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G = C3≀S3order 162 = 2·34

Wreath product of C3 by S3

non-abelian, supersoluble, monomial

Aliases: C3S3, He3⋊C6, C331S3, C3≀C32C2, He3⋊C21C3, C32.1(C3×S3), C3.6(C32⋊C6), SmallGroup(162,10)

Series: Derived Chief Lower central Upper central

C1C3He3 — C3≀S3
C1C3C32He3C3≀C3 — C3≀S3
He3 — C3≀S3
C1C3

Generators and relations for C3≀S3
 G = < a,b,c,d | a3=b3=c3=d6=1, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a-1c-1 >

9C2
3C3
3C3
3C3
3C3
9C3
3S3
9S3
9C6
9C6
9C6
9C6
3C32
3C32
3C32
3C32
3C32
6C9
3C3×S3
3C3×S3
3C3×S3
3C3×S3
9C3×C6
9C3×S3
23- 1+2
3S3×C32

Character table of C3≀S3

 class 123A3B3C3D3E3F3G3H3I3J6A6B6C6D6E6F6G6H9A9B
 size 1911333333618999999991818
ρ11111111111111111111111    trivial
ρ21-11111111111-1-1-1-1-1-1-1-111    linear of order 2
ρ31111ζ32ζ3ζ32ζ32ζ3ζ311ζ321ζ3ζ3ζ3ζ32ζ321ζ32ζ3    linear of order 3
ρ41-111ζ32ζ3ζ32ζ32ζ3ζ311ζ6-1ζ65ζ65ζ65ζ6ζ6-1ζ32ζ3    linear of order 6
ρ51111ζ3ζ32ζ3ζ3ζ32ζ3211ζ31ζ32ζ32ζ32ζ3ζ31ζ3ζ32    linear of order 3
ρ61-111ζ3ζ32ζ3ζ3ζ32ζ3211ζ65-1ζ6ζ6ζ6ζ65ζ65-1ζ3ζ32    linear of order 6
ρ720222222222-100000000-1-1    orthogonal lifted from S3
ρ82022-1--3-1+-3-1--3-1--3-1+-3-1+-32-100000000ζ6ζ65    complex lifted from C3×S3
ρ92022-1+-3-1--3-1+-3-1+-3-1--3-1--32-100000000ζ65ζ6    complex lifted from C3×S3
ρ1031-3-3-3/2-3+3-3/2--3-3--3/23+-3/2-3+-3/23--3/2-3001ζ32ζ321ζ3ζ3ζ32ζ300    complex faithful
ρ1131-3+3-3/2-3-3-3/2-3--3/23+-3/2-33--3/2--3-3+-3/200ζ3ζ31ζ32ζ31ζ32ζ3200    complex faithful
ρ123-1-3+3-3/2-3-3-3/2-3-3+-3/23--3/2-3--3/23+-3/2--300-1ζ65ζ65-1ζ6ζ6ζ65ζ600    complex faithful
ρ1331-3+3-3/2-3-3-3/2-3-3+-3/23--3/2-3--3/23+-3/2--3001ζ3ζ31ζ32ζ32ζ3ζ3200    complex faithful
ρ143-1-3-3-3/2-3+3-3/2-3+-3/23--3/2--33+-3/2-3-3--3/200ζ6ζ6-1ζ65ζ6-1ζ65ζ6500    complex faithful
ρ153-1-3+3-3/2-3-3-3/2-3--3/23+-3/2-33--3/2--3-3+-3/200ζ65ζ65-1ζ6ζ65-1ζ6ζ600    complex faithful
ρ163-1-3-3-3/2-3+3-3/2--3-3--3/23+-3/2-3+-3/23--3/2-300-1ζ6ζ6-1ζ65ζ65ζ6ζ6500    complex faithful
ρ173-1-3+3-3/2-3-3-3/23--3/2--3-3--3/2-3-3+-3/23+-3/200ζ6ζ65ζ6ζ65-1ζ65-1ζ600    complex faithful
ρ1831-3+3-3/2-3-3-3/23--3/2--3-3--3/2-3-3+-3/23+-3/200ζ32ζ3ζ32ζ31ζ31ζ3200    complex faithful
ρ1931-3-3-3/2-3+3-3/2-3+-3/23--3/2--33+-3/2-3-3--3/200ζ32ζ321ζ3ζ321ζ3ζ300    complex faithful
ρ2031-3-3-3/2-3+3-3/23+-3/2-3-3+-3/2--3-3--3/23--3/200ζ3ζ32ζ3ζ321ζ321ζ300    complex faithful
ρ213-1-3-3-3/2-3+3-3/23+-3/2-3-3+-3/2--3-3--3/23--3/200ζ65ζ6ζ65ζ6-1ζ6-1ζ6500    complex faithful
ρ226066000000-300000000000    orthogonal lifted from C32⋊C6

