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## G = C3×S3×D9order 324 = 22·34

### Direct product of C3, S3 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C3×S3×D9
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — S3×C3×C9 — C3×S3×D9
 Lower central C3×C9 — C3×S3×D9
 Upper central C1 — C3

Generators and relations for C3×S3×D9
G = < a,b,c,d,e | a3=b3=c2=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 424 in 86 conjugacy classes, 26 normal (all characteristic)
C1, C2 [×3], C3 [×3], C3 [×4], C22, S3, S3 [×4], C6 [×7], C9, C9 [×3], C32 [×3], C32 [×4], D6 [×2], C2×C6, D9, D9 [×2], C18 [×2], C3×S3 [×2], C3×S3 [×7], C3⋊S3, C3×C6 [×2], C3×C9 [×2], C3×C9 [×4], C33, D18, S32, S3×C6 [×2], C3×D9 [×2], C3×D9 [×3], S3×C9, S3×C9, C9⋊S3, C3×C18, S3×C32, S3×C32, C3×C3⋊S3, C32×C9, S3×D9, C6×D9, C3×S32, C32×D9, S3×C3×C9, C3×C9⋊S3, C3×S3×D9
Quotients: C1, C2 [×3], C3, C22, S3 [×2], C6 [×3], D6 [×2], C2×C6, D9, C3×S3 [×2], D18, S32, S3×C6 [×2], C3×D9, S3×D9, C6×D9, C3×S32, C3×S3×D9

Smallest permutation representation of C3×S3×D9
On 36 points
Generators in S36
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(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)
(1 12)(2 11)(3 10)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 36)(27 35)

G:=sub<Sym(36)| (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (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), (1,12)(2,11)(3,10)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,36)(27,35)>;

G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (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), (1,12)(2,11)(3,10)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,36)(27,35) );

G=PermutationGroup([(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33)], [(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)], [(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)], [(1,12),(2,11),(3,10),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,36),(27,35)])

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 9A ··· 9I 9J ··· 9R 18A ··· 18I order 1 2 2 2 3 3 3 ··· 3 3 3 3 6 6 6 6 6 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 9 27 1 1 2 ··· 2 4 4 4 3 3 6 6 6 9 9 18 18 18 27 27 2 ··· 2 4 ··· 4 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 S3 D6 D6 C3×S3 D9 C3×S3 S3×C6 D18 S3×C6 C3×D9 C6×D9 S32 S3×D9 C3×S32 C3×S3×D9 kernel C3×S3×D9 C32×D9 S3×C3×C9 C3×C9⋊S3 S3×D9 C3×D9 S3×C9 C9⋊S3 C3×D9 S3×C32 C3×C9 C33 D9 C3×S3 C3×S3 C9 C32 C32 S3 C3 C32 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 1 2 3 2 2 3 2 6 6 1 3 2 6

Matrix representation of C3×S3×D9 in GL4(𝔽19) generated by

 1 0 0 0 0 1 0 0 0 0 11 0 0 0 0 11
,
 18 1 0 0 18 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 1 0 0 0 0 17 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 0 9 0 0 17 0
G:=sub<GL(4,GF(19))| [1,0,0,0,0,1,0,0,0,0,11,0,0,0,0,11],[18,18,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,17,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,17,0,0,9,0] >;

C3×S3×D9 in GAP, Magma, Sage, TeX

C_3\times S_3\times D_9
% in TeX

G:=Group("C3xS3xD9");
// GroupNames label

G:=SmallGroup(324,114);
// by ID

G=gap.SmallGroup(324,114);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1593,453,2164,3899]);
// Polycyclic

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

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