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## G = C3×S32order 108 = 22·33

### Direct product of C3, S3 and S3

Aliases: C3×S32, C325D6, C331C22, (C3×S3)⋊C6, C3⋊S32C6, C31(S3×C6), C322(C2×C6), (S3×C32)⋊1C2, (C3×C3⋊S3)⋊1C2, SmallGroup(108,38)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×S32
G = < a,b,c,d,e | a3=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 152 in 54 conjugacy classes, 20 normal (8 characteristic)
C1, C2 [×3], C3, C3 [×2], C3 [×4], C22, S3 [×2], S3 [×3], C6 [×7], C32, C32 [×2], C32 [×4], D6 [×2], C2×C6, C3×S3 [×4], C3×S3 [×5], C3⋊S3, C3×C6 [×2], C33, S32, S3×C6 [×2], S3×C32 [×2], C3×C3⋊S3, C3×S32
Quotients: C1, C2 [×3], C3, C22, S3 [×2], C6 [×3], D6 [×2], C2×C6, C3×S3 [×2], S32, S3×C6 [×2], C3×S32

Character table of C3×S32

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L size 1 3 3 9 1 1 2 2 2 2 2 2 4 4 4 3 3 3 3 6 6 6 6 6 6 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 1 -1 1 1 1 -1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ5 1 -1 -1 1 ζ3 ζ32 ζ32 1 ζ32 ζ3 ζ3 1 ζ3 1 ζ32 ζ6 ζ65 ζ65 ζ6 -1 ζ6 ζ65 ζ65 ζ6 -1 ζ3 ζ32 linear of order 6 ρ6 1 -1 -1 1 ζ32 ζ3 ζ3 1 ζ3 ζ32 ζ32 1 ζ32 1 ζ3 ζ65 ζ6 ζ6 ζ65 -1 ζ65 ζ6 ζ6 ζ65 -1 ζ32 ζ3 linear of order 6 ρ7 1 -1 1 -1 ζ3 ζ32 ζ32 1 ζ32 ζ3 ζ3 1 ζ3 1 ζ32 ζ6 ζ3 ζ65 ζ32 1 ζ32 ζ65 ζ3 ζ6 -1 ζ65 ζ6 linear of order 6 ρ8 1 1 -1 -1 ζ32 ζ3 ζ3 1 ζ3 ζ32 ζ32 1 ζ32 1 ζ3 ζ3 ζ6 ζ32 ζ65 -1 ζ65 ζ32 ζ6 ζ3 1 ζ6 ζ65 linear of order 6 ρ9 1 1 -1 -1 ζ3 ζ32 ζ32 1 ζ32 ζ3 ζ3 1 ζ3 1 ζ32 ζ32 ζ65 ζ3 ζ6 -1 ζ6 ζ3 ζ65 ζ32 1 ζ65 ζ6 linear of order 6 ρ10 1 1 1 1 ζ32 ζ3 ζ3 1 ζ3 ζ32 ζ32 1 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ3 1 ζ32 ζ3 linear of order 3 ρ11 1 -1 1 -1 ζ32 ζ3 ζ3 1 ζ3 ζ32 ζ32 1 ζ32 1 ζ3 ζ65 ζ32 ζ6 ζ3 1 ζ3 ζ6 ζ32 ζ65 -1 ζ6 ζ65 linear of order 6 ρ12 1 1 1 1 ζ3 ζ32 ζ32 1 ζ32 ζ3 ζ3 1 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ32 1 ζ3 ζ32 linear of order 3 ρ13 2 -2 0 0 2 2 2 2 -1 2 -1 -1 -1 -1 -1 -2 0 -2 0 0 0 1 0 1 1 0 0 orthogonal lifted from D6 ρ14 2 2 0 0 2 2 2 2 -1 2 -1 -1 -1 -1 -1 2 0 2 0 0 0 -1 0 -1 -1 0 0 orthogonal lifted from S3 ρ15 2 0 -2 0 2 2 -1 -1 2 -1 2 2 -1 -1 -1 0 -2 0 -2 1 1 0 1 0 0 0 0 orthogonal lifted from D6 ρ16 2 0 2 0 2 2 -1 -1 2 -1 2 2 -1 -1 -1 0 2 0 2 -1 -1 0 -1 0 0 0 0 orthogonal lifted from S3 ρ17 2 0 -2 0 -1-√-3 -1+√-3 ζ65 -1 -1+√-3 ζ6 -1-√-3 2 ζ6 -1 ζ65 0 1+√-3 0 1-√-3 1 ζ3 0 ζ32 0 0 0 0 complex lifted from S3×C6 ρ18 2 0 -2 0 -1+√-3 -1-√-3 ζ6 -1 -1-√-3 ζ65 -1+√-3 2 ζ65 -1 ζ6 0 1-√-3 0 1+√-3 1 ζ32 0 ζ3 0 0 0 0 complex lifted from S3×C6 ρ19 2 2 0 0 -1-√-3 -1+√-3 -1+√-3 2 ζ65 -1-√-3 ζ6 -1 ζ6 -1 ζ65 -1+√-3 0 -1-√-3 0 0 0 ζ6 0 ζ65 -1 0 0 complex lifted from C3×S3 ρ20 2 0 2 0 -1-√-3 -1+√-3 ζ65 -1 -1+√-3 ζ6 -1-√-3 2 ζ6 -1 ζ65 0 -1-√-3 0 -1+√-3 -1 ζ65 0 ζ6 0 0 0 0 complex lifted from C3×S3 ρ21 2 0 2 0 -1+√-3 -1-√-3 ζ6 -1 -1-√-3 ζ65 -1+√-3 2 ζ65 -1 ζ6 0 -1+√-3 0 -1-√-3 -1 ζ6 0 ζ65 0 0 0 0 complex lifted from C3×S3 ρ22 2 2 0 0 -1+√-3 -1-√-3 -1-√-3 2 ζ6 -1+√-3 ζ65 -1 ζ65 -1 ζ6 -1-√-3 0 -1+√-3 0 0 0 ζ65 0 ζ6 -1 0 0 complex lifted from C3×S3 ρ23 2 -2 0 0 -1+√-3 -1-√-3 -1-√-3 2 ζ6 -1+√-3 ζ65 -1 ζ65 -1 ζ6 1+√-3 0 1-√-3 0 0 0 ζ3 0 ζ32 1 0 0 complex lifted from S3×C6 ρ24 2 -2 0 0 -1-√-3 -1+√-3 -1+√-3 2 ζ65 -1-√-3 ζ6 -1 ζ6 -1 ζ65 1-√-3 0 1+√-3 0 0 0 ζ32 0 ζ3 1 0 0 complex lifted from S3×C6 ρ25 4 0 0 0 4 4 -2 -2 -2 -2 -2 -2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ26 4 0 0 0 -2+2√-3 -2-2√-3 1+√-3 -2 1+√-3 1-√-3 1-√-3 -2 ζ3 1 ζ32 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ27 4 0 0 0 -2-2√-3 -2+2√-3 1-√-3 -2 1-√-3 1+√-3 1+√-3 -2 ζ32 1 ζ3 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C3×S32
On 12 points - transitive group 12T70
Generators in S12
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 3 2)(4 5 6)(7 9 8)(10 11 12)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)
(1 2 3)(4 5 6)(7 9 8)(10 12 11)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)

