,124,71)(57,125,72)(58,126,67)(59,121,68)(60,122,69)(73,152,134)(74,153,135)(75,154,136)(76,155,137)(77,156,138)(78,151,133)(79,104,143)(80,105,144)(81,106,139)(82,107,140)(83,108,141)(84,103,142)(91,117,109)(92,118,110)(93,119,111)(94,120,112)(95,115,113)(96,116,114), (1,79,127)(2,80,128)(3,81,129)(4,82,130)(5,83,131)(6,84,132)(7,105,136)(8,106,137)(9,107,138)(10,108,133)(11,103,134)(12,104,135)(13,88,36)(14,89,31)(15,90,32)(16,85,33)(17,86,34)(18,87,35)(19,38,113)(20,39,114)(21,40,109)(22,41,110)(23,42,111)(24,37,112)(25,141,161)(26,142,162)(27,143,157)(28,144,158)(29,139,159)(30,140,160)(43,123,156)(44,124,151)(45,125,152)(46,126,153)(47,121,154)(48,122,155)(49,70,95)(50,71,96)(51,72,91)(52,67,92)(53,68,93)(54,69,94)(55,75,150)(56,76,145)(57,77,146)(58,78,147)(59,73,148)(60,74,149)(61,101,115)(62,102,116)(63,97,117)(64,98,118)(65,99,119)(66,100,120)>;
G:=Group( (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), (1,22,46)(2,23,47)(3,24,48)(4,19,43)(5,20,44)(6,21,45)(7,49,86)(8,50,87)(9,51,88)(10,52,89)(11,53,90)(12,54,85)(13,93,136)(14,94,137)(15,95,138)(16,96,133)(17,91,134)(18,92,135)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,83,126)(38,84,121)(39,79,122)(40,80,123)(41,81,124)(42,82,125)(55,140,99)(56,141,100)(57,142,101)(58,143,102)(59,144,97)(60,139,98)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,119,162)(74,120,157)(75,115,158)(76,116,159)(77,117,160)(78,118,161)(103,107,105)(104,108,106)(109,113,111)(110,114,112)(127,131,129)(128,132,130)(145,149,147)(146,150,148)(151,155,153)(152,156,154), (1,44,24)(2,45,19)(3,46,20)(4,47,21)(5,48,22)(6,43,23)(7,51,90)(8,52,85)(9,53,86)(10,54,87)(11,49,88)(12,50,89)(13,134,95)(14,135,96)(15,136,91)(16,137,92)(17,138,93)(18,133,94)(25,149,62)(26,150,63)(27,145,64)(28,146,65)(29,147,66)(30,148,61)(31,104,71)(32,105,72)(33,106,67)(34,107,68)(35,108,69)(36,103,70)(37,79,124)(38,80,125)(39,81,126)(40,82,121)(41,83,122)(42,84,123)(55,97,142)(56,98,143)(57,99,144)(58,100,139)(59,101,140)(60,102,141)(73,115,160)(74,116,161)(75,117,162)(76,118,157)(77,119,158)(78,120,159)(109,130,154)(110,131,155)(111,132,156)(112,127,151)(113,128,152)(114,129,153), (1,8,149)(2,9,150)(3,10,145)(4,11,146)(5,12,147)(6,7,148)(13,160,128)(14,161,129)(15,162,130)(16,157,131)(17,158,132)(18,159,127)(19,86,26)(20,87,27)(21,88,28)(22,89,29)(23,90,30)(24,85,25)(31,98,37)(32,99,38)(33,100,39)(34,101,40)(35,102,41)(36,97,42)(43,51,61)(44,52,62)(45,53,63)(46,54,64)(47,49,65)(48,50,66)(55,123,70)(56,124,71)(57,125,72)(58,126,67)(59,121,68)(60,122,69)(73,152,134)(74,153,135)(75,154,136)(76,155,137)(77,156,138)(78,151,133)(79,104,143)(80,105,144)(81,106,139)(82,107,140)(83,108,141)(84,103,142)(91,117,109)(92,118,110)(93,119,111)(94,120,112)(95,115,113)(96,116,114), (1,79,127)(2,80,128)(3,81,129)(4,82,130)(5,83,131)(6,84,132)(7,105,136)(8,106,137)(9,107,138)(10,108,133)(11,103,134)(12,104,135)(13,88,36)(14,89,31)(15,90,32)(16,85,33)(17,86,34)(18,87,35)(19,38,113)(20,39,114)(21,40,109)(22,41,110)(23,42,111)(24,37,112)(25,141,161)(26,142,162)(27,143,157)(28,144,158)(29,139,159)(30,140,160)(43,123,156)(44,124,151)(45,125,152)(46,126,153)(47,121,154)(48,122,155)(49,70,95)(50,71,96)(51,72,91)(52,67,92)(53,68,93)(54,69,94)(55,75,150)(56,76,145)(57,77,146)(58,78,147)(59,73,148)(60,74,149)(61,101,115)(62,102,116)(63,97,117)(64,98,118)(65,99,119)(66,100,120) );
