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G = C3×C54order 162 = 2·34

Abelian group of type [3,54]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C54, SmallGroup(162,26)

Series: Derived Chief Lower central Upper central

C1 — C3×C54
C1C3C9C3×C9C3×C27 — 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 97 142)(2 98 143)(3 99 144)(4 100 145)(5 101 146)(6 102 147)(7 103 148)(8 104 149)(9 105 150)(10 106 151)(11 107 152)(12 108 153)(13 55 154)(14 56 155)(15 57 156)(16 58 157)(17 59 158)(18 60 159)(19 61 160)(20 62 161)(21 63 162)(22 64 109)(23 65 110)(24 66 111)(25 67 112)(26 68 113)(27 69 114)(28 70 115)(29 71 116)(30 72 117)(31 73 118)(32 74 119)(33 75 120)(34 76 121)(35 77 122)(36 78 123)(37 79 124)(38 80 125)(39 81 126)(40 82 127)(41 83 128)(42 84 129)(43 85 130)(44 86 131)(45 87 132)(46 88 133)(47 89 134)(48 90 135)(49 91 136)(50 92 137)(51 93 138)(52 94 139)(53 95 140)(54 96 141)
(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)| (1,97,142)(2,98,143)(3,99,144)(4,100,145)(5,101,146)(6,102,147)(7,103,148)(8,104,149)(9,105,150)(10,106,151)(11,107,152)(12,108,153)(13,55,154)(14,56,155)(15,57,156)(16,58,157)(17,59,158)(18,60,159)(19,61,160)(20,62,161)(21,63,162)(22,64,109)(23,65,110)(24,66,111)(25,67,112)(26,68,113)(27,69,114)(28,70,115)(29,71,116)(30,72,117)(31,73,118)(32,74,119)(33,75,120)(34,76,121)(35,77,122)(36,78,123)(37,79,124)(38,80,125)(39,81,126)(40,82,127)(41,83,128)(42,84,129)(43,85,130)(44,86,131)(45,87,132)(46,88,133)(47,89,134)(48,90,135)(49,91,136)(50,92,137)(51,93,138)(52,94,139)(53,95,140)(54,96,141), (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( (1,97,142)(2,98,143)(3,99,144)(4,100,145)(5,101,146)(6,102,147)(7,103,148)(8,104,149)(9,105,150)(10,106,151)(11,107,152)(12,108,153)(13,55,154)(14,56,155)(15,57,156)(16,58,157)(17,59,158)(18,60,159)(19,61,160)(20,62,161)(21,63,162)(22,64,109)(23,65,110)(24,66,111)(25,67,112)(26,68,113)(27,69,114)(28,70,115)(29,71,116)(30,72,117)(31,73,118)(32,74,119)(33,75,120)(34,76,121)(35,77,122)(36,78,123)(37,79,124)(38,80,125)(39,81,126)(40,82,127)(41,83,128)(42,84,129)(43,85,130)(44,86,131)(45,87,132)(46,88,133)(47,89,134)(48,90,135)(49,91,136)(50,92,137)(51,93,138)(52,94,139)(53,95,140)(54,96,141), (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([(1,97,142),(2,98,143),(3,99,144),(4,100,145),(5,101,146),(6,102,147),(7,103,148),(8,104,149),(9,105,150),(10,106,151),(11,107,152),(12,108,153),(13,55,154),(14,56,155),(15,57,156),(16,58,157),(17,59,158),(18,60,159),(19,61,160),(20,62,161),(21,63,162),(22,64,109),(23,65,110),(24,66,111),(25,67,112),(26,68,113),(27,69,114),(28,70,115),(29,71,116),(30,72,117),(31,73,118),(32,74,119),(33,75,120),(34,76,121),(35,77,122),(36,78,123),(37,79,124),(38,80,125),(39,81,126),(40,82,127),(41,83,128),(42,84,129),(43,85,130),(44,86,131),(45,87,132),(46,88,133),(47,89,134),(48,90,135),(49,91,136),(50,92,137),(51,93,138),(52,94,139),(53,95,140),(54,96,141)], [(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···3H6A···6H9A···9R18A···18R27A···27BB54A···54BB
order123···36···69···918···1827···2754···54
size111···11···11···11···11···11···1

162 irreducible representations

dim111111111111
type++
imageC1C2C3C3C6C6C9C9C18C18C27C54
kernelC3×C54C3×C27C54C3×C18C27C3×C9C18C3×C6C9C32C6C3
# reps1162621261265454

Matrix representation of C3×C54 in GL2(𝔽109) generated by

630
063
,
10
0104
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

Export

Subgroup lattice of C3×C54 in TeX

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