G:=sub<Sym(12)| (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,3,2)(4,5,6)(7,9,8)(10,11,12), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (1,2,3)(4,5,6)(7,9,8)(10,12,11), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)>;

G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,3,2)(4,5,6)(7,9,8)(10,11,12), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (1,2,3)(4,5,6)(7,9,8)(10,12,11), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(1,3,2),(4,5,6),(7,9,8),(10,11,12)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)], [(1,2,3),(4,5,6),(7,9,8),(10,12,11)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)])

G:=TransitiveGroup(12,70);

On 18 points - transitive group 18T43
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 14 10)(2 15 11)(3 13 12)(4 8 18)(5 9 16)(6 7 17)
(1 16)(2 17)(3 18)(4 12)(5 10)(6 11)(7 15)(8 13)(9 14)
(1 10 14)(2 11 15)(3 12 13)(4 8 18)(5 9 16)(6 7 17)
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 11)(8 12)(9 10)

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

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

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

G:=TransitiveGroup(18,43);

On 18 points - transitive group 18T46
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(4 7)(5 8)(6 9)(13 16)(14 17)(15 18)
(1 2 3)(4 5 6)(7 8 9)(10 12 11)(13 15 14)(16 18 17)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)

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

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

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

G:=TransitiveGroup(18,46);

On 27 points - transitive group 27T36
Generators in S27
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 14 6)(2 15 4)(3 13 5)(7 16 24)(8 17 22)(9 18 23)(10 19 27)(11 20 25)(12 21 26)
(4 15)(5 13)(6 14)(10 19)(11 20)(12 21)(16 24)(17 22)(18 23)
(1 8 26)(2 9 27)(3 7 25)(4 23 19)(5 24 20)(6 22 21)(10 15 18)(11 13 16)(12 14 17)
(7 25)(8 26)(9 27)(10 18)(11 16)(12 17)(19 23)(20 24)(21 22)

G:=sub<Sym(27)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (4,15)(5,13)(6,14)(10,19)(11,20)(12,21)(16,24)(17,22)(18,23), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (7,25)(8,26)(9,27)(10,18)(11,16)(12,17)(19,23)(20,24)(21,22)>;

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)(25,26,27), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (4,15)(5,13)(6,14)(10,19)(11,20)(12,21)(16,24)(17,22)(18,23), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (7,25)(8,26)(9,27)(10,18)(11,16)(12,17)(19,23)(20,24)(21,22) );

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),(25,26,27)], [(1,14,6),(2,15,4),(3,13,5),(7,16,24),(8,17,22),(9,18,23),(10,19,27),(11,20,25),(12,21,26)], [(4,15),(5,13),(6,14),(10,19),(11,20),(12,21),(16,24),(17,22),(18,23)], [(1,8,26),(2,9,27),(3,7,25),(4,23,19),(5,24,20),(6,22,21),(10,15,18),(11,13,16),(12,14,17)], [(7,25),(8,26),(9,27),(10,18),(11,16),(12,17),(19,23),(20,24),(21,22)])

G:=TransitiveGroup(27,36);

C3×S32 is a maximal subgroup of   C33⋊D4

Polynomial with Galois group C3×S32 over ℚ
actionf(x)Disc(f)
12T70x12+9x6-18x3+9212·334·56

Matrix representation of C3×S32 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 6 6
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 6 6
,
 0 1 0 0 6 6 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 6 6 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,6,0,0,1,6],[1,0,0,0,0,1,0,0,0,0,1,6,0,0,0,6],[0,6,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[1,6,0,0,0,6,0,0,0,0,1,0,0,0,0,1] >;

C3×S32 in GAP, Magma, Sage, TeX

C_3\times S_3^2
% in TeX

G:=Group("C3xS3^2");
// GroupNames label

G:=SmallGroup(108,38);
// by ID

G=gap.SmallGroup(108,38);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,248,1804]);
// Polycyclic

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

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