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G = C3×D53order 318 = 2·3·53

Direct product of C3 and D53

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×D53, C53⋊C6, C1592C2, SmallGroup(318,2)

Series: Derived Chief Lower central Upper central

C1C53 — C3×D53
C1C53C159 — C3×D53
C53 — C3×D53
C1C3

Generators and relations for C3×D53
 G = < a,b,c | a3=b53=c2=1, ab=ba, ac=ca, cbc=b-1 >

53C2
53C6

Smallest permutation representation of C3×D53
On 159 points
Generators in S159
(1 141 65)(2 142 66)(3 143 67)(4 144 68)(5 145 69)(6 146 70)(7 147 71)(8 148 72)(9 149 73)(10 150 74)(11 151 75)(12 152 76)(13 153 77)(14 154 78)(15 155 79)(16 156 80)(17 157 81)(18 158 82)(19 159 83)(20 107 84)(21 108 85)(22 109 86)(23 110 87)(24 111 88)(25 112 89)(26 113 90)(27 114 91)(28 115 92)(29 116 93)(30 117 94)(31 118 95)(32 119 96)(33 120 97)(34 121 98)(35 122 99)(36 123 100)(37 124 101)(38 125 102)(39 126 103)(40 127 104)(41 128 105)(42 129 106)(43 130 54)(44 131 55)(45 132 56)(46 133 57)(47 134 58)(48 135 59)(49 136 60)(50 137 61)(51 138 62)(52 139 63)(53 140 64)
(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)
(1 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(76 106)(77 105)(78 104)(79 103)(80 102)(81 101)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(107 121)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(122 159)(123 158)(124 157)(125 156)(126 155)(127 154)(128 153)(129 152)(130 151)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 143)(139 142)(140 141)

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

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

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

84 conjugacy classes

class 1  2 3A3B6A6B53A···53Z159A···159AZ
order12336653···53159···159
size1531153532···22···2

84 irreducible representations

dim111122
type+++
imageC1C2C3C6D53C3×D53
kernelC3×D53C159D53C53C3C1
# reps11222652

Matrix representation of C3×D53 in GL2(𝔽3181) generated by

4400
0440
,
24161
5741018
,
2872793
28032894
G:=sub<GL(2,GF(3181))| [440,0,0,440],[2416,574,1,1018],[287,2803,2793,2894] >;

C3×D53 in GAP, Magma, Sage, TeX

C_3\times D_{53}
% in TeX

G:=Group("C3xD53");
// GroupNames label

G:=SmallGroup(318,2);
// by ID

G=gap.SmallGroup(318,2);
# by ID

G:=PCGroup([3,-2,-3,-53,2810]);
// Polycyclic

G:=Group<a,b,c|a^3=b^53=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D53 in TeX

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