direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic3×D9, D18.S3, C6.2D18, C18.2D6, C6.2S32, (C3×D9)⋊C4, C3⋊3(C4×D9), (C6×D9).C2, C2.1(S3×D9), C9⋊Dic3⋊1C2, C9⋊1(C2×Dic3), (C3×C6).23D6, (C9×Dic3)⋊2C2, C32.2(C4×S3), C3.1(S3×Dic3), (C3×C18).2C22, (C3×Dic3).4S3, (C3×C9)⋊1(C2×C4), SmallGroup(216,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — Dic3×D9 |
Generators and relations for Dic3×D9
G = < a,b,c,d | a6=c9=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
9C2
9C2
2C3
3C4
9C22
27C4
2C6
3S3
3S3
9C6
9C6
2C9
27C2×C4
3D6
3C12
9Dic3
9Dic3
18Dic3
2C18
3C36
3Dic9
6Dic9
Smallest permutation representation of Dic3×D9
►On 72 points
Generators in S72
(1 17 7 14 4 11)(2 18 8 15 5 12)(3 10 9 16 6 13)(19 34 22 28 25 31)(20 35 23 29 26 32)(21 36 24 30 27 33)(37 52 40 46 43 49)(38 53 41 47 44 50)(39 54 42 48 45 51)(55 67 61 64 58 70)(56 68 62 65 59 71)(57 69 63 66 60 72)
(1 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)
(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 30)(2 29)(3 28)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(17 27)(18 26)(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)| (1,17,7,14,4,11)(2,18,8,15,5,12)(3,10,9,16,6,13)(19,34,22,28,25,31)(20,35,23,29,26,32)(21,36,24,30,27,33)(37,52,40,46,43,49)(38,53,41,47,44,50)(39,54,42,48,45,51)(55,67,61,64,58,70)(56,68,62,65,59,71)(57,69,63,66,60,72), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (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), (1,30)(2,29)(3,28)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,27)(18,26)(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( (1,17,7,14,4,11)(2,18,8,15,5,12)(3,10,9,16,6,13)(19,34,22,28,25,31)(20,35,23,29,26,32)(21,36,24,30,27,33)(37,52,40,46,43,49)(38,53,41,47,44,50)(39,54,42,48,45,51)(55,67,61,64,58,70)(56,68,62,65,59,71)(57,69,63,66,60,72), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (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), (1,30)(2,29)(3,28)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,27)(18,26)(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([[(1,17,7,14,4,11),(2,18,8,15,5,12),(3,10,9,16,6,13),(19,34,22,28,25,31),(20,35,23,29,26,32),(21,36,24,30,27,33),(37,52,40,46,43,49),(38,53,41,47,44,50),(39,54,42,48,45,51),(55,67,61,64,58,70),(56,68,62,65,59,71),(57,69,63,66,60,72)], [(1,50,14,41),(2,51,15,42),(3,52,16,43),(4,53,17,44),(5,54,18,45),(6,46,10,37),(7,47,11,38),(8,48,12,39),(9,49,13,40),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63)], [(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)], [(1,30),(2,29),(3,28),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(17,27),(18,26),(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)]])
Dic3×D9 is a maximal subgroup of
D18.D6 D36⋊5S3 C4×S3×D9 D18.3D6 D18.4D6
Dic3×D9 is a maximal quotient of C36.39D6 Dic9⋊Dic3 D18⋊Dic3
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 18A | 18B | 18C | 18D | 18E | 18F | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 3 | 3 | 27 | 27 | 2 | 2 | 4 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | Dic3 | D6 | D6 | D9 | C4×S3 | D18 | C4×D9 | S32 | S3×Dic3 | S3×D9 | Dic3×D9 |
| kernel | Dic3×D9 | C9×Dic3 | C9⋊Dic3 | C6×D9 | C3×D9 | D18 | C3×Dic3 | D9 | C18 | C3×C6 | Dic3 | C32 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of Dic3×D9 ►in GL4(𝔽37) generated by
| 0 | 1 | 0 | 0 |
| 36 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 6 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 20 | 31 |
| 0 | 0 | 6 | 26 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 20 | 31 |
| 0 | 0 | 11 | 17 |
G:=sub<GL(4,GF(37))| [0,36,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[0,6,0,0,6,0,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,20,6,0,0,31,26],[1,0,0,0,0,1,0,0,0,0,20,11,0,0,31,17] >;
Dic3×D9 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times D_9 % in TeX
G:=Group("Dic3xD9"); // GroupNames label
G:=SmallGroup(216,27);
// by ID
G=gap.SmallGroup(216,27);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,1065,453,1444,2603]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^9=d^2=1,b^2=a^3,b*a*b^-1=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