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## G = C3×S3×S4order 432 = 24·33

### Direct product of C3, S3 and S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C3×S3×S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C32×A4 — C3×S3×A4 — C3×S3×S4
 Lower central C3×A4 — C3×S3×S4
 Upper central C1 — C3

Generators and relations for C3×S3×S4
G = < a,b,c,d,e,f,g | a3=b3=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, cbc=b-1, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 904 in 163 conjugacy classes, 26 normal (all characteristic)
C1, C2 [×5], C3 [×2], C3 [×5], C4 [×2], C22, C22 [×6], S3, S3 [×5], C6 [×11], C2×C4, D4 [×4], C23 [×2], C32, C32 [×6], Dic3, C12 [×4], A4, A4 [×3], D6 [×6], C2×C6 [×2], C2×C6 [×9], C2×D4, C3×S3, C3×S3 [×9], C3⋊S3, C3×C6 [×3], C4×S3, D12, C3⋊D4 [×2], C2×C12, C3×D4 [×6], S4, S4 [×2], C2×A4 [×2], C22×S3, C22×S3, C22×C6 [×2], C33, C3×Dic3, C3×C12, S32, C3×A4 [×2], C3×A4 [×4], S3×C6 [×6], C62, C62, S3×D4, C6×D4, C2×S4, S3×C32 [×2], C3×C3⋊S3, S3×C12, C3×D12, C3×C3⋊D4 [×2], D4×C32, C3×S4 [×2], C3×S4 [×3], C3⋊S4, S3×A4, S3×A4, C6×A4, S3×C2×C6, S3×C2×C6, C3×S32, C32×A4, C3×S3×D4, S3×S4, C6×S4, C32×S4, C3×C3⋊S4, C3×S3×A4, C3×S3×S4
Quotients: C1, C2 [×3], C3, C22, S3 [×2], C6 [×3], D6 [×2], C2×C6, C3×S3 [×2], S4, S32, S3×C6 [×2], C2×S4, C3×S4, C3×S32, S3×S4, C6×S4, C3×S3×S4

Permutation representations of C3×S3×S4
On 24 points - transitive group 24T1328
Generators in S24
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 2 3)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 18 17)(19 20 21)(22 24 23)
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 22)(11 23)(12 24)(13 18)(14 16)(15 17)
(1 14)(2 15)(3 13)(4 24)(5 22)(6 23)(7 16)(8 17)(9 18)(10 20)(11 21)(12 19)
(1 21)(2 19)(3 20)(4 8)(5 9)(6 7)(10 13)(11 14)(12 15)(16 23)(17 24)(18 22)
(4 24 17)(5 22 18)(6 23 16)(10 13 20)(11 14 21)(12 15 19)
(10 13)(11 14)(12 15)(16 23)(17 24)(18 22)

G:=sub<Sym(24)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2,3)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,18,17)(19,20,21)(22,24,23), (1,7)(2,8)(3,9)(4,19)(5,20)(6,21)(10,22)(11,23)(12,24)(13,18)(14,16)(15,17), (1,14)(2,15)(3,13)(4,24)(5,22)(6,23)(7,16)(8,17)(9,18)(10,20)(11,21)(12,19), (1,21)(2,19)(3,20)(4,8)(5,9)(6,7)(10,13)(11,14)(12,15)(16,23)(17,24)(18,22), (4,24,17)(5,22,18)(6,23,16)(10,13,20)(11,14,21)(12,15,19), (10,13)(11,14)(12,15)(16,23)(17,24)(18,22)>;

G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2,3)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,18,17)(19,20,21)(22,24,23), (1,7)(2,8)(3,9)(4,19)(5,20)(6,21)(10,22)(11,23)(12,24)(13,18)(14,16)(15,17), (1,14)(2,15)(3,13)(4,24)(5,22)(6,23)(7,16)(8,17)(9,18)(10,20)(11,21)(12,19), (1,21)(2,19)(3,20)(4,8)(5,9)(6,7)(10,13)(11,14)(12,15)(16,23)(17,24)(18,22), (4,24,17)(5,22,18)(6,23,16)(10,13,20)(11,14,21)(12,15,19), (10,13)(11,14)(12,15)(16,23)(17,24)(18,22) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,2,3),(4,6,5),(7,9,8),(10,11,12),(13,14,15),(16,18,17),(19,20,21),(22,24,23)], [(1,7),(2,8),(3,9),(4,19),(5,20),(6,21),(10,22),(11,23),(12,24),(13,18),(14,16),(15,17)], [(1,14),(2,15),(3,13),(4,24),(5,22),(6,23),(7,16),(8,17),(9,18),(10,20),(11,21),(12,19)], [(1,21),(2,19),(3,20),(4,8),(5,9),(6,7),(10,13),(11,14),(12,15),(16,23),(17,24),(18,22)], [(4,24,17),(5,22,18),(6,23,16),(10,13,20),(11,14,21),(12,15,19)], [(10,13),(11,14),(12,15),(16,23),(17,24),(18,22)])

G:=TransitiveGroup(24,1328);

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 4A 4B 6A 6B 6C 6D 6E ··· 6I 6J 6K 6L 6M 6N 6O 6P 6Q 6R 6S 12A 12B 12C 12D 12E 12F 12G order 1 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 4 4 6 6 6 6 6 ··· 6 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 12 12 size 1 3 3 6 9 18 1 1 2 2 2 8 8 8 16 16 16 6 18 3 3 3 3 6 ··· 6 9 9 12 12 12 18 18 24 24 24 6 6 12 12 12 18 18

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 4 4 6 6 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 S3 D6 D6 C3×S3 C3×S3 S3×C6 S3×C6 S4 C2×S4 C3×S4 C6×S4 S32 C3×S32 S3×S4 C3×S3×S4 kernel C3×S3×S4 C32×S4 C3×C3⋊S4 C3×S3×A4 S3×S4 C3×S4 C3⋊S4 S3×A4 C3×S4 S3×C2×C6 C3×A4 C62 S4 C22×S3 A4 C2×C6 C3×S3 C32 S3 C3 C2×C6 C22 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 4 4 1 2 2 4

Matrix representation of C3×S3×S4 in GL7(𝔽13)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 9 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 12 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 12 12 12 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12 12 0 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 1 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 12 0 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[9,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,1,12,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,1,0,0,0,0,12,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C3×S3×S4 in GAP, Magma, Sage, TeX

C_3\times S_3\times S_4
% in TeX

G:=Group("C3xS3xS4");
// GroupNames label

G:=SmallGroup(432,745);
// by ID

G=gap.SmallGroup(432,745);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,346,2524,4548,782,2659,1350]);
// Polycyclic

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

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