direct product, abelian, monomial, 3-elementary
Aliases: C9×C18, SmallGroup(162,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C9×C18 |
| C1 — C9×C18 |
| C1 — C9×C18 |
Generators and relations for C9×C18
G = < a,b | a9=b18=1, ab=ba >
Smallest permutation representation of C9×C18
►Regular action on 162 points
Generators in S162
(1 102 135 45 117 78 30 62 152)(2 103 136 46 118 79 31 63 153)(3 104 137 47 119 80 32 64 154)(4 105 138 48 120 81 33 65 155)(5 106 139 49 121 82 34 66 156)(6 107 140 50 122 83 35 67 157)(7 108 141 51 123 84 36 68 158)(8 91 142 52 124 85 19 69 159)(9 92 143 53 125 86 20 70 160)(10 93 144 54 126 87 21 71 161)(11 94 127 37 109 88 22 72 162)(12 95 128 38 110 89 23 55 145)(13 96 129 39 111 90 24 56 146)(14 97 130 40 112 73 25 57 147)(15 98 131 41 113 74 26 58 148)(16 99 132 42 114 75 27 59 149)(17 100 133 43 115 76 28 60 150)(18 101 134 44 116 77 29 61 151)
(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)| (1,102,135,45,117,78,30,62,152)(2,103,136,46,118,79,31,63,153)(3,104,137,47,119,80,32,64,154)(4,105,138,48,120,81,33,65,155)(5,106,139,49,121,82,34,66,156)(6,107,140,50,122,83,35,67,157)(7,108,141,51,123,84,36,68,158)(8,91,142,52,124,85,19,69,159)(9,92,143,53,125,86,20,70,160)(10,93,144,54,126,87,21,71,161)(11,94,127,37,109,88,22,72,162)(12,95,128,38,110,89,23,55,145)(13,96,129,39,111,90,24,56,146)(14,97,130,40,112,73,25,57,147)(15,98,131,41,113,74,26,58,148)(16,99,132,42,114,75,27,59,149)(17,100,133,43,115,76,28,60,150)(18,101,134,44,116,77,29,61,151), (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( (1,102,135,45,117,78,30,62,152)(2,103,136,46,118,79,31,63,153)(3,104,137,47,119,80,32,64,154)(4,105,138,48,120,81,33,65,155)(5,106,139,49,121,82,34,66,156)(6,107,140,50,122,83,35,67,157)(7,108,141,51,123,84,36,68,158)(8,91,142,52,124,85,19,69,159)(9,92,143,53,125,86,20,70,160)(10,93,144,54,126,87,21,71,161)(11,94,127,37,109,88,22,72,162)(12,95,128,38,110,89,23,55,145)(13,96,129,39,111,90,24,56,146)(14,97,130,40,112,73,25,57,147)(15,98,131,41,113,74,26,58,148)(16,99,132,42,114,75,27,59,149)(17,100,133,43,115,76,28,60,150)(18,101,134,44,116,77,29,61,151), (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([[(1,102,135,45,117,78,30,62,152),(2,103,136,46,118,79,31,63,153),(3,104,137,47,119,80,32,64,154),(4,105,138,48,120,81,33,65,155),(5,106,139,49,121,82,34,66,156),(6,107,140,50,122,83,35,67,157),(7,108,141,51,123,84,36,68,158),(8,91,142,52,124,85,19,69,159),(9,92,143,53,125,86,20,70,160),(10,93,144,54,126,87,21,71,161),(11,94,127,37,109,88,22,72,162),(12,95,128,38,110,89,23,55,145),(13,96,129,39,111,90,24,56,146),(14,97,130,40,112,73,25,57,147),(15,98,131,41,113,74,26,58,148),(16,99,132,42,114,75,27,59,149),(17,100,133,43,115,76,28,60,150),(18,101,134,44,116,77,29,61,151)], [(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)]])
C9×C18 is a maximal subgroup of
C9⋊Dic9
162 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 9A | ··· | 9BT | 18A | ··· | 18BT |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C6 | C9 | C18 |
| kernel | C9×C18 | C92 | C3×C18 | C3×C9 | C18 | C9 |
| # reps | 1 | 1 | 8 | 8 | 72 | 72 |
Matrix representation of C9×C18 ►in GL2(𝔽19) generated by
| 11 | 0 |
| 0 | 17 |
| 10 | 0 |
| 0 | 10 |
G:=sub<GL(2,GF(19))| [11,0,0,17],[10,0,0,10] >;
C9×C18 in GAP, Magma, Sage, TeX
C_9\times C_{18} % in TeX
G:=Group("C9xC18"); // GroupNames label
G:=SmallGroup(162,23);
// by ID
G=gap.SmallGroup(162,23);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,96,147]);
// Polycyclic
G:=Group<a,b|a^9=b^18=1,a*b=b*a>;
// generators/relations
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