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G = C9×C18order 162 = 2·34

Abelian group of type [9,18]

direct product, abelian, monomial, 3-elementary

Aliases: C9×C18, SmallGroup(162,23)

Series: Derived Chief Lower central Upper central

C1 — C9×C18
C1C3C32C3×C9C92 — 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 73 99 141 29 70 44 118 155)(2 74 100 142 30 71 45 119 156)(3 75 101 143 31 72 46 120 157)(4 76 102 144 32 55 47 121 158)(5 77 103 127 33 56 48 122 159)(6 78 104 128 34 57 49 123 160)(7 79 105 129 35 58 50 124 161)(8 80 106 130 36 59 51 125 162)(9 81 107 131 19 60 52 126 145)(10 82 108 132 20 61 53 109 146)(11 83 91 133 21 62 54 110 147)(12 84 92 134 22 63 37 111 148)(13 85 93 135 23 64 38 112 149)(14 86 94 136 24 65 39 113 150)(15 87 95 137 25 66 40 114 151)(16 88 96 138 26 67 41 115 152)(17 89 97 139 27 68 42 116 153)(18 90 98 140 28 69 43 117 154)
(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)| (1,73,99,141,29,70,44,118,155)(2,74,100,142,30,71,45,119,156)(3,75,101,143,31,72,46,120,157)(4,76,102,144,32,55,47,121,158)(5,77,103,127,33,56,48,122,159)(6,78,104,128,34,57,49,123,160)(7,79,105,129,35,58,50,124,161)(8,80,106,130,36,59,51,125,162)(9,81,107,131,19,60,52,126,145)(10,82,108,132,20,61,53,109,146)(11,83,91,133,21,62,54,110,147)(12,84,92,134,22,63,37,111,148)(13,85,93,135,23,64,38,112,149)(14,86,94,136,24,65,39,113,150)(15,87,95,137,25,66,40,114,151)(16,88,96,138,26,67,41,115,152)(17,89,97,139,27,68,42,116,153)(18,90,98,140,28,69,43,117,154), (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( (1,73,99,141,29,70,44,118,155)(2,74,100,142,30,71,45,119,156)(3,75,101,143,31,72,46,120,157)(4,76,102,144,32,55,47,121,158)(5,77,103,127,33,56,48,122,159)(6,78,104,128,34,57,49,123,160)(7,79,105,129,35,58,50,124,161)(8,80,106,130,36,59,51,125,162)(9,81,107,131,19,60,52,126,145)(10,82,108,132,20,61,53,109,146)(11,83,91,133,21,62,54,110,147)(12,84,92,134,22,63,37,111,148)(13,85,93,135,23,64,38,112,149)(14,86,94,136,24,65,39,113,150)(15,87,95,137,25,66,40,114,151)(16,88,96,138,26,67,41,115,152)(17,89,97,139,27,68,42,116,153)(18,90,98,140,28,69,43,117,154), (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([(1,73,99,141,29,70,44,118,155),(2,74,100,142,30,71,45,119,156),(3,75,101,143,31,72,46,120,157),(4,76,102,144,32,55,47,121,158),(5,77,103,127,33,56,48,122,159),(6,78,104,128,34,57,49,123,160),(7,79,105,129,35,58,50,124,161),(8,80,106,130,36,59,51,125,162),(9,81,107,131,19,60,52,126,145),(10,82,108,132,20,61,53,109,146),(11,83,91,133,21,62,54,110,147),(12,84,92,134,22,63,37,111,148),(13,85,93,135,23,64,38,112,149),(14,86,94,136,24,65,39,113,150),(15,87,95,137,25,66,40,114,151),(16,88,96,138,26,67,41,115,152),(17,89,97,139,27,68,42,116,153),(18,90,98,140,28,69,43,117,154)], [(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···3H6A···6H9A···9BT18A···18BT
order123···36···69···918···18
size111···11···11···11···1

162 irreducible representations

dim111111
type++
imageC1C2C3C6C9C18
kernelC9×C18C92C3×C18C3×C9C18C9
# reps11887272

Matrix representation of C9×C18 in GL2(𝔽19) generated by

110
017
,
100
010
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

Export

Subgroup lattice of C9×C18 in TeX

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