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G = C3×S3×S4order 432 = 24·33

Direct product of C3, S3 and S4

direct product, non-abelian, soluble, monomial

Aliases: C3×S3×S4, C627D6, (C3×S4)⋊C6, (S3×A4)⋊C6, C3⋊S42C6, C31(C6×S4), A41(S3×C6), (C3×A4)⋊7D6, C326(C2×S4), (C32×S4)⋊1C2, (C32×A4)⋊1C22, (C2×C6)⋊3S32, (C2×C6)⋊(S3×C6), C22⋊(C3×S32), (C3×S3×A4)⋊1C2, (S3×C2×C6)⋊1S3, (C3×C3⋊S4)⋊3C2, (C3×A4)⋊2(C2×C6), (C22×S3)⋊(C3×S3), SmallGroup(432,745)

Series: Derived Chief Lower central Upper central

C1C22C3×A4 — C3×S3×S4
C1C22C2×C6C3×A4C32×A4C3×S3×A4 — C3×S3×S4
C3×A4 — C3×S3×S4
C1C3

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 2A2B2C2D2E3A3B3C3D3E3F3G3H3I3J3K4A4B6A6B6C6D6E···6I6J6K6L6M6N6O6P6Q6R6S12A12B12C12D12E12F12G
order122222333333333334466666···6666666666612121212121212
size13369181122288816161661833336···6991212121818242424661212121818

45 irreducible representations

dim111111112222222233334466
type++++++++++++
imageC1C2C2C2C3C6C6C6S3S3D6D6C3×S3C3×S3S3×C6S3×C6S4C2×S4C3×S4C6×S4S32C3×S32S3×S4C3×S3×S4
kernelC3×S3×S4C32×S4C3×C3⋊S4C3×S3×A4S3×S4C3×S4C3⋊S4S3×A4C3×S4S3×C2×C6C3×A4C62S4C22×S3A4C2×C6C3×S3C32S3C3C2×C6C22C3C1
# reps111122221111222222441224

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

1000000
0100000
0030000
0003000
0000100
0000010
0000001
,
9000000
3300000
0010000
0001000
0000100
0000010
0000001
,
12200000
0100000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
0000001
0000121212
0000100
,
1000000
0100000
0010000
0001000
0000121212
0000001
0000010
,
1000000
0100000
00012000
00112000
0000100
0000121212
0000010
,
12000000
01200000
00120000
00121000
0000100
0000001
0000010

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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