Copied to
clipboard

## G = D9×D13order 468 = 22·32·13

### Direct product of D9 and D13

Aliases: D9×D13, C91D26, C39.D6, D117⋊C2, C131D18, C117⋊C22, (C9×D13)⋊C2, (C13×D9)⋊C2, C3.(S3×D13), (C3×D13).1S3, SmallGroup(468,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C117 — D9×D13
 Chief series C1 — C3 — C39 — C117 — C9×D13 — D9×D13
 Lower central C117 — D9×D13
 Upper central C1

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 >

9C2
13C2
117C2
117C22
3S3
13C6
39S3
9C26
9D13
39D6
13C18
13D9
9D26
3D39
13D18

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

׿
×
𝔽