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## G = S3×C3×C9order 162 = 2·34

### Direct product of C3×C9 and S3

Aliases: S3×C3×C9, C323C18, C33.5C6, C3⋊(C3×C18), (C3×C9)⋊13C6, (C32×C9)⋊1C2, C3.4(S3×C32), C32.6(C3×C6), (C3×S3).1C32, (S3×C32).2C3, C32.19(C3×S3), SmallGroup(162,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C3×C9
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — S3×C3×C9
 Lower central C3 — S3×C3×C9
 Upper central C1 — C3×C9

Generators and relations for S3×C3×C9
G = < a,b,c,d | a3=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 90 in 55 conjugacy classes, 30 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, C33, S3×C9, C3×C18, S3×C32, C32×C9, S3×C3×C9
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, S3×C9, C3×C18, S3×C32, S3×C3×C9

Smallest permutation representation of S3×C3×C9
On 54 points
Generators in S54
(1 31 41)(2 32 42)(3 33 43)(4 34 44)(5 35 45)(6 36 37)(7 28 38)(8 29 39)(9 30 40)(10 20 51)(11 21 52)(12 22 53)(13 23 54)(14 24 46)(15 25 47)(16 26 48)(17 27 49)(18 19 50)
(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 28 44)(2 29 45)(3 30 37)(4 31 38)(5 32 39)(6 33 40)(7 34 41)(8 35 42)(9 36 43)(10 54 26)(11 46 27)(12 47 19)(13 48 20)(14 49 21)(15 50 22)(16 51 23)(17 52 24)(18 53 25)
(1 52)(2 53)(3 54)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 28)(18 29)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 37)(27 38)

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

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

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

S3×C3×C9 is a maximal subgroup of   He3⋊C18  C3≀S33C3

81 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 6A ··· 6H 9A ··· 9R 9S ··· 9AJ 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C3 C6 C6 C9 C18 S3 C3×S3 C3×S3 S3×C9 kernel S3×C3×C9 C32×C9 S3×C9 S3×C32 C3×C9 C33 C3×S3 C32 C3×C9 C9 C32 C3 # reps 1 1 6 2 6 2 18 18 1 6 2 18

Matrix representation of S3×C3×C9 in GL3(𝔽19) generated by

 11 0 0 0 7 0 0 0 7
,
 6 0 0 0 11 0 0 0 11
,
 1 0 0 0 7 0 0 8 11
,
 1 0 0 0 1 10 0 0 18
G:=sub<GL(3,GF(19))| [11,0,0,0,7,0,0,0,7],[6,0,0,0,11,0,0,0,11],[1,0,0,0,7,8,0,0,11],[1,0,0,0,1,0,0,10,18] >;

S3×C3×C9 in GAP, Magma, Sage, TeX

S_3\times C_3\times C_9
% in TeX

G:=Group("S3xC3xC9");
// GroupNames label

G:=SmallGroup(162,33);
// by ID

G=gap.SmallGroup(162,33);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,57,2704]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=c^3=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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