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## G = C14×D9order 252 = 22·32·7

### Direct product of C14 and D9

Aliases: C14×D9, C18⋊C14, C1262C2, C42.6S3, C633C22, C21.3D6, C9⋊(C2×C14), C3.(S3×C14), C6.2(S3×C7), SmallGroup(252,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C14×D9
 Chief series C1 — C3 — C9 — C63 — C7×D9 — C14×D9
 Lower central C9 — C14×D9
 Upper central C1 — C14

Generators and relations for C14×D9
G = < a,b,c | a14=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C14×D9
On 126 points
Generators in S126
(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)
(1 34 112 15 77 98 45 69 126)(2 35 99 16 78 85 46 70 113)(3 36 100 17 79 86 47 57 114)(4 37 101 18 80 87 48 58 115)(5 38 102 19 81 88 49 59 116)(6 39 103 20 82 89 50 60 117)(7 40 104 21 83 90 51 61 118)(8 41 105 22 84 91 52 62 119)(9 42 106 23 71 92 53 63 120)(10 29 107 24 72 93 54 64 121)(11 30 108 25 73 94 55 65 122)(12 31 109 26 74 95 56 66 123)(13 32 110 27 75 96 43 67 124)(14 33 111 28 76 97 44 68 125)
(1 126)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 121)(11 122)(12 123)(13 124)(14 125)(15 98)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 110)(44 111)(45 112)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3 6 7A ··· 7F 9A 9B 9C 14A ··· 14F 14G ··· 14R 18A 18B 18C 21A ··· 21F 42A ··· 42F 63A ··· 63R 126A ··· 126R order 1 2 2 2 3 6 7 ··· 7 9 9 9 14 ··· 14 14 ··· 14 18 18 18 21 ··· 21 42 ··· 42 63 ··· 63 126 ··· 126 size 1 1 9 9 2 2 1 ··· 1 2 2 2 1 ··· 1 9 ··· 9 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C7 C14 C14 S3 D6 D9 D18 S3×C7 S3×C14 C7×D9 C14×D9 kernel C14×D9 C7×D9 C126 D18 D9 C18 C42 C21 C14 C7 C6 C3 C2 C1 # reps 1 2 1 6 12 6 1 1 3 3 6 6 18 18

Matrix representation of C14×D9 in GL3(𝔽127) generated by

 126 0 0 0 2 0 0 0 2
,
 1 0 0 0 9 22 0 105 31
,
 126 0 0 0 31 118 0 22 96
G:=sub<GL(3,GF(127))| [126,0,0,0,2,0,0,0,2],[1,0,0,0,9,105,0,22,31],[126,0,0,0,31,22,0,118,96] >;

C14×D9 in GAP, Magma, Sage, TeX

C_{14}\times D_9
% in TeX

G:=Group("C14xD9");
// GroupNames label

G:=SmallGroup(252,13);
// by ID

G=gap.SmallGroup(252,13);
# by ID

G:=PCGroup([5,-2,-2,-7,-3,-3,2803,138,4204]);
// Polycyclic

G:=Group<a,b,c|a^14=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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