direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D9×C27, C9⋊3C54, C92.5C6, (C9×C27)⋊1C2, C9.7(C3×D9), C3.4(C9×D9), C3.1(S3×C27), (C3×C27).3S3, (C9×D9).2C3, (C3×D9).2C9, (C3×C9).11C18, C32.12(S3×C9), (C3×C9).47(C3×S3), SmallGroup(486,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — D9×C27 |
Generators and relations for D9×C27
G = < a,b,c | a27=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D9×C27
►On 54 points
Generators in S54
(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 7 13 19 25 4 10 16 22)(2 8 14 20 26 5 11 17 23)(3 9 15 21 27 6 12 18 24)(28 49 43 37 31 52 46 40 34)(29 50 44 38 32 53 47 41 35)(30 51 45 39 33 54 48 42 36)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)
G:=sub<Sym(54)| (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), (1,7,13,19,25,4,10,16,22)(2,8,14,20,26,5,11,17,23)(3,9,15,21,27,6,12,18,24)(28,49,43,37,31,52,46,40,34)(29,50,44,38,32,53,47,41,35)(30,51,45,39,33,54,48,42,36), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)>;
G:=Group( (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), (1,7,13,19,25,4,10,16,22)(2,8,14,20,26,5,11,17,23)(3,9,15,21,27,6,12,18,24)(28,49,43,37,31,52,46,40,34)(29,50,44,38,32,53,47,41,35)(30,51,45,39,33,54,48,42,36), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44) );
G=PermutationGroup([[(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)], [(1,7,13,19,25,4,10,16,22),(2,8,14,20,26,5,11,17,23),(3,9,15,21,27,6,12,18,24),(28,49,43,37,31,52,46,40,34),(29,50,44,38,32,53,47,41,35),(30,51,45,39,33,54,48,42,36)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44)]])
162 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9AM | 18A | ··· | 18F | 27A | ··· | 27R | 27S | ··· | 27CL | 54A | ··· | 54R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 9 | 1 | 1 | 2 | 2 | 2 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 | S3 | D9 | C3×S3 | C3×D9 | S3×C9 | C9×D9 | S3×C27 | D9×C27 |
| kernel | D9×C27 | C9×C27 | C9×D9 | C92 | C3×D9 | C3×C9 | D9 | C9 | C3×C27 | C27 | C3×C9 | C9 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 1 | 3 | 2 | 6 | 6 | 18 | 18 | 54 |
Matrix representation of D9×C27 ►in GL2(𝔽109) generated by
| 35 | 0 |
| 0 | 35 |
| 27 | 0 |
| 0 | 105 |
| 0 | 105 |
| 27 | 0 |
G:=sub<GL(2,GF(109))| [35,0,0,35],[27,0,0,105],[0,27,105,0] >;
D9×C27 in GAP, Magma, Sage, TeX
D_9\times C_{27} % in TeX
G:=Group("D9xC27"); // GroupNames label
G:=SmallGroup(486,14);
// by ID
G=gap.SmallGroup(486,14);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,68,8104,208,11669]);
// Polycyclic
G:=Group<a,b,c|a^27=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export