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G = C3×D59order 354 = 2·3·59

Direct product of C3 and D59

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

Aliases: C3×D59, C59⋊C6, C1772C2, SmallGroup(354,2)

Series: Derived Chief Lower central Upper central

C1C59 — C3×D59
C1C59C177 — C3×D59
C59 — C3×D59
C1C3

Generators and relations for C3×D59
 G = < a,b,c | a3=b59=c2=1, ab=ba, ac=ca, cbc=b-1 >

59C2
59C6

Smallest permutation representation of C3×D59
On 177 points
Generators in S177
(1 169 87)(2 170 88)(3 171 89)(4 172 90)(5 173 91)(6 174 92)(7 175 93)(8 176 94)(9 177 95)(10 119 96)(11 120 97)(12 121 98)(13 122 99)(14 123 100)(15 124 101)(16 125 102)(17 126 103)(18 127 104)(19 128 105)(20 129 106)(21 130 107)(22 131 108)(23 132 109)(24 133 110)(25 134 111)(26 135 112)(27 136 113)(28 137 114)(29 138 115)(30 139 116)(31 140 117)(32 141 118)(33 142 60)(34 143 61)(35 144 62)(36 145 63)(37 146 64)(38 147 65)(39 148 66)(40 149 67)(41 150 68)(42 151 69)(43 152 70)(44 153 71)(45 154 72)(46 155 73)(47 156 74)(48 157 75)(49 158 76)(50 159 77)(51 160 78)(52 161 79)(53 162 80)(54 163 81)(55 164 82)(56 165 83)(57 166 84)(58 167 85)(59 168 86)
(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)(119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(60 113)(61 112)(62 111)(63 110)(64 109)(65 108)(66 107)(67 106)(68 105)(69 104)(70 103)(71 102)(72 101)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(114 118)(115 117)(119 159)(120 158)(121 157)(122 156)(123 155)(124 154)(125 153)(126 152)(127 151)(128 150)(129 149)(130 148)(131 147)(132 146)(133 145)(134 144)(135 143)(136 142)(137 141)(138 140)(160 177)(161 176)(162 175)(163 174)(164 173)(165 172)(166 171)(167 170)(168 169)

G:=sub<Sym(177)| (1,169,87)(2,170,88)(3,171,89)(4,172,90)(5,173,91)(6,174,92)(7,175,93)(8,176,94)(9,177,95)(10,119,96)(11,120,97)(12,121,98)(13,122,99)(14,123,100)(15,124,101)(16,125,102)(17,126,103)(18,127,104)(19,128,105)(20,129,106)(21,130,107)(22,131,108)(23,132,109)(24,133,110)(25,134,111)(26,135,112)(27,136,113)(28,137,114)(29,138,115)(30,139,116)(31,140,117)(32,141,118)(33,142,60)(34,143,61)(35,144,62)(36,145,63)(37,146,64)(38,147,65)(39,148,66)(40,149,67)(41,150,68)(42,151,69)(43,152,70)(44,153,71)(45,154,72)(46,155,73)(47,156,74)(48,157,75)(49,158,76)(50,159,77)(51,160,78)(52,161,79)(53,162,80)(54,163,81)(55,164,82)(56,165,83)(57,166,84)(58,167,85)(59,168,86), (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)(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(114,118)(115,117)(119,159)(120,158)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140)(160,177)(161,176)(162,175)(163,174)(164,173)(165,172)(166,171)(167,170)(168,169)>;

G:=Group( (1,169,87)(2,170,88)(3,171,89)(4,172,90)(5,173,91)(6,174,92)(7,175,93)(8,176,94)(9,177,95)(10,119,96)(11,120,97)(12,121,98)(13,122,99)(14,123,100)(15,124,101)(16,125,102)(17,126,103)(18,127,104)(19,128,105)(20,129,106)(21,130,107)(22,131,108)(23,132,109)(24,133,110)(25,134,111)(26,135,112)(27,136,113)(28,137,114)(29,138,115)(30,139,116)(31,140,117)(32,141,118)(33,142,60)(34,143,61)(35,144,62)(36,145,63)(37,146,64)(38,147,65)(39,148,66)(40,149,67)(41,150,68)(42,151,69)(43,152,70)(44,153,71)(45,154,72)(46,155,73)(47,156,74)(48,157,75)(49,158,76)(50,159,77)(51,160,78)(52,161,79)(53,162,80)(54,163,81)(55,164,82)(56,165,83)(57,166,84)(58,167,85)(59,168,86), (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)(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(114,118)(115,117)(119,159)(120,158)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140)(160,177)(161,176)(162,175)(163,174)(164,173)(165,172)(166,171)(167,170)(168,169) );

G=PermutationGroup([(1,169,87),(2,170,88),(3,171,89),(4,172,90),(5,173,91),(6,174,92),(7,175,93),(8,176,94),(9,177,95),(10,119,96),(11,120,97),(12,121,98),(13,122,99),(14,123,100),(15,124,101),(16,125,102),(17,126,103),(18,127,104),(19,128,105),(20,129,106),(21,130,107),(22,131,108),(23,132,109),(24,133,110),(25,134,111),(26,135,112),(27,136,113),(28,137,114),(29,138,115),(30,139,116),(31,140,117),(32,141,118),(33,142,60),(34,143,61),(35,144,62),(36,145,63),(37,146,64),(38,147,65),(39,148,66),(40,149,67),(41,150,68),(42,151,69),(43,152,70),(44,153,71),(45,154,72),(46,155,73),(47,156,74),(48,157,75),(49,158,76),(50,159,77),(51,160,78),(52,161,79),(53,162,80),(54,163,81),(55,164,82),(56,165,83),(57,166,84),(58,167,85),(59,168,86)], [(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),(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177)], [(1,59),(2,58),(3,57),(4,56),(5,55),(6,54),(7,53),(8,52),(9,51),(10,50),(11,49),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,40),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(60,113),(61,112),(62,111),(63,110),(64,109),(65,108),(66,107),(67,106),(68,105),(69,104),(70,103),(71,102),(72,101),(73,100),(74,99),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(114,118),(115,117),(119,159),(120,158),(121,157),(122,156),(123,155),(124,154),(125,153),(126,152),(127,151),(128,150),(129,149),(130,148),(131,147),(132,146),(133,145),(134,144),(135,143),(136,142),(137,141),(138,140),(160,177),(161,176),(162,175),(163,174),(164,173),(165,172),(166,171),(167,170),(168,169)])

93 conjugacy classes

class 1  2 3A3B6A6B59A···59AC177A···177BF
order12336659···59177···177
size1591159592···22···2

93 irreducible representations

dim111122
type+++
imageC1C2C3C6D59C3×D59
kernelC3×D59C177D59C59C3C1
# reps11222958

Matrix representation of C3×D59 in GL2(𝔽709) generated by

2270
0227
,
01
70865
,
01
10
G:=sub<GL(2,GF(709))| [227,0,0,227],[0,708,1,65],[0,1,1,0] >;

C3×D59 in GAP, Magma, Sage, TeX

C_3\times D_{59}
% in TeX

G:=Group("C3xD59");
// GroupNames label

G:=SmallGroup(354,2);
// by ID

G=gap.SmallGroup(354,2);
# by ID

G:=PCGroup([3,-2,-3,-59,3134]);
// Polycyclic

G:=Group<a,b,c|a^3=b^59=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D59 in TeX

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