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G = C32×C18order 162 = 2·34

Abelian group of type [3,3,18]

direct product, abelian, monomial, 3-elementary

Aliases: C32×C18, SmallGroup(162,47)

Series: Derived Chief Lower central Upper central

C1 — C32×C18
C1C3C32C33C32×C9 — C32×C18
C1 — C32×C18
C1 — C32×C18

Generators and relations for C32×C18
 G = < a,b,c | a3=b3=c18=1, ab=ba, ac=ca, bc=cb >

Subgroups: 100, all normal (8 characteristic)
C1, C2, C3, C3 [×12], C6, C6 [×12], C9 [×9], C32 [×13], C18 [×9], C3×C6 [×13], C3×C9 [×12], C33, C3×C18 [×12], C32×C6, C32×C9, C32×C18
Quotients: C1, C2, C3 [×13], C6 [×13], C9 [×9], C32 [×13], C18 [×9], C3×C6 [×13], C3×C9 [×12], C33, C3×C18 [×12], C32×C6, C32×C9, C32×C18

Smallest permutation representation of C32×C18
Regular action on 162 points
Generators in S162
(1 54 110)(2 37 111)(3 38 112)(4 39 113)(5 40 114)(6 41 115)(7 42 116)(8 43 117)(9 44 118)(10 45 119)(11 46 120)(12 47 121)(13 48 122)(14 49 123)(15 50 124)(16 51 125)(17 52 126)(18 53 109)(19 96 158)(20 97 159)(21 98 160)(22 99 161)(23 100 162)(24 101 145)(25 102 146)(26 103 147)(27 104 148)(28 105 149)(29 106 150)(30 107 151)(31 108 152)(32 91 153)(33 92 154)(34 93 155)(35 94 156)(36 95 157)(55 88 129)(56 89 130)(57 90 131)(58 73 132)(59 74 133)(60 75 134)(61 76 135)(62 77 136)(63 78 137)(64 79 138)(65 80 139)(66 81 140)(67 82 141)(68 83 142)(69 84 143)(70 85 144)(71 86 127)(72 87 128)
(1 140 92)(2 141 93)(3 142 94)(4 143 95)(5 144 96)(6 127 97)(7 128 98)(8 129 99)(9 130 100)(10 131 101)(11 132 102)(12 133 103)(13 134 104)(14 135 105)(15 136 106)(16 137 107)(17 138 108)(18 139 91)(19 114 85)(20 115 86)(21 116 87)(22 117 88)(23 118 89)(24 119 90)(25 120 73)(26 121 74)(27 122 75)(28 123 76)(29 124 77)(30 125 78)(31 126 79)(32 109 80)(33 110 81)(34 111 82)(35 112 83)(36 113 84)(37 67 155)(38 68 156)(39 69 157)(40 70 158)(41 71 159)(42 72 160)(43 55 161)(44 56 162)(45 57 145)(46 58 146)(47 59 147)(48 60 148)(49 61 149)(50 62 150)(51 63 151)(52 64 152)(53 65 153)(54 66 154)
(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)| (1,54,110)(2,37,111)(3,38,112)(4,39,113)(5,40,114)(6,41,115)(7,42,116)(8,43,117)(9,44,118)(10,45,119)(11,46,120)(12,47,121)(13,48,122)(14,49,123)(15,50,124)(16,51,125)(17,52,126)(18,53,109)(19,96,158)(20,97,159)(21,98,160)(22,99,161)(23,100,162)(24,101,145)(25,102,146)(26,103,147)(27,104,148)(28,105,149)(29,106,150)(30,107,151)(31,108,152)(32,91,153)(33,92,154)(34,93,155)(35,94,156)(36,95,157)(55,88,129)(56,89,130)(57,90,131)(58,73,132)(59,74,133)(60,75,134)(61,76,135)(62,77,136)(63,78,137)(64,79,138)(65,80,139)(66,81,140)(67,82,141)(68,83,142)(69,84,143)(70,85,144)(71,86,127)(72,87,128), (1,140,92)(2,141,93)(3,142,94)(4,143,95)(5,144,96)(6,127,97)(7,128,98)(8,129,99)(9,130,100)(10,131,101)(11,132,102)(12,133,103)(13,134,104)(14,135,105)(15,136,106)(16,137,107)(17,138,108)(18,139,91)(19,114,85)(20,115,86)(21,116,87)(22,117,88)(23,118,89)(24,119,90)(25,120,73)(26,121,74)(27,122,75)(28,123,76)(29,124,77)(30,125,78)(31,126,79)(32,109,80)(33,110,81)(34,111,82)(35,112,83)(36,113,84)(37,67,155)(38,68,156)(39,69,157)(40,70,158)(41,71,159)(42,72,160)(43,55,161)(44,56,162)(45,57,145)(46,58,146)(47,59,147)(48,60,148)(49,61,149)(50,62,150)(51,63,151)(52,64,152)(53,65,153)(54,66,154), (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( (1,54,110)(2,37,111)(3,38,112)(4,39,113)(5,40,114)(6,41,115)(7,42,116)(8,43,117)(9,44,118)(