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G = C3×S3≀C2order 216 = 23·33

Direct product of C3 and S3≀C2

direct product, non-abelian, soluble, monomial

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

C1C32C3⋊S3 — C3×S3≀C2
C1C32C3⋊S3C3×C3⋊S3C3×S32 — C3×S3≀C2
C32C3⋊S3 — C3×S3≀C2
C1C3

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 12A2B2C3A3B3C3D3E3F3G3H46A6B6C6D6E6F6G6H6I6J6K6L12A12B
 size 166911444444186666991212121212121818
ρ1111111111111111111111111111    trivial
ρ211-1111111111-11-1-1111-1-11-111-1-1    linear of order 2
ρ31-11111111111-1-111-11111-11-1-1-1-1    linear of order 2
ρ41-1-11111111111-1-1-1-111-1-1-1-1-1-111    linear of order 2
ρ511-11ζ3ζ32ζ32ζ32ζ3ζ311-1ζ3ζ6ζ65ζ32ζ32ζ3-1ζ65ζ32ζ6ζ31ζ6ζ65    linear of order 6
ρ61111ζ3ζ32ζ32ζ32ζ3ζ3111ζ3ζ32ζ3ζ32ζ32ζ31ζ3ζ32ζ32ζ31ζ32ζ3    linear of order 3
ρ71-111ζ32ζ3ζ3ζ3ζ32ζ3211-1ζ6ζ3ζ32ζ65ζ3ζ321ζ32ζ65ζ3ζ6-1ζ65ζ6    linear of order 6
ρ811-11ζ32ζ3ζ3ζ3ζ32ζ3211-1ζ32ζ65ζ6ζ3ζ3ζ32-1ζ6ζ3ζ65ζ321ζ65ζ6    linear of order 6
ρ91-111ζ3ζ32ζ32ζ32ζ3ζ311-1ζ65ζ32ζ3ζ6ζ32ζ31ζ3ζ6ζ32ζ65-1ζ6ζ65    linear of order 6
ρ101-1-11ζ3ζ32ζ32ζ32ζ3ζ3111ζ65ζ6ζ65ζ6ζ32ζ3-1ζ65ζ6ζ6ζ65-1ζ32ζ3    linear of order 6
ρ111-1-11ζ32ζ3ζ3ζ3ζ32ζ32111ζ6ζ65ζ6ζ65ζ3ζ32-1ζ6ζ65ζ65ζ6-1ζ3ζ32    linear of order 6
ρ121111ζ32ζ3ζ3ζ3ζ32ζ32111ζ32ζ3ζ32ζ3ζ3ζ321ζ32ζ3ζ3ζ321ζ3ζ32    linear of order 3
ρ13200-22222222200000-2-200000000    orthogonal lifted from D4
ρ14200-2-1--3-1+-3-1+-3-1+-3-1--3-1--322000001--31+-300000000    complex lifted from C3×D4
ρ15200-2-1+-3-1--3-1--3-1--3-1+-3-1+-322000001+-31--300000000    complex lifted from C3×D4
ρ1640-20441-2-21-2100-2-200011010000    orthogonal lifted from S3≀C2
ρ174020441-2-21-210022000-1-10-10000    orthogonal lifted from S3≀C2
ρ184-20044-211-21-20-200-20000101100    orthogonal lifted from S3≀C2
ρ19420044-211-21-2020020000-10-1-100    orthogonal lifted from S3≀C2
ρ204200-2+2-3-2-2-31+-3ζ32ζ31--31-20-1+-300-1--30000ζ60ζ65-100    complex faithful
ρ214020-2+2-3-2-2-3ζ321+-31--3ζ3-2100-1--3-1+-3000-1ζ650ζ60000    complex faithful
ρ224-200-2-2-3-2+2-31--3ζ3ζ321+-31-201+-3001--30000ζ30ζ32100    complex faithful
ρ2340-20-2-2-3-2+2-3ζ31--31+-3ζ32-21001--31+-30001ζ320ζ30000    complex faithful
ρ244-200-2+2-3-2-2-31+-3ζ32ζ31--31-201--3001+-30000ζ320ζ3100    complex faithful
ρ254020-2-2-3-2+2-3ζ31--31+-3ζ32-2100-1+-3-1--3000-1ζ60ζ650000    complex faithful
ρ2640-20-2+2-3-2-2-3ζ321+-31--3ζ3-21001+-31--30001ζ30ζ320000    complex faithful
ρ274200-2-2-3-2+2-31--3ζ3ζ321+-31-20-1--300-1+-30000ζ650ζ6-100    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

2000
0200
0020
0002
,
4250
2352
2262
2432
,
3551
3115
6463
1665
,
2463
4265
4351
1665
,
4253
3336
3426
2523
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

Export

Character table of C3×S3≀C2 in TeX

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