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G = D9×C27order 486 = 2·35

Direct product of C27 and D9

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: D9×C27, C93C54, 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

C1C9 — D9×C27
C1C3C9C3×C9C92C9×C27 — D9×C27
C9 — D9×C27
C1C27

Generators and relations for D9×C27
 G = < a,b,c | a27=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
2C3
3S3
9C6
2C9
2C9
2C9
2C9
2C9
3C3×S3
9C18
2C3×C9
2C27
2C27
2C27
2C27
3S3×C9
9C54
2C3×C27
3S3×C27

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 53)(2 54)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)

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), (1,53)(2,54)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)>;

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), (1,53)(2,54)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52) );

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)], [(1,53),(2,54),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52)])

162 conjugacy classes

class 1  2 3A3B3C3D3E6A6B9A···9F9G···9AM18A···18F27A···27R27S···27CL54A···54R
order1233333669···99···918···1827···2727···2754···54
size1911222991···12···29···91···12···29···9

162 irreducible representations

dim1111111122222222
type++++
imageC1C2C3C6C9C18C27C54S3D9C3×S3C3×D9S3×C9C9×D9S3×C27D9×C27
kernelD9×C27C9×C27C9×D9C92C3×D9C3×C9D9C9C3×C27C27C3×C9C9C32C3C3C1
# reps112266181813266181854

Matrix representation of D9×C27 in GL2(𝔽109) generated by

350
035
,
270
0105
,
0105
270
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

Subgroup lattice of D9×C27 in TeX

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