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G = C14×D9order 252 = 22·32·7

Direct product of C14 and D9

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

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

C1C9 — C14×D9
C1C3C9C63C7×D9 — C14×D9
C9 — C14×D9
C1C14

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

9C2
9C2
9C22
3S3
3S3
9C14
9C14
3D6
9C2×C14
3S3×C7
3S3×C7
3S3×C14

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 2A2B2C 3  6 7A···7F9A9B9C14A···14F14G···14R18A18B18C21A···21F42A···42F63A···63R126A···126R
order1222367···799914···1414···1418181821···2142···4263···63126···126
size1199221···12221···19···92222···22···22···22···2

84 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C7C14C14S3D6D9D18S3×C7S3×C14C7×D9C14×D9
kernelC14×D9C7×D9C126D18D9C18C42C21C14C7C6C3C2C1
# reps12161261133661818

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

12600
020
002
,
100
0922
010531
,
12600
031118
02296
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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Subgroup lattice of C14×D9 in TeX

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