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G = C17×D9order 306 = 2·32·17

Direct product of C17 and D9

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

Aliases: C17×D9, C9⋊C34, C1533C2, C51.2S3, C3.(S3×C17), SmallGroup(306,1)

Series: Derived Chief Lower central Upper central

C1C9 — C17×D9
C1C3C9C153 — C17×D9
C9 — C17×D9
C1C17

Generators and relations for C17×D9
 G = < a,b,c | a17=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
3S3
9C34
3S3×C17

Smallest permutation representation of C17×D9
On 153 points
Generators in S153
(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)
(1 28 152 131 100 58 42 113 78)(2 29 153 132 101 59 43 114 79)(3 30 137 133 102 60 44 115 80)(4 31 138 134 86 61 45 116 81)(5 32 139 135 87 62 46 117 82)(6 33 140 136 88 63 47 118 83)(7 34 141 120 89 64 48 119 84)(8 18 142 121 90 65 49 103 85)(9 19 143 122 91 66 50 104 69)(10 20 144 123 92 67 51 105 70)(11 21 145 124 93 68 35 106 71)(12 22 146 125 94 52 36 107 72)(13 23 147 126 95 53 37 108 73)(14 24 148 127 96 54 38 109 74)(15 25 149 128 97 55 39 110 75)(16 26 150 129 98 56 40 111 76)(17 27 151 130 99 57 41 112 77)
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 85)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 103)(19 104)(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 151)(42 152)(43 153)(44 137)(45 138)(46 139)(47 140)(48 141)(49 142)(50 143)(51 144)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 120)(65 121)(66 122)(67 123)(68 124)

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

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

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

102 conjugacy classes

class 1  2  3 9A9B9C17A···17P34A···34P51A···51P153A···153AV
order12399917···1734···3451···51153···153
size1922221···19···92···22···2

102 irreducible representations

dim11112222
type++++
imageC1C2C17C34S3D9S3×C17C17×D9
kernelC17×D9C153D9C9C51C17C3C1
# reps111616131648

Matrix representation of C17×D9 in GL2(𝔽307) generated by

2730
0273
,
91156
151242
,
151242
91156
G:=sub<GL(2,GF(307))| [273,0,0,273],[91,151,156,242],[151,91,242,156] >;

C17×D9 in GAP, Magma, Sage, TeX

C_{17}\times D_9
% in TeX

G:=Group("C17xD9");
// GroupNames label

G:=SmallGroup(306,1);
// by ID

G=gap.SmallGroup(306,1);
# by ID

G:=PCGroup([4,-2,-17,-3,-3,2042,82,3267]);
// Polycyclic

G:=Group<a,b,c|a^17=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C17×D9 in TeX

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