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G = C3×D81order 486 = 2·35

Direct product of C3 and D81

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C3×D81, C813C6, C32.2D27, (C3×C81)⋊2C2, C9.2(C3×D9), (C3×C9).7D9, C27.1(C3×S3), (C3×C27).4S3, C3.2(C3×D27), SmallGroup(486,32)

Series: Derived Chief Lower central Upper central

C1C81 — C3×D81
C1C3C9C27C81C3×C81 — C3×D81
C81 — C3×D81
C1C3

Generators and relations for C3×D81
 G = < a,b,c | a3=b81=c2=1, ab=ba, ac=ca, cbc=b-1 >

81C2
2C3
27S3
81C6
2C9
9D9
27C3×S3
2C27
3D27
9C3×D9
2C81
3C3×D27

Smallest permutation representation of C3×D81
On 162 points
Generators in S162
(1 28 55)(2 29 56)(3 30 57)(4 31 58)(5 32 59)(6 33 60)(7 34 61)(8 35 62)(9 36 63)(10 37 64)(11 38 65)(12 39 66)(13 40 67)(14 41 68)(15 42 69)(16 43 70)(17 44 71)(18 45 72)(19 46 73)(20 47 74)(21 48 75)(22 49 76)(23 50 77)(24 51 78)(25 52 79)(26 53 80)(27 54 81)(82 136 109)(83 137 110)(84 138 111)(85 139 112)(86 140 113)(87 141 114)(88 142 115)(89 143 116)(90 144 117)(91 145 118)(92 146 119)(93 147 120)(94 148 121)(95 149 122)(96 150 123)(97 151 124)(98 152 125)(99 153 126)(100 154 127)(101 155 128)(102 156 129)(103 157 130)(104 158 131)(105 159 132)(106 160 133)(107 161 134)(108 162 135)
(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)
(1 118)(2 117)(3 116)(4 115)(5 114)(6 113)(7 112)(8 111)(9 110)(10 109)(11 108)(12 107)(13 106)(14 105)(15 104)(16 103)(17 102)(18 101)(19 100)(20 99)(21 98)(22 97)(23 96)(24 95)(25 94)(26 93)(27 92)(28 91)(29 90)(30 89)(31 88)(32 87)(33 86)(34 85)(35 84)(36 83)(37 82)(38 162)(39 161)(40 160)(41 159)(42 158)(43 157)(44 156)(45 155)(46 154)(47 153)(48 152)(49 151)(50 150)(51 149)(52 148)(53 147)(54 146)(55 145)(56 144)(57 143)(58 142)(59 141)(60 140)(61 139)(62 138)(63 137)(64 136)(65 135)(66 134)(67 133)(68 132)(69 131)(70 130)(71 129)(72 128)(73 127)(74 126)(75 125)(76 124)(77 123)(78 122)(79 121)(80 120)(81 119)

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

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

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

126 conjugacy classes

class 1  2 3A3B3C3D3E6A6B9A···9I27A···27AA81A···81CC
order1233333669···927···2781···81
size1811122281812···22···22···2

126 irreducible representations

dim111122222222
type++++++
imageC1C2C3C6S3C3×S3D9C3×D9D27D81C3×D27C3×D81
kernelC3×D81C3×C81D81C81C3×C27C27C3×C9C9C32C3C3C1
# reps112212369271854

Matrix representation of C3×D81 in GL2(𝔽163) generated by

1040
0104
,
1520
074
,
01
10
G:=sub<GL(2,GF(163))| [104,0,0,104],[152,0,0,74],[0,1,1,0] >;

C3×D81 in GAP, Magma, Sage, TeX

C_3\times D_{81}
% in TeX

G:=Group("C3xD81");
// GroupNames label

G:=SmallGroup(486,32);
// by ID

G=gap.SmallGroup(486,32);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,542,284,2163,381,8104,208,11669]);
// Polycyclic

G:=Group<a,b,c|a^3=b^81=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D81 in TeX

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