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## G = C13×D9order 234 = 2·32·13

### Direct product of C13 and D9

Aliases: C13×D9, C9⋊C26, C1173C2, C39.2S3, C3.(S3×C13), SmallGroup(234,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C13×D9
 Chief series C1 — C3 — C9 — C117 — C13×D9
 Lower central C9 — C13×D9
 Upper central C1 — C13

Generators and relations for C13×D9
G = < a,b,c | a13=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C13×D9
On 117 points
Generators in S117
(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)
(1 60 17 109 67 79 36 41 99)(2 61 18 110 68 80 37 42 100)(3 62 19 111 69 81 38 43 101)(4 63 20 112 70 82 39 44 102)(5 64 21 113 71 83 27 45 103)(6 65 22 114 72 84 28 46 104)(7 53 23 115 73 85 29 47 92)(8 54 24 116 74 86 30 48 93)(9 55 25 117 75 87 31 49 94)(10 56 26 105 76 88 32 50 95)(11 57 14 106 77 89 33 51 96)(12 58 15 107 78 90 34 52 97)(13 59 16 108 66 91 35 40 98)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 27)(22 28)(23 29)(24 30)(25 31)(26 32)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 53)(48 54)(49 55)(50 56)(51 57)(52 58)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 105)(89 106)(90 107)(91 108)

G:=sub<Sym(117)| (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), (1,60,17,109,67,79,36,41,99)(2,61,18,110,68,80,37,42,100)(3,62,19,111,69,81,38,43,101)(4,63,20,112,70,82,39,44,102)(5,64,21,113,71,83,27,45,103)(6,65,22,114,72,84,28,46,104)(7,53,23,115,73,85,29,47,92)(8,54,24,116,74,86,30,48,93)(9,55,25,117,75,87,31,49,94)(10,56,26,105,76,88,32,50,95)(11,57,14,106,77,89,33,51,96)(12,58,15,107,78,90,34,52,97)(13,59,16,108,66,91,35,40,98), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,27)(22,28)(23,29)(24,30)(25,31)(26,32)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,53)(48,54)(49,55)(50,56)(51,57)(52,58)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,105)(89,106)(90,107)(91,108)>;

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), (1,60,17,109,67,79,36,41,99)(2,61,18,110,68,80,37,42,100)(3,62,19,111,69,81,38,43,101)(4,63,20,112,70,82,39,44,102)(5,64,21,113,71,83,27,45,103)(6,65,22,114,72,84,28,46,104)(7,53,23,115,73,85,29,47,92)(8,54,24,116,74,86,30,48,93)(9,55,25,117,75,87,31,49,94)(10,56,26,105,76,88,32,50,95)(11,57,14,106,77,89,33,51,96)(12,58,15,107,78,90,34,52,97)(13,59,16,108,66,91,35,40,98), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,27)(22,28)(23,29)(24,30)(25,31)(26,32)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,53)(48,54)(49,55)(50,56)(51,57)(52,58)(79,109)(80,110)(81,111)(82,112)(83,113)(84,114)(85,115)(86,116)(87,117)(88,105)(89,106)(90,107)(91,108) );

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)], [(1,60,17,109,67,79,36,41,99),(2,61,18,110,68,80,37,42,100),(3,62,19,111,69,81,38,43,101),(4,63,20,112,70,82,39,44,102),(5,64,21,113,71,83,27,45,103),(6,65,22,114,72,84,28,46,104),(7,53,23,115,73,85,29,47,92),(8,54,24,116,74,86,30,48,93),(9,55,25,117,75,87,31,49,94),(10,56,26,105,76,88,32,50,95),(11,57,14,106,77,89,33,51,96),(12,58,15,107,78,90,34,52,97),(13,59,16,108,66,91,35,40,98)], [(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,92),(8,93),(9,94),(10,95),(11,96),(12,97),(13,98),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,27),(22,28),(23,29),(24,30),(25,31),(26,32),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,53),(48,54),(49,55),(50,56),(51,57),(52,58),(79,109),(80,110),(81,111),(82,112),(83,113),(84,114),(85,115),(86,116),(87,117),(88,105),(89,106),(90,107),(91,108)])

78 conjugacy classes

 class 1 2 3 9A 9B 9C 13A ··· 13L 26A ··· 26L 39A ··· 39L 117A ··· 117AJ order 1 2 3 9 9 9 13 ··· 13 26 ··· 26 39 ··· 39 117 ··· 117 size 1 9 2 2 2 2 1 ··· 1 9 ··· 9 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C13 C26 S3 D9 S3×C13 C13×D9 kernel C13×D9 C117 D9 C9 C39 C13 C3 C1 # reps 1 1 12 12 1 3 12 36

Matrix representation of C13×D9 in GL2(𝔽937) generated by

 676 0 0 676
,
 465 262 675 203
,
 465 262 734 472
G:=sub<GL(2,GF(937))| [676,0,0,676],[465,675,262,203],[465,734,262,472] >;

C13×D9 in GAP, Magma, Sage, TeX

C_{13}\times D_9
% in TeX

G:=Group("C13xD9");
// GroupNames label

G:=SmallGroup(234,3);
// by ID

G=gap.SmallGroup(234,3);
# by ID

G:=PCGroup([4,-2,-13,-3,-3,1562,82,2499]);
// Polycyclic

G:=Group<a,b,c|a^13=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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