metabelian, supersoluble, monomial
Aliases: D6:1D9, D18:2S3, C6.6D18, C18.6D6, C6.6S32, (C3xC9):2D4, (C6xD9):2C2, C2.6(S3xD9), (S3xC18):3C2, (S3xC6).2S3, C9:2(C3:D4), C3:2(C9:D4), C9:Dic3:3C2, (C3xC6).27D6, (C3xC18).6C22, C3.2(D6:S3), C32.2(C3:D4), SmallGroup(216,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6:D9
G = < a,b,c,d | a6=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=c-1 >
Quotients: C1, C2, C22, S3, D4, D6, D9, C3:D4, D18, S32, C9:D4, D6:S3, S3xD9, D6:D9
6C2
18C2
2C3
3C22
9C22
27C4
2C6
2S3
6S3
6C6
18C6
2C9
27D4
3D6
9Dic3
9Dic3
18Dic3
2D9
2C18
6C18
3Dic9
6Dic9
Smallest permutation representation of D6:D9
►On 72 points
Generators in S72
(1 11 4 14 7 17)(2 12 5 15 8 18)(3 13 6 16 9 10)(19 31 25 28 22 34)(20 32 26 29 23 35)(21 33 27 30 24 36)(37 49 43 46 40 52)(38 50 44 47 41 53)(39 51 45 48 42 54)(55 70 58 64 61 67)(56 71 59 65 62 68)(57 72 60 66 63 69)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 46)(10 43)(11 44)(12 45)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 64)(26 65)(27 66)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 55)(35 56)(36 57)
(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)
(1 21)(2 20)(3 19)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 36)(18 35)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 72)(45 71)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 63)(54 62)
G:=sub<Sym(72)| (1,11,4,14,7,17)(2,12,5,15,8,18)(3,13,6,16,9,10)(19,31,25,28,22,34)(20,32,26,29,23,35)(21,33,27,30,24,36)(37,49,43,46,40,52)(38,50,44,47,41,53)(39,51,45,48,42,54)(55,70,58,64,61,67)(56,71,59,65,62,68)(57,72,60,66,63,69), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,46)(10,43)(11,44)(12,45)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,64)(26,65)(27,66)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,55)(35,56)(36,57), (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), (1,21)(2,20)(3,19)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,36)(18,35)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,72)(45,71)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,63)(54,62)>;
G:=Group( (1,11,4,14,7,17)(2,12,5,15,8,18)(3,13,6,16,9,10)(19,31,25,28,22,34)(20,32,26,29,23,35)(21,33,27,30,24,36)(37,49,43,46,40,52)(38,50,44,47,41,53)(39,51,45,48,42,54)(55,70,58,64,61,67)(56,71,59,65,62,68)(57,72,60,66,63,69), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,46)(10,43)(11,44)(12,45)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,64)(26,65)(27,66)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,55)(35,56)(36,57), (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), (1,21)(2,20)(3,19)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,36)(18,35)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,72)(45,71)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,63)(54,62) );
G=PermutationGroup([[(1,11,4,14,7,17),(2,12,5,15,8,18),(3,13,6,16,9,10),(19,31,25,28,22,34),(20,32,26,29,23,35),(21,33,27,30,24,36),(37,49,43,46,40,52),(38,50,44,47,41,53),(39,51,45,48,42,54),(55,70,58,64,61,67),(56,71,59,65,62,68),(57,72,60,66,63,69)], [(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,46),(10,43),(11,44),(12,45),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,64),(26,65),(27,66),(28,58),(29,59),(30,60),(31,61),(32,62),(33,63),(34,55),(35,56),(36,57)], [(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)], [(1,21),(2,20),(3,19),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,36),(18,35),(37,70),(38,69),(39,68),(40,67),(41,66),(42,65),(43,64),(44,72),(45,71),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,63),(54,62)]])
D6:D9 is a maximal subgroup of
D12:5D9 D6.D18 D36:5S3 C36:D6 D18.4D6 S3xC9:D4 D9xC3:D4
D6:D9 is a maximal quotient of D36:S3 D12.D9 Dic6:D9 C12.D18 C18.Dic6 D18:Dic3 D6:Dic9
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 9D | 9E | 9F | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 6 | 18 | 2 | 2 | 4 | 54 | 2 | 2 | 4 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D9 | C3:D4 | C3:D4 | D18 | C9:D4 | S32 | D6:S3 | S3xD9 | D6:D9 |
| kernel | D6:D9 | C9:Dic3 | C6xD9 | S3xC18 | D18 | S3xC6 | C3xC9 | C18 | C3xC6 | D6 | C9 | C32 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of D6:D9 ►in GL4(F37) generated by
| 1 | 36 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 26 |
| 0 | 0 | 11 | 6 |
| 7 | 23 | 0 | 0 |
| 14 | 30 | 0 | 0 |
| 0 | 0 | 26 | 31 |
| 0 | 0 | 20 | 11 |
G:=sub<GL(4,GF(37))| [1,1,0,0,36,0,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,17,11,0,0,26,6],[7,14,0,0,23,30,0,0,0,0,26,20,0,0,31,11] >;
D6:D9 in GAP, Magma, Sage, TeX
D_6\rtimes D_9
% in TeX
G:=Group("D6:D9"); // GroupNames label
G:=SmallGroup(216,31);
// by ID
G=gap.SmallGroup(216,31);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,1065,453,1444,2603]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations
Export