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## G = C2×S33order 432 = 24·33

### Direct product of C2, S3, S3 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×S33
 Chief series C1 — C3 — C32 — C33 — S3×C32 — C3×S32 — S33 — C2×S33
 Lower central C33 — C2×S33
 Upper central C1 — C2

Generators and relations for C2×S33
G = < a,b,c,d,e,f,g | a2=b3=c2=d3=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, ede=d-1, df=fd, dg=gd, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 3916 in 642 conjugacy classes, 132 normal (7 characteristic)
C1, C2, C2 [×14], C3 [×3], C3 [×4], C22 [×35], S3 [×6], S3 [×32], C6 [×3], C6 [×28], C23 [×15], C32 [×3], C32 [×4], D6 [×3], D6 [×100], C2×C6 [×24], C24, C3×S3 [×12], C3×S3 [×24], C3⋊S3 [×6], C3⋊S3 [×14], C3×C6 [×3], C3×C6 [×10], C22×S3 [×45], C22×C6 [×3], C33, S32 [×12], S32 [×48], S3×C6 [×6], S3×C6 [×36], C2×C3⋊S3 [×3], C2×C3⋊S3 [×19], C62 [×3], S3×C23 [×3], S3×C32 [×6], C3×C3⋊S3 [×6], C33⋊C2 [×2], C32×C6, C2×S32 [×3], C2×S32 [×36], S3×C2×C6 [×6], C22×C3⋊S3 [×3], C3×S32 [×12], S3×C3⋊S3 [×12], C324D6 [×4], S3×C3×C6 [×3], C6×C3⋊S3 [×3], C2×C33⋊C2, C22×S32 [×3], S33 [×8], S32×C6 [×3], C2×S3×C3⋊S3 [×3], C2×C324D6, C2×S33
Quotients: C1, C2 [×15], C22 [×35], S3 [×3], C23 [×15], D6 [×21], C24, C22×S3 [×21], S32 [×3], S3×C23 [×3], C2×S32 [×9], C22×S32 [×3], S33, C2×S33

Permutation representations of C2×S33
On 24 points - transitive group 24T1296
Generators in S24
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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 16)(2 18)(3 17)(4 14)(5 13)(6 15)(7 23)(8 22)(9 24)(10 20)(11 19)(12 21)
(1 2 3)(4 5 6)(7 9 8)(10 12 11)(13 15 14)(16 18 17)(19 20 21)(22 23 24)
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(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 19)(2 20)(3 21)(4 24)(5 22)(6 23)(7 15)(8 13)(9 14)(10 18)(11 16)(12 17)

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

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

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

G:=TransitiveGroup(24,1296);

54 conjugacy classes

 class 1 2A 2B ··· 2G 2H ··· 2M 2N 2O 3A 3B 3C 3D 3E 3F 3G 6A 6B 6C 6D 6E 6F 6G ··· 6R 6S 6T ··· 6Y 6Z ··· 6AE order 1 2 2 ··· 2 2 ··· 2 2 2 3 3 3 3 3 3 3 6 6 6 6 6 6 6 ··· 6 6 6 ··· 6 6 ··· 6 size 1 1 3 ··· 3 9 ··· 9 27 27 2 2 2 4 4 4 8 2 2 2 4 4 4 6 ··· 6 8 12 ··· 12 18 ··· 18

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 4 8 8 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D6 D6 D6 S32 C2×S32 C2×S32 S33 C2×S33 kernel C2×S33 S33 S32×C6 C2×S3×C3⋊S3 C2×C32⋊4D6 C2×S32 S32 S3×C6 C2×C3⋊S3 D6 S3 C6 C2 C1 # reps 1 8 3 3 1 3 12 6 3 3 6 3 1 1

Matrix representation of C2×S33 in GL6(ℤ)

 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×S33 in GAP, Magma, Sage, TeX

C_2\times S_3^3
% in TeX

G:=Group("C2xS3^3");
// GroupNames label

G:=SmallGroup(432,759);
// by ID

G=gap.SmallGroup(432,759);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,298,2028,14118]);
// Polycyclic

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

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