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G = C163⋊C3order 489 = 3·163

The semidirect product of C163 and C3 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C163⋊C3, SmallGroup(489,1)

Series: Derived Chief Lower central Upper central

C1C163 — C163⋊C3
C1C163 — C163⋊C3
C163 — C163⋊C3
C1

Generators and relations for C163⋊C3
 G = < a,b | a163=b3=1, bab-1=a58 >

163C3

Smallest permutation representation of C163⋊C3
On 163 points: primitive
Generators in S163
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163)
(2 105 59)(3 46 117)(4 150 12)(5 91 70)(6 32 128)(7 136 23)(8 77 81)(9 18 139)(10 122 34)(11 63 92)(13 108 45)(14 49 103)(15 153 161)(16 94 56)(17 35 114)(19 80 67)(20 21 125)(22 66 78)(24 111 31)(25 52 89)(26 156 147)(27 97 42)(28 38 100)(29 142 158)(30 83 53)(33 69 64)(36 55 75)(37 159 133)(39 41 86)(40 145 144)(43 131 155)(44 72 50)(47 58 61)(48 162 119)(51 148 130)(54 134 141)(57 120 152)(60 106 163)(62 151 116)(65 137 127)(68 123 138)(71 109 149)(73 154 102)(74 95 160)(76 140 113)(79 126 124)(82 112 135)(84 157 88)(85 98 146)(87 143 99)(90 129 110)(93 115 121)(96 101 132)(104 118 107)

G:=sub<Sym(163)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163), (2,105,59)(3,46,117)(4,150,12)(5,91,70)(6,32,128)(7,136,23)(8,77,81)(9,18,139)(10,122,34)(11,63,92)(13,108,45)(14,49,103)(15,153,161)(16,94,56)(17,35,114)(19,80,67)(20,21,125)(22,66,78)(24,111,31)(25,52,89)(26,156,147)(27,97,42)(28,38,100)(29,142,158)(30,83,53)(33,69,64)(36,55,75)(37,159,133)(39,41,86)(40,145,144)(43,131,155)(44,72,50)(47,58,61)(48,162,119)(51,148,130)(54,134,141)(57,120,152)(60,106,163)(62,151,116)(65,137,127)(68,123,138)(71,109,149)(73,154,102)(74,95,160)(76,140,113)(79,126,124)(82,112,135)(84,157,88)(85,98,146)(87,143,99)(90,129,110)(93,115,121)(96,101,132)(104,118,107)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163), (2,105,59)(3,46,117)(4,150,12)(5,91,70)(6,32,128)(7,136,23)(8,77,81)(9,18,139)(10,122,34)(11,63,92)(13,108,45)(14,49,103)(15,153,161)(16,94,56)(17,35,114)(19,80,67)(20,21,125)(22,66,78)(24,111,31)(25,52,89)(26,156,147)(27,97,42)(28,38,100)(29,142,158)(30,83,53)(33,69,64)(36,55,75)(37,159,133)(39,41,86)(40,145,144)(43,131,155)(44,72,50)(47,58,61)(48,162,119)(51,148,130)(54,134,141)(57,120,152)(60,106,163)(62,151,116)(65,137,127)(68,123,138)(71,109,149)(73,154,102)(74,95,160)(76,140,113)(79,126,124)(82,112,135)(84,157,88)(85,98,146)(87,143,99)(90,129,110)(93,115,121)(96,101,132)(104,118,107) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163)], [(2,105,59),(3,46,117),(4,150,12),(5,91,70),(6,32,128),(7,136,23),(8,77,81),(9,18,139),(10,122,34),(11,63,92),(13,108,45),(14,49,103),(15,153,161),(16,94,56),(17,35,114),(19,80,67),(20,21,125),(22,66,78),(24,111,31),(25,52,89),(26,156,147),(27,97,42),(28,38,100),(29,142,158),(30,83,53),(33,69,64),(36,55,75),(37,159,133),(39,41,86),(40,145,144),(43,131,155),(44,72,50),(47,58,61),(48,162,119),(51,148,130),(54,134,141),(57,120,152),(60,106,163),(62,151,116),(65,137,127),(68,123,138),(71,109,149),(73,154,102),(74,95,160),(76,140,113),(79,126,124),(82,112,135),(84,157,88),(85,98,146),(87,143,99),(90,129,110),(93,115,121),(96,101,132),(104,118,107)])

57 conjugacy classes

class 1 3A3B163A···163BB
order133163···163
size11631633···3

57 irreducible representations

dim113
type+
imageC1C3C163⋊C3
kernelC163⋊C3C163C1
# reps1254

Matrix representation of C163⋊C3 in GL3(𝔽5869) generated by

010
001
14223593
,
100
21346432138
8329625225
G:=sub<GL(3,GF(5869))| [0,0,1,1,0,422,0,1,3593],[1,2134,832,0,643,962,0,2138,5225] >;

C163⋊C3 in GAP, Magma, Sage, TeX

C_{163}\rtimes C_3
% in TeX

G:=Group("C163:C3");
// GroupNames label

G:=SmallGroup(489,1);
// by ID

G=gap.SmallGroup(489,1);
# by ID

G:=PCGroup([2,-3,-163,1249]);
// Polycyclic

G:=Group<a,b|a^163=b^3=1,b*a*b^-1=a^58>;
// generators/relations

Export

Subgroup lattice of C163⋊C3 in TeX

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