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## G = S32×A4order 432 = 24·33

### Direct product of S3, S3 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — S32×A4
 Chief series C1 — C3 — C32 — C62 — C32×A4 — C3×S3×A4 — S32×A4
 Lower central C62 — S32×A4
 Upper central 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, C3, C3, C22, C22, S3, S3, C6, C23, C32, C32, A4, A4, D6, C2×C6, C2×C6, C24, C3×S3, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C2×A4, C22×S3, C22×S3, C22×C6, C33, S32, S32, C3×A4, C3×A4, S3×C6, C2×C3⋊S3, C62, C22×A4, S3×C23, S3×C32, C3×C3⋊S3, S3×A4, S3×A4, C2×S32, C6×A4, S3×C2×C6, C22×C3⋊S3, C3×S32, C32×A4, C2×S3×A4, C22×S32, C3×S3×A4, A4×C3⋊S3, S32×A4
Quotients: C1, C2, C3, C22, S3, C6, A4, D6, C2×C6, C3×S3, C2×A4, S32, S3×C6, C22×A4, S3×A4, C3×S32, C2×S3×A4, 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 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 6A 6B 6C 6D 6E ··· 6I 6J 6K 6L 6M 6N 6O 6P 6Q order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 6 6 6 6 6 ··· 6 6 6 6 6 6 6 6 6 size 1 3 3 3 9 9 9 27 2 2 4 4 4 8 8 8 8 16 16 6 6 6 6 12 ··· 12 18 18 24 24 24 24 36 36

36 irreducible representations

 dim 1 1 1 1 1 1 12 2 2 2 2 3 3 3 4 4 6 6 type + + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S32×A4 S3 D6 C3×S3 S3×C6 A4 C2×A4 C2×A4 S32 C3×S32 S3×A4 C2×S3×A4 kernel S32×A4 C3×S3×A4 A4×C3⋊S3 C22×S32 S3×C2×C6 C22×C3⋊S3 C1 S3×A4 C3×A4 C22×S3 C2×C6 S32 C3×S3 C3⋊S3 A4 C22 S3 C3 # reps 1 2 1 2 4 2 1 2 2 4 4 1 2 1 1 2 2 2

Matrix representation of S32×A4 in GL7(𝔽7)

 0 1 0 0 0 0 0 6 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 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 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 1 0 0 0 0 0 6 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 0 0 0 0 0 0 6 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 6 0 0 0 0 1 0 6 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 6 0 0 0 0 0 0 6 0 1 0 0 0 0 6 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 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0

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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