10,45,119)(11,46,120)(12,47,121)(13,48,122)(14,49,123)(15,50,124)(16,51,125)(17,52,126)(18,53,109)(19,96,158)(20,97,159)(21,98,160)(22,99,161)(23,100,162)(24,101,145)(25,102,146)(26,103,147)(27,104,148)(28,105,149)(29,106,150)(30,107,151)(31,108,152)(32,91,153)(33,92,154)(34,93,155)(35,94,156)(36,95,157)(55,88,129)(56,89,130)(57,90,131)(58,73,132)(59,74,133)(60,75,134)(61,76,135)(62,77,136)(63,78,137)(64,79,138)(65,80,139)(66,81,140)(67,82,141)(68,83,142)(69,84,143)(70,85,144)(71,86,127)(72,87,128), (1,140,92)(2,141,93)(3,142,94)(4,143,95)(5,144,96)(6,127,97)(7,128,98)(8,129,99)(9,130,100)(10,131,101)(11,132,102)(12,133,103)(13,134,104)(14,135,105)(15,136,106)(16,137,107)(17,138,108)(18,139,91)(19,114,85)(20,115,86)(21,116,87)(22,117,88)(23,118,89)(24,119,90)(25,120,73)(26,121,74)(27,122,75)(28,123,76)(29,124,77)(30,125,78)(31,126,79)(32,109,80)(33,110,81)(34,111,82)(35,112,83)(36,113,84)(37,67,155)(38,68,156)(39,69,157)(40,70,158)(41,71,159)(42,72,160)(43,55,161)(44,56,162)(45,57,145)(46,58,146)(47,59,147)(48,60,148)(49,61,149)(50,62,150)(51,63,151)(52,64,152)(53,65,153)(54,66,154), (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([(1,54,110),(2,37,111),(3,38,112),(4,39,113),(5,40,114),(6,41,115),(7,42,116),(8,43,117),(9,44,118),(10,45,119),(11,46,120),(12,47,121),(13,48,122),(14,49,123),(15,50,124),(16,51,125),(17,52,126),(18,53,109),(19,96,158),(20,97,159),(21,98,160),(22,99,161),(23,100,162),(24,101,145),(25,102,146),(26,103,147),(27,104,148),(28,105,149),(29,106,150),(30,107,151),(31,108,152),(32,91,153),(33,92,154),(34,93,155),(35,94,156),(36,95,157),(55,88,129),(56,89,130),(57,90,131),(58,73,132),(59,74,133),(60,75,134),(61,76,135),(62,77,136),(63,78,137),(64,79,138),(65,80,139),(66,81,140),(67,82,141),(68,83,142),(69,84,143),(70,85,144),(71,86,127),(72,87,128)], [(1,140,92),(2,141,93),(3,142,94),(4,143,95),(5,144,96),(6,127,97),(7,128,98),(8,129,99),(9,130,100),(10,131,101),(11,132,102),(12,133,103),(13,134,104),(14,135,105),(15,136,106),(16,137,107),(17,138,108),(18,139,91),(19,114,85),(20,115,86),(21,116,87),(22,117,88),(23,118,89),(24,119,90),(25,120,73),(26,121,74),(27,122,75),(28,123,76),(29,124,77),(30,125,78),(31,126,79),(32,109,80),(33,110,81),(34,111,82),(35,112,83),(36,113,84),(37,67,155),(38,68,156),(39,69,157),(40,70,158),(41,71,159),(42,72,160),(43,55,161),(44,56,162),(45,57,145),(46,58,146),(47,59,147),(48,60,148),(49,61,149),(50,62,150),(51,63,151),(52,64,152),(53,65,153),(54,66,154)], [(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)])

C32×C18 is a maximal subgroup of   C325Dic9

162 conjugacy classes

class 1  2 3A···3Z6A···6Z9A···9BB18A···18BB
order123···36···69···918···18
size111···11···11···11···1

162 irreducible representations

dim11111111
type++
imageC1C2C3C3C6C6C9C18
kernelC32×C18C32×C9C3×C18C32×C6C3×C9C33C3×C6C32
# reps112422425454

Matrix representation of C32×C18 in GL3(𝔽19) generated by

700
0110
0011
,
100
0110
001
,
100
030
0014
G:=sub<GL(3,GF(19))| [7,0,0,0,11,0,0,0,11],[1,0,0,0,11,0,0,0,1],[1,0,0,0,3,0,0,0,14] >;

C32×C18 in GAP, Magma, Sage, TeX

C_3^2\times C_{18}
% in TeX

G:=Group("C3^2xC18");
// GroupNames label

G:=SmallGroup(162,47);
// by ID

G=gap.SmallGroup(162,47);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,276]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^18=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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