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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
(1 6 3)(8 10 12)(13 15 17)(14 16 18)
(1 6 3)(2 5 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)
(1 9 13)(2 16 10)(3 7 15)(4 18 8)(5 14 12)(6 11 17)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14 15 16 17 18)

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

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

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