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G = C9×D19order 342 = 2·32·19

Direct product of C9 and D19

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

Aliases: C9×D19, C1712C2, C195C18, C57.3C6, C3.(C3×D19), (C3×D19).2C3, SmallGroup(342,4)

Series: Derived Chief Lower central Upper central

C1C19 — C9×D19
C1C19C57C171 — C9×D19
C19 — C9×D19
C1C9

Generators and relations for C9×D19
 G = < a,b,c | a9=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

19C2
19C6
19C18

Smallest permutation representation of C9×D19
On 171 points
Generators in S171
(1 155 108 41 151 93 29 124 75)(2 156 109 42 152 94 30 125 76)(3 157 110 43 134 95 31 126 58)(4 158 111 44 135 77 32 127 59)(5 159 112 45 136 78 33 128 60)(6 160 113 46 137 79 34 129 61)(7 161 114 47 138 80 35 130 62)(8 162 96 48 139 81 36 131 63)(9 163 97 49 140 82 37 132 64)(10 164 98 50 141 83 38 133 65)(11 165 99 51 142 84 20 115 66)(12 166 100 52 143 85 21 116 67)(13 167 101 53 144 86 22 117 68)(14 168 102 54 145 87 23 118 69)(15 169 103 55 146 88 24 119 70)(16 170 104 56 147 89 25 120 71)(17 171 105 57 148 90 26 121 72)(18 153 106 39 149 91 27 122 73)(19 154 107 40 150 92 28 123 74)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(39 42)(40 41)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(73 76)(74 75)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(90 95)(91 94)(92 93)(96 100)(97 99)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 132)(116 131)(117 130)(118 129)(119 128)(120 127)(121 126)(122 125)(123 124)(134 148)(135 147)(136 146)(137 145)(138 144)(139 143)(140 142)(149 152)(150 151)(153 156)(154 155)(157 171)(158 170)(159 169)(160 168)(161 167)(162 166)(163 165)

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

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

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

99 conjugacy classes

class 1  2 3A3B6A6B9A···9F18A···18F19A···19I57A···57R171A···171BB
order1233669···918···1819···1957···57171···171
size1191119191···119···192···22···22···2

99 irreducible representations

dim111111222
type+++
imageC1C2C3C6C9C18D19C3×D19C9×D19
kernelC9×D19C171C3×D19C57D19C19C9C3C1
# reps11226691854

Matrix representation of C9×D19 in GL2(𝔽37) generated by

340
034
,
271
418
,
1816
3319
G:=sub<GL(2,GF(37))| [34,0,0,34],[27,4,1,18],[18,33,16,19] >;

C9×D19 in GAP, Magma, Sage, TeX

C_9\times D_{19}
% in TeX

G:=Group("C9xD19");
// GroupNames label

G:=SmallGroup(342,4);
// by ID

G=gap.SmallGroup(342,4);
# by ID

G:=PCGroup([4,-2,-3,-3,-19,29,5187]);
// Polycyclic

G:=Group<a,b,c|a^9=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C9×D19 in TeX

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