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G = C2×C81⋊C3order 486 = 2·35

Direct product of C2 and C81⋊C3

direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary

Aliases: C2×C81⋊C3, C162⋊C3, C9.C54, C54.C9, C812C6, C27.C18, C18.C27, C32.C54, C54.2C32, (C3×C6).C27, (C3×C18).5C9, (C3×C54).3C3, C27.1(C3×C6), C6.3(C3×C27), C3.3(C3×C54), (C3×C27).4C6, C18.6(C3×C9), C9.6(C3×C18), (C3×C9).7C18, SmallGroup(486,84)

Series: Derived Chief Lower central Upper central

C1C3 — C2×C81⋊C3
C1C3C9C27C3×C27C81⋊C3 — C2×C81⋊C3
C1C3 — C2×C81⋊C3
C1C54 — C2×C81⋊C3

Generators and relations for C2×C81⋊C3
 G = < a,b,c | a2=b81=c3=1, ab=ba, ac=ca, cbc-1=b28 >

3C3
3C6

Smallest permutation representation of C2×C81⋊C3
On 162 points
Generators in S162
(1 162)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)(81 161)
(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)
(2 56 29)(3 30 57)(5 59 32)(6 33 60)(8 62 35)(9 36 63)(11 65 38)(12 39 66)(14 68 41)(15 42 69)(17 71 44)(18 45 72)(20 74 47)(21 48 75)(23 77 50)(24 51 78)(26 80 53)(27 54 81)(82 136 109)(83 110 137)(85 139 112)(86 113 140)(88 142 115)(89 116 143)(91 145 118)(92 119 146)(94 148 121)(95 122 149)(97 151 124)(98 125 152)(100 154 127)(101 128 155)(103 157 130)(104 131 158)(106 160 133)(107 134 161)

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

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

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

198 conjugacy classes

class 1  2 3A3B3C3D6A6B6C6D9A···9F9G9H9I9J18A···18F18G18H18I18J27A···27R27S···27AD54A···54R54S···54AD81A···81BB162A···162BB
order12333366669···9999918···181818181827···2727···2754···5454···5481···81162···162
size11113311331···133331···133331···13···31···13···33···33···3

198 irreducible representations

dim1111111111111133
type++
imageC1C2C3C3C6C6C9C9C18C18C27C27C54C54C81⋊C3C2×C81⋊C3
kernelC2×C81⋊C3C81⋊C3C162C3×C54C81C3×C27C54C3×C18C27C3×C9C18C3×C6C9C32C2C1
# reps116262126126361836181818

Matrix representation of C2×C81⋊C3 in GL4(𝔽163) generated by

162000
0100
0010
0001
,
58000
015125
014606
0960148
,
58000
01014
00104127
00058
G:=sub<GL(4,GF(163))| [162,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[58,0,0,0,0,15,146,96,0,1,0,0,0,25,6,148],[58,0,0,0,0,1,0,0,0,0,104,0,0,14,127,58] >;

C2×C81⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{81}\rtimes C_3
% in TeX

G:=Group("C2xC81:C3");
// GroupNames label

G:=SmallGroup(486,84);
// by ID

G=gap.SmallGroup(486,84);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,1520,93,118]);
// Polycyclic

G:=Group<a,b,c|a^2=b^81=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^28>;
// generators/relations

Export

Subgroup lattice of C2×C81⋊C3 in TeX

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