direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C14×D9, C18⋊C14, C126⋊2C2, C42.6S3, C63⋊3C22, C21.3D6, C9⋊(C2×C14), C3.(S3×C14), C6.2(S3×C7), SmallGroup(252,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C14×D9 |
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)
(1 124 110 57 95 35 55 79 16)(2 125 111 58 96 36 56 80 17)(3 126 112 59 97 37 43 81 18)(4 113 99 60 98 38 44 82 19)(5 114 100 61 85 39 45 83 20)(6 115 101 62 86 40 46 84 21)(7 116 102 63 87 41 47 71 22)(8 117 103 64 88 42 48 72 23)(9 118 104 65 89 29 49 73 24)(10 119 105 66 90 30 50 74 25)(11 120 106 67 91 31 51 75 26)(12 121 107 68 92 32 52 76 27)(13 122 108 69 93 33 53 77 28)(14 123 109 70 94 34 54 78 15)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 15)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 112)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(71 116)(72 117)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 113)(83 114)(84 115)
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), (1,124,110,57,95,35,55,79,16)(2,125,111,58,96,36,56,80,17)(3,126,112,59,97,37,43,81,18)(4,113,99,60,98,38,44,82,19)(5,114,100,61,85,39,45,83,20)(6,115,101,62,86,40,46,84,21)(7,116,102,63,87,41,47,71,22)(8,117,103,64,88,42,48,72,23)(9,118,104,65,89,29,49,73,24)(10,119,105,66,90,30,50,74,25)(11,120,106,67,91,31,51,75,26)(12,121,107,68,92,32,52,76,27)(13,122,108,69,93,33,53,77,28)(14,123,109,70,94,34,54,78,15), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,15)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,112)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,113)(83,114)(84,115)>;
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), (1,124,110,57,95,35,55,79,16)(2,125,111,58,96,36,56,80,17)(3,126,112,59,97,37,43,81,18)(4,113,99,60,98,38,44,82,19)(5,114,100,61,85,39,45,83,20)(6,115,101,62,86,40,46,84,21)(7,116,102,63,87,41,47,71,22)(8,117,103,64,88,42,48,72,23)(9,118,104,65,89,29,49,73,24)(10,119,105,66,90,30,50,74,25)(11,120,106,67,91,31,51,75,26)(12,121,107,68,92,32,52,76,27)(13,122,108,69,93,33,53,77,28)(14,123,109,70,94,34,54,78,15), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,15)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,112)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,113)(83,114)(84,115) );
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)], [(1,124,110,57,95,35,55,79,16),(2,125,111,58,96,36,56,80,17),(3,126,112,59,97,37,43,81,18),(4,113,99,60,98,38,44,82,19),(5,114,100,61,85,39,45,83,20),(6,115,101,62,86,40,46,84,21),(7,116,102,63,87,41,47,71,22),(8,117,103,64,88,42,48,72,23),(9,118,104,65,89,29,49,73,24),(10,119,105,66,90,30,50,74,25),(11,120,106,67,91,31,51,75,26),(12,121,107,68,92,32,52,76,27),(13,122,108,69,93,33,53,77,28),(14,123,109,70,94,34,54,78,15)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,15),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,63),(42,64),(43,112),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(71,116),(72,117),(73,118),(74,119),(75,120),(76,121),(77,122),(78,123),(79,124),(80,125),(81,126),(82,113),(83,114),(84,115)]])
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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