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## G = D9×C27order 486 = 2·35

### Direct product of C27 and D9

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

 Derived series C1 — C9 — D9×C27
 Chief series C1 — C3 — C9 — C3×C9 — C92 — C9×C27 — D9×C27
 Lower central C9 — D9×C27
 Upper central C1 — 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

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