direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D9×D13, C9⋊1D26, C39.D6, D117⋊C2, C13⋊1D18, C117⋊C22, (C9×D13)⋊C2, (C13×D9)⋊C2, C3.(S3×D13), (C3×D13).1S3, SmallGroup(468,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C117 — D9×D13 |
Generators and relations for D9×D13
G = < a,b,c,d | a9=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of D9×D13
►On 117 points
Generators in S117
(1 96 105 60 27 76 41 25 88)(2 97 106 61 28 77 42 26 89)(3 98 107 62 29 78 43 14 90)(4 99 108 63 30 66 44 15 91)(5 100 109 64 31 67 45 16 79)(6 101 110 65 32 68 46 17 80)(7 102 111 53 33 69 47 18 81)(8 103 112 54 34 70 48 19 82)(9 104 113 55 35 71 49 20 83)(10 92 114 56 36 72 50 21 84)(11 93 115 57 37 73 51 22 85)(12 94 116 58 38 74 52 23 86)(13 95 117 59 39 75 40 24 87)
(1 88)(2 89)(3 90)(4 91)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(40 117)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 66)(64 67)(65 68)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(40 41)(42 52)(43 51)(44 50)(45 49)(46 48)(54 65)(55 64)(56 63)(57 62)(58 61)(59 60)(66 72)(67 71)(68 70)(73 78)(74 77)(75 76)(79 83)(80 82)(84 91)(85 90)(86 89)(87 88)(92 99)(93 98)(94 97)(95 96)(100 104)(101 103)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)
G:=sub<Sym(117)| (1,96,105,60,27,76,41,25,88)(2,97,106,61,28,77,42,26,89)(3,98,107,62,29,78,43,14,90)(4,99,108,63,30,66,44,15,91)(5,100,109,64,31,67,45,16,79)(6,101,110,65,32,68,46,17,80)(7,102,111,53,33,69,47,18,81)(8,103,112,54,34,70,48,19,82)(9,104,113,55,35,71,49,20,83)(10,92,114,56,36,72,50,21,84)(11,93,115,57,37,73,51,22,85)(12,94,116,58,38,74,52,23,86)(13,95,117,59,39,75,40,24,87), (1,88)(2,89)(3,90)(4,91)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(40,117)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,66)(64,67)(65,68), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(40,41)(42,52)(43,51)(44,50)(45,49)(46,48)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(66,72)(67,71)(68,70)(73,78)(74,77)(75,76)(79,83)(80,82)(84,91)(85,90)(86,89)(87,88)(92,99)(93,98)(94,97)(95,96)(100,104)(101,103)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)>;
G:=Group( (1,96,105,60,27,76,41,25,88)(2,97,106,61,28,77,42,26,89)(3,98,107,62,29,78,43,14,90)(4,99,108,63,30,66,44,15,91)(5,100,109,64,31,67,45,16,79)(6,101,110,65,32,68,46,17,80)(7,102,111,53,33,69,47,18,81)(8,103,112,54,34,70,48,19,82)(9,104,113,55,35,71,49,20,83)(10,92,114,56,36,72,50,21,84)(11,93,115,57,37,73,51,22,85)(12,94,116,58,38,74,52,23,86)(13,95,117,59,39,75,40,24,87), (1,88)(2,89)(3,90)(4,91)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(40,117)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,66)(64,67)(65,68), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(40,41)(42,52)(43,51)(44,50)(45,49)(46,48)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(66,72)(67,71)(68,70)(73,78)(74,77)(75,76)(79,83)(80,82)(84,91)(85,90)(86,89)(87,88)(92,99)(93,98)(94,97)(95,96)(100,104)(101,103)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112) );
G=PermutationGroup([[(1,96,105,60,27,76,41,25,88),(2,97,106,61,28,77,42,26,89),(3,98,107,62,29,78,43,14,90),(4,99,108,63,30,66,44,15,91),(5,100,109,64,31,67,45,16,79),(6,101,110,65,32,68,46,17,80),(7,102,111,53,33,69,47,18,81),(8,103,112,54,34,70,48,19,82),(9,104,113,55,35,71,49,20,83),(10,92,114,56,36,72,50,21,84),(11,93,115,57,37,73,51,22,85),(12,94,116,58,38,74,52,23,86),(13,95,117,59,39,75,40,24,87)], [(1,88),(2,89),(3,90),(4,91),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,85),(12,86),(13,87),(14,98),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(40,117),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,113),(50,114),(51,115),(52,116),(53,69),(54,70),(55,71),(56,72),(57,73),(58,74),(59,75),(60,76),(61,77),(62,78),(63,66),(64,67),(65,68)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,22),(15,21),(16,20),(17,19),(23,26),(24,25),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(40,41),(42,52),(43,51),(44,50),(45,49),(46,48),(54,65),(55,64),(56,63),(57,62),(58,61),(59,60),(66,72),(67,71),(68,70),(73,78),(74,77),(75,76),(79,83),(80,82),(84,91),(85,90),(86,89),(87,88),(92,99),(93,98),(94,97),(95,96),(100,104),(101,103),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 9A | 9B | 9C | 13A | ··· | 13F | 18A | 18B | 18C | 26A | ··· | 26F | 39A | ··· | 39F | 117A | ··· | 117R |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 9 | 9 | 9 | 13 | ··· | 13 | 18 | 18 | 18 | 26 | ··· | 26 | 39 | ··· | 39 | 117 | ··· | 117 |
| size | 1 | 9 | 13 | 117 | 2 | 26 | 2 | 2 | 2 | 2 | ··· | 2 | 26 | 26 | 26 | 18 | ··· | 18 | 4 | ··· | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D6 | D9 | D13 | D18 | D26 | S3×D13 | D9×D13 |
| kernel | D9×D13 | C13×D9 | C9×D13 | D117 | C3×D13 | C39 | D13 | D9 | C13 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 6 | 3 | 6 | 6 | 18 |
Matrix representation of D9×D13 ►in GL4(𝔽937) generated by
| 262 | 472 | 0 | 0 |
| 465 | 734 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 675 | 734 | 0 | 0 |
| 472 | 262 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 839 | 1 |
| 0 | 0 | 659 | 108 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 108 | 936 |
| 0 | 0 | 419 | 829 |
G:=sub<GL(4,GF(937))| [262,465,0,0,472,734,0,0,0,0,1,0,0,0,0,1],[675,472,0,0,734,262,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,839,659,0,0,1,108],[1,0,0,0,0,1,0,0,0,0,108,419,0,0,936,829] >;
D9×D13 in GAP, Magma, Sage, TeX
D_9\times D_{13} % in TeX
G:=Group("D9xD13"); // GroupNames label
G:=SmallGroup(468,11);
// by ID
G=gap.SmallGroup(468,11);
# by ID
G:=PCGroup([5,-2,-2,-3,-13,-3,1237,1182,2883,3909]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^2=c^13=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export