direct product, abelian, monomial, 3-elementary
Aliases: C3×C54, SmallGroup(162,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C54 |
| C1 — C3×C54 |
| C1 — C3×C54 |
Generators and relations for C3×C54
G = < a,b | a3=b54=1, ab=ba >
Smallest permutation representation of C3×C54
►Regular action on 162 points
Generators in S162
(1 56 115)(2 57 116)(3 58 117)(4 59 118)(5 60 119)(6 61 120)(7 62 121)(8 63 122)(9 64 123)(10 65 124)(11 66 125)(12 67 126)(13 68 127)(14 69 128)(15 70 129)(16 71 130)(17 72 131)(18 73 132)(19 74 133)(20 75 134)(21 76 135)(22 77 136)(23 78 137)(24 79 138)(25 80 139)(26 81 140)(27 82 141)(28 83 142)(29 84 143)(30 85 144)(31 86 145)(32 87 146)(33 88 147)(34 89 148)(35 90 149)(36 91 150)(37 92 151)(38 93 152)(39 94 153)(40 95 154)(41 96 155)(42 97 156)(43 98 157)(44 99 158)(45 100 159)(46 101 160)(47 102 161)(48 103 162)(49 104 109)(50 105 110)(51 106 111)(52 107 112)(53 108 113)(54 55 114)
(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)| (1,56,115)(2,57,116)(3,58,117)(4,59,118)(5,60,119)(6,61,120)(7,62,121)(8,63,122)(9,64,123)(10,65,124)(11,66,125)(12,67,126)(13,68,127)(14,69,128)(15,70,129)(16,71,130)(17,72,131)(18,73,132)(19,74,133)(20,75,134)(21,76,135)(22,77,136)(23,78,137)(24,79,138)(25,80,139)(26,81,140)(27,82,141)(28,83,142)(29,84,143)(30,85,144)(31,86,145)(32,87,146)(33,88,147)(34,89,148)(35,90,149)(36,91,150)(37,92,151)(38,93,152)(39,94,153)(40,95,154)(41,96,155)(42,97,156)(43,98,157)(44,99,158)(45,100,159)(46,101,160)(47,102,161)(48,103,162)(49,104,109)(50,105,110)(51,106,111)(52,107,112)(53,108,113)(54,55,114), (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( (1,56,115)(2,57,116)(3,58,117)(4,59,118)(5,60,119)(6,61,120)(7,62,121)(8,63,122)(9,64,123)(10,65,124)(11,66,125)(12,67,126)(13,68,127)(14,69,128)(15,70,129)(16,71,130)(17,72,131)(18,73,132)(19,74,133)(20,75,134)(21,76,135)(22,77,136)(23,78,137)(24,79,138)(25,80,139)(26,81,140)(27,82,141)(28,83,142)(29,84,143)(30,85,144)(31,86,145)(32,87,146)(33,88,147)(34,89,148)(35,90,149)(36,91,150)(37,92,151)(38,93,152)(39,94,153)(40,95,154)(41,96,155)(42,97,156)(43,98,157)(44,99,158)(45,100,159)(46,101,160)(47,102,161)(48,103,162)(49,104,109)(50,105,110)(51,106,111)(52,107,112)(53,108,113)(54,55,114), (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([[(1,56,115),(2,57,116),(3,58,117),(4,59,118),(5,60,119),(6,61,120),(7,62,121),(8,63,122),(9,64,123),(10,65,124),(11,66,125),(12,67,126),(13,68,127),(14,69,128),(15,70,129),(16,71,130),(17,72,131),(18,73,132),(19,74,133),(20,75,134),(21,76,135),(22,77,136),(23,78,137),(24,79,138),(25,80,139),(26,81,140),(27,82,141),(28,83,142),(29,84,143),(30,85,144),(31,86,145),(32,87,146),(33,88,147),(34,89,148),(35,90,149),(36,91,150),(37,92,151),(38,93,152),(39,94,153),(40,95,154),(41,96,155),(42,97,156),(43,98,157),(44,99,158),(45,100,159),(46,101,160),(47,102,161),(48,103,162),(49,104,109),(50,105,110),(51,106,111),(52,107,112),(53,108,113),(54,55,114)], [(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)]])
C3×C54 is a maximal subgroup of
C27⋊Dic3
162 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 9A | ··· | 9R | 18A | ··· | 18R | 27A | ··· | 27BB | 54A | ··· | 54BB |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C9 | C18 | C18 | C27 | C54 |
| kernel | C3×C54 | C3×C27 | C54 | C3×C18 | C27 | C3×C9 | C18 | C3×C6 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 12 | 6 | 12 | 6 | 54 | 54 |
Matrix representation of C3×C54 ►in GL2(𝔽109) generated by
| 63 | 0 |
| 0 | 63 |
| 1 | 0 |
| 0 | 104 |
G:=sub<GL(2,GF(109))| [63,0,0,63],[1,0,0,104] >;
C3×C54 in GAP, Magma, Sage, TeX
C_3\times C_{54} % in TeX
G:=Group("C3xC54"); // GroupNames label
G:=SmallGroup(162,26);
// by ID
G=gap.SmallGroup(162,26);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,96,78]);
// Polycyclic
G:=Group<a,b|a^3=b^54=1,a*b=b*a>;
// generators/relations
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