direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×D27, C27⋊1D6, C3⋊1D54, C32.2D18, C27⋊S3⋊C2, C9.1S32, (S3×C27)⋊C2, (C3×D27)⋊C2, (S3×C9).S3, (C3×C27)⋊C22, (C3×S3).D9, C3.2(S3×D9), (C3×C9).5D6, SmallGroup(324,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C27 — S3×D27 |
Generators and relations for S3×D27
G = < a,b,c,d | a3=b2=c27=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
3C2
27C2
81C2
2C3
81C22
3C6
9S3
27S3
27S3
27C6
54S3
2C9
27D6
27D6
3C18
3D9
9D9
18D9
2C27
9D18
9S32
3D27
3C54
6D27
3D54
Smallest permutation representation of S3×D27
►On 54 points
Generators in S54
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)
(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)
(1 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 40)
G:=sub<Sym(54)| (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)>;
G:=Group( (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40) );
G=PermutationGroup([[(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,25),(8,17,26),(9,18,27),(28,46,37),(29,47,38),(30,48,39),(31,49,40),(32,50,41),(33,51,42),(34,52,43),(35,53,44),(36,54,45)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(23,34),(24,35),(25,36),(26,37),(27,38)], [(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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,54),(14,53),(15,52),(16,51),(17,50),(18,49),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,42),(26,41),(27,40)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 18A | 18B | 18C | 27A | ··· | 27I | 27J | ··· | 27R | 54A | ··· | 54I |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 27 | ··· | 27 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 3 | 27 | 81 | 2 | 2 | 4 | 6 | 54 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D9 | D18 | D27 | D54 | S32 | S3×D9 | S3×D27 |
| kernel | S3×D27 | C3×D27 | S3×C27 | C27⋊S3 | D27 | S3×C9 | C27 | C3×C9 | C3×S3 | C32 | S3 | C3 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 9 | 9 | 1 | 3 | 9 |
Matrix representation of S3×D27 ►in GL4(𝔽109) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 108 | 1 |
| 0 | 0 | 108 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 7 | 63 | 0 | 0 |
| 75 | 99 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 108 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(109))| [1,0,0,0,0,1,0,0,0,0,108,108,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[7,75,0,0,63,99,0,0,0,0,1,0,0,0,0,1],[108,2,0,0,0,1,0,0,0,0,1,0,0,0,0,1] >;
S3×D27 in GAP, Magma, Sage, TeX
S_3\times D_{27} % in TeX
G:=Group("S3xD27"); // GroupNames label
G:=SmallGroup(324,38);
// by ID
G=gap.SmallGroup(324,38);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,404,824,579,2710,208,3899]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^27=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