Permutation representations of C3≀S3
On 9 points - transitive group 9T20
Generators in S9
(2 3 6)(7 9 8)
(1 4 5)(2 3 6)(7 8 9)
(1 7 2)(3 4 8)(5 9 6)
(1 2)(3 4)(5 6)(7 8 9)

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

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

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

G:=TransitiveGroup(9,20);

On 18 points - transitive group 18T86
Generators in S18
(2 6 3)(7 10 11)(13 15 17)(14 16 18)
(1 5 4)(2 6 3)(7 10 11)(8 9 12)(13 17 15)(14 18 16)
(1 15 10)(2 9 18)(3 8 14)(4 17 7)(5 13 11)(6 12 16)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,86);

On 27 points - transitive group 27T37
Generators in S27
(1 4 7)(2 5 8)(3 6 9)(11 27 19)(13 23 21)(15 25 17)
(1 7 4)(2 8 5)(3 9 6)(10 26 18)(11 27 19)(12 22 20)(13 23 21)(14 24 16)(15 25 17)
(1 25 22)(2 21 18)(3 11 14)(4 15 12)(5 23 26)(6 19 16)(7 17 20)(8 13 10)(9 27 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 25 26 27)

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

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

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

G:=TransitiveGroup(27,37);

On 27 points - transitive group 27T50
Generators in S27
(1 5 4)(2 9 8)(3 7 6)(10 14 12)(11 16 27)(13 18 23)(15 20 25)(17 21 19)(22 26 24)
(1 2 3)(4 8 6)(5 9 7)(10 19 22)(11 20 23)(12 21 24)(13 16 25)(14 17 26)(15 18 27)
(1 23 26)(2 11 14)(3 20 17)(4 15 10)(5 16 21)(6 27 22)(7 13 12)(8 18 19)(9 25 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 25 26 27)

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

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

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

G:=TransitiveGroup(27,50);

On 27 points - transitive group 27T70
Generators in S27
(1 15 20)(2 13 18)(3 11 16)(4 21 25)(5 19 23)(6 17 27)(7 24 12)(8 22 10)(9 26 14)
(1 7 6)(2 8 4)(3 9 5)(10 25 18)(11 26 19)(12 27 20)(13 22 21)(14 23 16)(15 24 17)
(1 15 12)(2 18 21)(4 25 22)(6 17 20)(7 24 27)(8 10 13)(11 26 19)(14 16 23)
(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)

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

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

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

G:=TransitiveGroup(27,70);

C3≀S3 is a maximal subgroup of   He3⋊D6  C3≀S33C3  C3≀C3⋊C6  C3≀C3.C6  C345S3
C3≀S3 is a maximal quotient of   He3⋊C12  C3.C3≀S3  C32⋊C9⋊C6  C3.3C3≀S3  C331D9  He3⋊C18  C345S3

Polynomial with Galois group C3≀S3 over ℚ
actionf(x)Disc(f)
9T20x9-4x8-4x7+22x6-x5-31x4+4x3+15x2-1372·2293

Matrix representation of C3≀S3 in GL3(𝔽7) generated by

065
605
560
,
200
020
002
,
065
164
001
,
140
041
050
G:=sub<GL(3,GF(7))| [0,6,5,6,0,6,5,5,0],[2,0,0,0,2,0,0,0,2],[0,1,0,6,6,0,5,4,1],[1,0,0,4,4,5,0,1,0] >;

C3≀S3 in GAP, Magma, Sage, TeX

C_3\wr S_3
% in TeX

G:=Group("C3wrS3");
// GroupNames label

G:=SmallGroup(162,10);
// by ID

G=gap.SmallGroup(162,10);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,182,187,1803,253]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^6=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*c^-1>;
// generators/relations

Export

Subgroup lattice of C3≀S3 in TeX
Character table of C3≀S3 in TeX

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