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G = S32×A4order 432 = 24·33

Direct product of S3, S3 and A4

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

Aliases: S32×A4, (C3×A4)⋊8D6, C622(C2×C6), C322(C22×A4), (C32×A4)⋊5C22, (S3×C2×C6)⋊C6, C31(C2×S3×A4), (C3×S3)⋊(C2×A4), (C3×S3×A4)⋊3C2, (C22×S32)⋊C3, C3⋊S32(C2×A4), (A4×C3⋊S3)⋊3C2, (C2×C6)⋊2(S3×C6), C222(C3×S32), (C22×C3⋊S3)⋊4C6, (C22×S3)⋊2(C3×S3), SmallGroup(432,749)

Series: Derived Chief Lower central Upper central

C1C62 — S32×A4
C1C3C32C62C32×A4C3×S3×A4 — S32×A4
C62 — S32×A4
C1

Generators and relations for S32×A4
 G = < a,b,c,d,e,f,g | a3=b2=c3=d2=e2=f2=g3=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, dcd=c-1, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 1372 in 188 conjugacy classes, 30 normal (12 characteristic)
C1, C2 [×7], C3 [×2], C3 [×5], C22, C22 [×12], S3 [×2], S3 [×8], C6 [×12], C23 [×7], C32, C32 [×6], A4, A4 [×3], D6 [×22], C2×C6 [×2], C2×C6 [×6], C24, C3×S3 [×2], C3×S3 [×9], C3⋊S3, C3⋊S3, C3×C6 [×3], C2×A4 [×5], C22×S3 [×2], C22×S3 [×11], C22×C6 [×2], C33, S32, S32 [×5], C3×A4 [×2], C3×A4 [×4], S3×C6 [×6], C2×C3⋊S3 [×2], C62, C22×A4, S3×C23 [×2], S3×C32 [×2], C3×C3⋊S3, S3×A4 [×2], S3×A4 [×5], C2×S32 [×4], C6×A4 [×2], S3×C2×C6 [×2], C22×C3⋊S3, C3×S32, C32×A4, C2×S3×A4 [×2], C22×S32, C3×S3×A4 [×2], A4×C3⋊S3, S32×A4
Quotients: C1, C2 [×3], C3, C22, S3 [×2], C6 [×3], A4, D6 [×2], C2×C6, C3×S3 [×2], C2×A4 [×3], S32, S3×C6 [×2], C22×A4, S3×A4 [×2], C3×S32, C2×S3×A4 [×2], S32×A4

Permutation representations of S32×A4
On 24 points - transitive group 24T1338
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 13)(2 15)(3 14)(4 16)(5 18)(6 17)(7 19)(8 21)(9 20)(10 22)(11 24)(12 23)
(1 3 2)(4 6 5)(7 9 8)(10 12 11)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(4 7 10)(5 8 11)(6 9 12)(16 19 22)(17 20 23)(18 21 24)

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

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

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

G:=TransitiveGroup(24,1338);

36 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F3G3H3I3J3K6A6B6C6D6E···6I6J6K6L6M6N6O6P6Q
order122222223333333333366666···666666666
size1333999272244488881616666612···121818242424243636

36 irreducible representations

dim1111111222223334466
type++++++++++++
imageC1C2C2C3C6C6S32×A4S3D6C3×S3S3×C6A4C2×A4C2×A4S32C3×S32S3×A4C2×S3×A4
kernelS32×A4C3×S3×A4A4×C3⋊S3C22×S32S3×C2×C6C22×C3⋊S3C1S3×A4C3×A4C22×S3C2×C6S32C3×S3C3⋊S3A4C22S3C3
# reps121242122441211222

Matrix representation of S32×A4 in GL7(𝔽7)

0100000
6600000
0010000
0001000
0000100
0000010
0000001
,
6000000
1100000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
0001000
0066000
0000100
0000010
0000001
,
6000000
0600000
0010000
0066000
0000600
0000060
0000006
,
1000000
0100000
0010000
0001000
0000016
0000106
0000006
,
1000000
0100000
0010000
0001000
0000600
0000601
0000610
,
2000000
0200000
0020000
0002000
0000001
0000100
0000010

G:=sub<GL(7,GF(7))| [0,6,0,0,0,0,0,1,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,1,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,6,6,6],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,6,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0] >;

S32×A4 in GAP, Magma, Sage, TeX

S_3^2\times A_4
% in TeX

G:=Group("S3^2xA4");
// GroupNames label

G:=SmallGroup(432,749);
// by ID

G=gap.SmallGroup(432,749);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-3,-3,269,123,2028,14118]);
// Polycyclic

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

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