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## G = C3×S3≀C2order 216 = 23·33

### Direct product of C3 and S3≀C2

Aliases: C3×S3≀C2, C331D4, S32⋊C6, C32⋊C4⋊C6, C32⋊(C3×D4), (C3×S32)⋊1C2, C3⋊S3.1(C2×C6), (C3×C32⋊C4)⋊3C2, (C3×C3⋊S3).1C22, SmallGroup(216,157)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C3×S3≀C2
 Chief series C1 — C32 — C3⋊S3 — C3×C3⋊S3 — C3×S32 — C3×S3≀C2
 Lower central C32 — C3⋊S3 — C3×S3≀C2
 Upper central C1 — C3

Generators and relations for C3×S3≀C2
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 272 in 60 conjugacy classes, 14 normal (10 characteristic)
C1, C2 [×3], C3, C3 [×4], C4, C22 [×2], S3 [×4], C6 [×7], D4, C32, C32 [×4], C12, D6 [×2], C2×C6 [×2], C3×S3 [×8], C3⋊S3, C3×C6 [×2], C3×D4, C33, C32⋊C4, S32 [×2], S3×C6 [×2], S3×C32 [×2], C3×C3⋊S3, S3≀C2, C3×C32⋊C4, C3×S32 [×2], C3×S3≀C2
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, C2×C6, C3×D4, S3≀C2, C3×S3≀C2

Character table of C3×S3≀C2

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 4 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 12A 12B size 1 6 6 9 1 1 4 4 4 4 4 4 18 6 6 6 6 9 9 12 12 12 12 12 12 18 18 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 ζ32 ζ3 ζ3 1 1 -1 ζ3 ζ6 ζ65 ζ32 ζ32 ζ3 -1 ζ65 ζ32 ζ6 ζ3 1 ζ6 ζ65 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ3 1 ζ32 ζ3 linear of order 3 ρ7 1 -1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 -1 ζ6 ζ3 ζ32 ζ65 ζ3 ζ32 1 ζ32 ζ65 ζ3 ζ6 -1 ζ65 ζ6 linear of order 6 ρ8 1 1 -1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 -1 ζ32 ζ65 ζ6 ζ3 ζ3 ζ32 -1 ζ6 ζ3 ζ65 ζ32 1 ζ65 ζ6 linear of order 6 ρ9 1 -1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 -1 ζ65 ζ32 ζ3 ζ6 ζ32 ζ3 1 ζ3 ζ6 ζ32 ζ65 -1 ζ6 ζ65 linear of order 6 ρ10 1 -1 -1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 ζ65 ζ6 ζ65 ζ6 ζ32 ζ3 -1 ζ65 ζ6 ζ6 ζ65 -1 ζ32 ζ3 linear of order 6 ρ11 1 -1 -1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 ζ6 ζ65 ζ6 ζ65 ζ3 ζ32 -1 ζ6 ζ65 ζ65 ζ6 -1 ζ3 ζ32 linear of order 6 ρ12 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ32 1 ζ3 ζ32 linear of order 3 ρ13 2 0 0 -2 2 2 2 2 2 2 2 2 0 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 0 0 -2 -1-√-3 -1+√-3 -1+√-3 -1+√-3 -1-√-3 -1-√-3 2 2 0 0 0 0 0 1-√-3 1+√-3 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ15 2 0 0 -2 -1+√-3 -1-√-3 -1-√-3 -1-√-3 -1+√-3 -1+√-3 2 2 0 0 0 0 0 1+√-3 1-√-3 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ16 4 0 -2 0 4 4 1 -2 -2 1 -2 1 0 0 -2 -2 0 0 0 1 1 0 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ17 4 0 2 0 4 4 1 -2 -2 1 -2 1 0 0 2 2 0 0 0 -1 -1 0 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ18 4 -2 0 0 4 4 -2 1 1 -2 1 -2 0 -2 0 0 -2 0 0 0 0 1 0 1 1 0 0 orthogonal lifted from S3≀C2 ρ19 4 2 0 0 4 4 -2 1 1 -2 1 -2 0 2 0 0 2 0 0 0 0 -1 0 -1 -1 0 0 orthogonal lifted from S3≀C2 ρ20 4 2 0 0 -2+2√-3 -2-2√-3 1+√-3 ζ32 ζ3 1-√-3 1 -2 0 -1+√-3 0 0 -1-√-3 0 0 0 0 ζ6 0 ζ65 -1 0 0 complex faithful ρ21 4 0 2 0 -2+2√-3 -2-2√-3 ζ32 1+√-3 1-√-3 ζ3 -2 1 0 0 -1-√-3 -1+√-3 0 0 0 -1 ζ65 0 ζ6 0 0 0 0 complex faithful ρ22 4 -2 0 0 -2-2√-3 -2+2√-3 1-√-3 ζ3 ζ32 1+√-3 1 -2 0 1+√-3 0 0 1-√-3 0 0 0 0 ζ3 0 ζ32 1 0 0 complex faithful ρ23 4 0 -2 0 -2-2√-3 -2+2√-3 ζ3 1-√-3 1+√-3 ζ32 -2 1 0 0 1-√-3 1+√-3 0 0 0 1 ζ32 0 ζ3 0 0 0 0 complex faithful ρ24 4 -2 0 0 -2+2√-3 -2-2√-3 1+√-3 ζ32 ζ3 1-√-3 1 -2 0 1-√-3 0 0 1+√-3 0 0 0 0 ζ32 0 ζ3 1 0 0 complex faithful ρ25 4 0 2 0 -2-2√-3 -2+2√-3 ζ3 1-√-3 1+√-3 ζ32 -2 1 0 0 -1+√-3 -1-√-3 0 0 0 -1 ζ6 0 ζ65 0 0 0 0 complex faithful ρ26 4 0 -2 0 -2+2√-3 -2-2√-3 ζ32 1+√-3 1-√-3 ζ3 -2 1 0 0 1+√-3 1-√-3 0 0 0 1 ζ3 0 ζ32 0 0 0 0 complex faithful ρ27 4 2 0 0 -2-2√-3 -2+2√-3 1-√-3 ζ3 ζ32 1+√-3 1 -2 0 -1-√-3 0 0 -1+√-3 0 0 0 0 ζ65 0 ζ6 -1 0 0 complex faithful

