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## G = D9×C3×C9order 486 = 2·35

### Direct product of C3×C9 and D9

Series: Derived Chief Lower central Upper central

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

Generators and relations for D9×C3×C9
G = < a,b,c,d | a3=b9=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 256 in 108 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C9, C32, C32, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, C33, C3×D9, C3×D9, S3×C9, C3×C18, S3×C32, C92, C92, C32×C9, C32×C9, C9×D9, C32×D9, S3×C3×C9, C3×C92, D9×C3×C9
Quotients: C1, C2, C3, S3, C6, C9, C32, D9, C18, C3×S3, C3×C6, C3×C9, C3×D9, S3×C9, C3×C18, S3×C32, C9×D9, C32×D9, S3×C3×C9, D9×C3×C9

Smallest permutation representation of D9×C3×C9
On 54 points
Generators in S54
(1 29 45)(2 30 37)(3 31 38)(4 32 39)(5 33 40)(6 34 41)(7 35 42)(8 36 43)(9 28 44)(10 53 20)(11 54 21)(12 46 22)(13 47 23)(14 48 24)(15 49 25)(16 50 26)(17 51 27)(18 52 19)
(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 41 30 7 38 36 4 44 33)(2 42 31 8 39 28 5 45 34)(3 43 32 9 40 29 6 37 35)(10 48 19 13 51 22 16 54 25)(11 49 20 14 52 23 17 46 26)(12 50 21 15 53 24 18 47 27)
(1 26)(2 27)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 32)(11 33)(12 34)(13 35)(14 36)(15 28)(16 29)(17 30)(18 31)(37 51)(38 52)(39 53)(40 54)(41 46)(42 47)(43 48)(44 49)(45 50)

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

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

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

162 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 6A ··· 6H 9A ··· 9R 9S ··· 9DM 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 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 C3 C6 C6 C9 C18 S3 D9 C3×S3 C3×S3 C3×D9 C3×D9 S3×C9 C9×D9 kernel D9×C3×C9 C3×C92 C9×D9 C32×D9 C92 C32×C9 C3×D9 C3×C9 C32×C9 C3×C9 C3×C9 C33 C9 C32 C32 C3 # reps 1 1 6 2 6 2 18 18 1 3 6 2 18 6 18 54

Matrix representation of D9×C3×C9 in GL3(𝔽19) generated by

 11 0 0 0 7 0 0 0 7
,
 11 0 0 0 9 0 0 0 9
,
 1 0 0 0 5 0 0 0 4
,
 1 0 0 0 0 4 0 5 0
G:=sub<GL(3,GF(19))| [11,0,0,0,7,0,0,0,7],[11,0,0,0,9,0,0,0,9],[1,0,0,0,5,0,0,0,4],[1,0,0,0,0,5,0,4,0] >;

D9×C3×C9 in GAP, Magma, Sage, TeX

D_9\times C_3\times C_9
% in TeX

G:=Group("D9xC3xC9");
// GroupNames label

G:=SmallGroup(486,91);
// by ID

G=gap.SmallGroup(486,91);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,68,8104,208,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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