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G = C9×D17order 306 = 2·32·17

Direct product of C9 and D17

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

Aliases: C9×D17, C17⋊C18, C51.C6, C1532C2, C3.(C3×D17), (C3×D17).C3, SmallGroup(306,2)

Series: Derived Chief Lower central Upper central

C1C17 — C9×D17
C1C17C51C153 — C9×D17
C17 — C9×D17
C1C9

Generators and relations for C9×D17
 G = < a,b,c | a9=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

17C2
17C6
17C18

Smallest permutation representation of C9×D17
On 153 points
Generators in S153
(1 146 88 47 123 70 24 113 59)(2 147 89 48 124 71 25 114 60)(3 148 90 49 125 72 26 115 61)(4 149 91 50 126 73 27 116 62)(5 150 92 51 127 74 28 117 63)(6 151 93 35 128 75 29 118 64)(7 152 94 36 129 76 30 119 65)(8 153 95 37 130 77 31 103 66)(9 137 96 38 131 78 32 104 67)(10 138 97 39 132 79 33 105 68)(11 139 98 40 133 80 34 106 52)(12 140 99 41 134 81 18 107 53)(13 141 100 42 135 82 19 108 54)(14 142 101 43 136 83 20 109 55)(15 143 102 44 120 84 21 110 56)(16 144 86 45 121 85 22 111 57)(17 145 87 46 122 69 23 112 58)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(30 34)(31 33)(35 41)(36 40)(37 39)(42 51)(43 50)(44 49)(45 48)(46 47)(52 65)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(66 68)(69 70)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(86 89)(87 88)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)(103 105)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(120 125)(121 124)(122 123)(126 136)(127 135)(128 134)(129 133)(130 132)(138 153)(139 152)(140 151)(141 150)(142 149)(143 148)(144 147)(145 146)

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

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

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

90 conjugacy classes

class 1  2 3A3B6A6B9A···9F17A···17H18A···18F51A···51P153A···153AV
order1233669···917···1718···1851···51153···153
size1171117171···12···217···172···22···2

90 irreducible representations

dim111111222
type+++
imageC1C2C3C6C9C18D17C3×D17C9×D17
kernelC9×D17C153C3×D17C51D17C17C9C3C1
# reps11226681648

Matrix representation of C9×D17 in GL2(𝔽307) generated by

930
093
,
2691
11913
,
301149
766
G:=sub<GL(2,GF(307))| [93,0,0,93],[269,119,1,13],[301,76,149,6] >;

C9×D17 in GAP, Magma, Sage, TeX

C_9\times D_{17}
% in TeX

G:=Group("C9xD17");
// GroupNames label

G:=SmallGroup(306,2);
// by ID

G=gap.SmallGroup(306,2);
# by ID

G:=PCGroup([4,-2,-3,-3,-17,29,4611]);
// Polycyclic

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

Export

Subgroup lattice of C9×D17 in TeX

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