Permutation representations of C3×S3≀C2
On 12 points - transitive group 12T121
Generators in S12
(1 9 8)(2 10 5)(3 11 6)(4 12 7)
(2 5 10)(4 12 7)
(1 8 9)(3 11 6)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 7)(10 12)

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

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

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

G:=TransitiveGroup(12,121);

On 18 points - transitive group 18T93
Generators in S18
(1 4 6)(2 3 5)(7 13 18)(8 14 15)(9 11 16)(10 12 17)
(1 15 17)(2 16 18)(3 9 7)(4 8 10)(5 11 13)(6 14 12)
(1 15 17)(2 18 16)(3 7 9)(4 8 10)(5 13 11)(6 14 12)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 18)(16 17)

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

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

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

G:=TransitiveGroup(18,93);

On 24 points - transitive group 24T561
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 21 20)(2 22 17)(3 18 23)(4 19 24)(5 14 10)(6 15 11)(7 12 16)(8 9 13)
(1 21 20)(2 17 22)(3 18 23)(4 24 19)(5 14 10)(6 11 15)(7 12 16)(8 13 9)
(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 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,561);

On 27 points - transitive group 27T84
Generators in S27
(1 2 3)(4 20 13)(5 21 14)(6 22 15)(7 23 12)(8 25 18)(9 26 19)(10 27 16)(11 24 17)
(1 15 13)(2 6 4)(3 22 20)(5 10 9)(7 11 8)(12 17 18)(14 16 19)(21 27 26)(23 24 25)
(1 14 12)(2 5 7)(3 21 23)(4 9 8)(6 10 11)(13 19 18)(15 16 17)(20 26 25)(22 27 24)
(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)
(4 6)(8 11)(9 10)(13 15)(16 19)(17 18)(20 22)(24 25)(26 27)

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

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

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

G:=TransitiveGroup(27,84);

C3×S3≀C2 is a maximal subgroup of   C33⋊SD16

Polynomial with Galois group C3×S3≀C2 over ℚ
actionf(x)Disc(f)
12T121x12-x9+2x6-2x3+1318·133

Matrix representation of C3×S3≀C2 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 4 2 5 0 2 3 5 2 2 2 6 2 2 4 3 2
,
 3 5 5 1 3 1 1 5 6 4 6 3 1 6 6 5
,
 2 4 6 3 4 2 6 5 4 3 5 1 1 6 6 5
,
 4 2 5 3 3 3 3 6 3 4 2 6 2 5 2 3
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,2,2,2,2,3,2,4,5,5,6,3,0,2,2,2],[3,3,6,1,5,1,4,6,5,1,6,6,1,5,3,5],[2,4,4,1,4,2,3,6,6,6,5,6,3,5,1,5],[4,3,3,2,2,3,4,5,5,3,2,2,3,6,6,3] >;

C3×S3≀C2 in GAP, Magma, Sage, TeX

C_3\times S_3\wr C_2
% in TeX

G:=Group("C3xS3wrC2");
// GroupNames label

G:=SmallGroup(216,157);
// by ID

G=gap.SmallGroup(216,157);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,3,169,1444,1090,142,443,455]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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