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## G = C9×D27order 486 = 2·35

### Direct product of C9 and D27

Aliases: C9×D27, C275C18, C92.3S3, (C9×C27)⋊2C2, C9.3(S3×C9), C3.2(C9×D9), (C3×C9).6D9, (C3×C27).6C6, C3.4(C3×D27), (C3×D27).2C3, C32.12(C3×D9), (C3×C9).52(C3×S3), SmallGroup(486,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C9×D27
 Chief series C1 — C3 — C9 — C27 — C3×C27 — C9×C27 — C9×D27
 Lower central C27 — C9×D27
 Upper central C1 — C9

Generators and relations for C9×D27
G = < a,b,c | a9=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C9×D27
On 54 points
Generators in S54
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)(28 52 49 46 43 40 37 34 31)(29 53 50 47 44 41 38 35 32)(30 54 51 48 45 42 39 36 33)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)

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

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

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

135 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 6A 6B 9A ··· 9F 9G ··· 9AM 18A ··· 18F 27A ··· 27CC order 1 2 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 27 ··· 27 size 1 27 1 1 2 2 2 27 27 1 ··· 1 2 ··· 2 27 ··· 27 2 ··· 2

135 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 C9 C18 S3 D9 C3×S3 D27 S3×C9 C3×D9 C9×D9 C3×D27 C9×D27 kernel C9×D27 C9×C27 C3×D27 C3×C27 D27 C27 C92 C3×C9 C3×C9 C9 C9 C32 C3 C3 C1 # reps 1 1 2 2 6 6 1 3 2 9 6 6 18 18 54

Matrix representation of C9×D27 in GL2(𝔽109) generated by

 16 0 0 16
,
 5 34 0 22
,
 88 99 44 21
G:=sub<GL(2,GF(109))| [16,0,0,16],[5,0,34,22],[88,44,99,21] >;

C9×D27 in GAP, Magma, Sage, TeX

C_9\times D_{27}
% in TeX

G:=Group("C9xD27");
// GroupNames label

G:=SmallGroup(486,13);
// by ID

G=gap.SmallGroup(486,13);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,2163,381,8104,208,11669]);
// Polycyclic

G:=Group<a,b,c|a^9=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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