direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13xD9, C9:C26, C117:3C2, C39.2S3, C3.(S3xC13), SmallGroup(234,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C13xD9 |
Generators and relations for C13xD9
G = < a,b,c | a13=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, S3, C13, D9, C26, S3xC13, C13xD9
Smallest permutation representation of C13xD9
►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 78 93 32 16 88 44 106 60)(2 66 94 33 17 89 45 107 61)(3 67 95 34 18 90 46 108 62)(4 68 96 35 19 91 47 109 63)(5 69 97 36 20 79 48 110 64)(6 70 98 37 21 80 49 111 65)(7 71 99 38 22 81 50 112 53)(8 72 100 39 23 82 51 113 54)(9 73 101 27 24 83 52 114 55)(10 74 102 28 25 84 40 115 56)(11 75 103 29 26 85 41 116 57)(12 76 104 30 14 86 42 117 58)(13 77 92 31 15 87 43 105 59)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 79)(37 80)(38 81)(39 82)(40 102)(41 103)(42 104)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(52 101)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 113)(73 114)(74 115)(75 116)(76 117)(77 105)(78 106)
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), (1,78,93,32,16,88,44,106,60)(2,66,94,33,17,89,45,107,61)(3,67,95,34,18,90,46,108,62)(4,68,96,35,19,91,47,109,63)(5,69,97,36,20,79,48,110,64)(6,70,98,37,21,80,49,111,65)(7,71,99,38,22,81,50,112,53)(8,72,100,39,23,82,51,113,54)(9,73,101,27,24,83,52,114,55)(10,74,102,28,25,84,40,115,56)(11,75,103,29,26,85,41,116,57)(12,76,104,30,14,86,42,117,58)(13,77,92,31,15,87,43,105,59), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,79)(37,80)(38,81)(39,82)(40,102)(41,103)(42,104)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(52,101)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,105)(78,106)>;
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), (1,78,93,32,16,88,44,106,60)(2,66,94,33,17,89,45,107,61)(3,67,95,34,18,90,46,108,62)(4,68,96,35,19,91,47,109,63)(5,69,97,36,20,79,48,110,64)(6,70,98,37,21,80,49,111,65)(7,71,99,38,22,81,50,112,53)(8,72,100,39,23,82,51,113,54)(9,73,101,27,24,83,52,114,55)(10,74,102,28,25,84,40,115,56)(11,75,103,29,26,85,41,116,57)(12,76,104,30,14,86,42,117,58)(13,77,92,31,15,87,43,105,59), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,79)(37,80)(38,81)(39,82)(40,102)(41,103)(42,104)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(52,101)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,105)(78,106) );
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)], [(1,78,93,32,16,88,44,106,60),(2,66,94,33,17,89,45,107,61),(3,67,95,34,18,90,46,108,62),(4,68,96,35,19,91,47,109,63),(5,69,97,36,20,79,48,110,64),(6,70,98,37,21,80,49,111,65),(7,71,99,38,22,81,50,112,53),(8,72,100,39,23,82,51,113,54),(9,73,101,27,24,83,52,114,55),(10,74,102,28,25,84,40,115,56),(11,75,103,29,26,85,41,116,57),(12,76,104,30,14,86,42,117,58),(13,77,92,31,15,87,43,105,59)], [(1,60),(2,61),(3,62),(4,63),(5,64),(6,65),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,79),(37,80),(38,81),(39,82),(40,102),(41,103),(42,104),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,99),(51,100),(52,101),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,113),(73,114),(74,115),(75,116),(76,117),(77,105),(78,106)]])
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 | S3xC13 | C13xD9 |
| kernel | C13xD9 | C117 | D9 | C9 | C39 | C13 | C3 | C1 |
| # reps | 1 | 1 | 12 | 12 | 1 | 3 | 12 | 36 |
Matrix representation of C13xD9 ►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] >;
C13xD9 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
Export