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G = C3×S3×D9order 324 = 22·34

Direct product of C3, S3 and D9

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C3×S3×D9, C326D18, C33.7D6, C9⋊S37C6, C94(S3×C6), C31(C6×D9), (S3×C9)⋊3C6, (C3×D9)⋊3C6, (C3×C9)⋊16D6, C32.10S32, (C32×D9)⋊1C2, C32.9(S3×C6), (C32×C9)⋊2C22, (S3×C32).2S3, (S3×C3×C9)⋊2C2, C3.1(C3×S32), (C3×C9⋊S3)⋊1C2, (C3×C9)⋊9(C2×C6), (C3×S3).1(C3×S3), SmallGroup(324,114)

Series: Derived Chief Lower central Upper central

C1C3×C9 — C3×S3×D9
C1C3C32C3×C9C32×C9S3×C3×C9 — C3×S3×D9
C3×C9 — C3×S3×D9
C1C3

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 2A2B2C3A3B3C···3H3I3J3K6A6B6C6D6E6F6G6H6I6J6K6L9A···9I9J···9R18A···18I
order1222333···33336666666666669···99···918···18
size13927112···2444336669918181827272···24···46···6

54 irreducible representations

dim111111112222222222224444
type++++++++++++
imageC1C2C2C2C3C6C6C6S3S3D6D6C3×S3D9C3×S3S3×C6D18S3×C6C3×D9C6×D9S32S3×D9C3×S32C3×S3×D9
kernelC3×S3×D9C32×D9S3×C3×C9C3×C9⋊S3S3×D9C3×D9S3×C9C9⋊S3C3×D9S3×C32C3×C9C33D9C3×S3C3×S3C9C32C32S3C3C32C3C3C1
# reps111122221111232232661326

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

1000
0100
00110
00011
,
18100
18000
0010
0001
,
0100
1000
00180
00018
,
1000
0100
00170
0009
,
1000
0100
0009
00170
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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