G=PermutationGroup([[(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)], [(1,22,46),(2,23,47),(3,24,48),(4,19,43),(5,20,44),(6,21,45),(7,49,86),(8,50,87),(9,51,88),(10,52,89),(11,53,90),(12,54,85),(13,93,136),(14,94,137),(15,95,138),(16,96,133),(17,91,134),(18,92,135),(25,29,27),(26,30,28),(31,35,33),(32,36,34),(37,83,126),(38,84,121),(39,79,122),(40,80,123),(41,81,124),(42,82,125),(55,140,99),(56,141,100),(57,142,101),(58,143,102),(59,144,97),(60,139,98),(61,65,63),(62,66,64),(67,71,69),(68,72,70),(73,119,162),(74,120,157),(75,115,158),(76,116,159),(77,117,160),(78,118,161),(103,107,105),(104,108,106),(109,113,111),(110,114,112),(127,131,129),(128,132,130),(145,149,147),(146,150,148),(151,155,153),(152,156,154)], [(1,44,24),(2,45,19),(3,46,20),(4,47,21),(5,48,22),(6,43,23),(7,51,90),(8,52,85),(9,53,86),(10,54,87),(11,49,88),(12,50,89),(13,134,95),(14,135,96),(15,136,91),(16,137,92),(17,138,93),(18,133,94),(25,149,62),(26,150,63),(27,145,64),(28,146,65),(29,147,66),(30,148,61),(31,104,71),(32,105,72),(33,106,67),(34,107,68),(35,108,69),(36,103,70),(37,79,124),(38,80,125),(39,81,126),(40,82,121),(41,83,122),(42,84,123),(55,97,142),(56,98,143),(57,99,144),(58,100,139),(59,101,140),(60,102,141),(73,115,160),(74,116,161),(75,117,162),(76,118,157),(77,119,158),(78,120,159),(109,130,154),(110,131,155),(111,132,156),(112,127,151),(113,128,152),(114,129,153)], [(1,8,149),(2,9,150),(3,10,145),(4,11,146),(5,12,147),(6,7,148),(13,160,128),(14,161,129),(15,162,130),(16,157,131),(17,158,132),(18,159,127),(19,86,26),(20,87,27),(21,88,28),(22,89,29),(23,90,30),(24,85,25),(31,98,37),(32,99,38),(33,100,39),(34,101,40),(35,102,41),(36,97,42),(43,51,61),(44,52,62),(45,53,63),(46,54,64),(47,49,65),(48,50,66),(55,123,70),(56,124,71),(57,125,72),(58,126,67),(59,121,68),(60,122,69),(73,152,134),(74,153,135),(75,154,136),(76,155,137),(77,156,138),(78,151,133),(79,104,143),(80,105,144),(81,106,139),(82,107,140),(83,108,141),(84,103,142),(91,117,109),(92,118,110),(93,119,111),(94,120,112),(95,115,113),(96,116,114)], [(1,79,127),(2,80,128),(3,81,129),(4,82,130),(5,83,131),(6,84,132),(7,105,136),(8,106,137),(9,107,138),(10,108,133),(11,103,134),(12,104,135),(13,88,36),(14,89,31),(15,90,32),(16,85,33),(17,86,34),(18,87,35),(19,38,113),(20,39,114),(21,40,109),(22,41,110),(23,42,111),(24,37,112),(25,141,161),(26,142,162),(27,143,157),(28,144,158),(29,139,159),(30,140,160),(43,123,156),(44,124,151),(45,125,152),(46,126,153),(47,121,154),(48,122,155),(49,70,95),(50,71,96),(51,72,91),(52,67,92),(53,68,93),(54,69,94),(55,75,150),(56,76,145),(57,77,146),(58,78,147),(59,73,148),(60,74,149),(61,101,115),(62,102,116),(63,97,117),(64,98,118),(65,99,119),(66,100,120)]])
102 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3N | 3O | ··· | 3AF | 6A | ··· | 6H | 6I | ··· | 6N | 6O | ··· | 6AF | 9A | ··· | 9R | 18A | ··· | 18R |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 3 | ··· | 3 | 3 | ··· | 3 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | He3 | C2×He3 | He3⋊C3 | C2×He3⋊C3 |
| kernel | C6×He3⋊C3 | C3×He3⋊C3 | C2×He3⋊C3 | C32×C18 | C6×He3 | He3⋊C3 | C32×C9 | C3×He3 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 18 | 2 | 6 | 18 | 2 | 6 | 6 | 6 | 18 | 18 |
Matrix representation of C6×He3⋊C3 ►in GL4(𝔽19) generated by
| 11 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 7 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 6 | 0 | 0 |
| 7 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 17 | 0 | 0 |
G:=sub<GL(4,GF(19))| [11,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,11,0,0,0,0,7],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[7,0,0,0,0,0,0,6,0,4,0,0,0,0,4,0],[7,0,0,0,0,0,0,17,0,16,0,0,0,0,16,0] >;
C6×He3⋊C3 in GAP, Magma, Sage, TeX
C_6\times {\rm He}_3\rtimes C_3 % in TeX
G:=Group("C6xHe3:C3"); // GroupNames label
G:=SmallGroup(486,212);
// by ID
G=gap.SmallGroup(486,212);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,986,3250]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^3=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,d*b*d^-1=e*b*e^-1=b*c^-1,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d>;
// generators/relations