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G = S3≀C2order 72 = 23·32

Wreath product of S3 by C2

non-abelian, soluble, monomial, rational

Aliases: S3C2, SO+4(𝔽2), O+4(𝔽2), PSO+4(𝔽2), PO+4(𝔽2), CO+4(𝔽2), CSO+4(𝔽2), PCO+4(𝔽2), PCSO+4(𝔽2), C32⋊D4, S32⋊C2, C32⋊C4⋊C2, C3⋊S3.1C22, SmallGroup(72,40)

Series: Derived Chief Lower central Upper central

C1C32C3⋊S3 — S3≀C2
C1C32C3⋊S3S32 — S3≀C2
C32C3⋊S3 — S3≀C2
C1

Generators and relations for S3≀C2
 G = < a,b,c,d | a3=b3=c4=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >

6C2
6C2
9C2
2C3
2C3
9C4
9C22
9C22
2S3
2S3
6S3
6S3
6C6
6C6
9D4
6D6
6D6
2C3×S3
2C3×S3

Character table of S3≀C2

 class 12A2B2C3A3B46A6B
 size 166944181212
ρ1111111111    trivial
ρ21-11111-11-1    linear of order 2
ρ311-1111-1-11    linear of order 2
ρ41-1-11111-1-1    linear of order 2
ρ5200-222000    orthogonal lifted from D4
ρ640-20-21010    orthogonal faithful
ρ74-2001-2001    orthogonal faithful
ρ84020-210-10    orthogonal faithful
ρ942001-200-1    orthogonal faithful

Permutation representations of S3≀C2
On 6 points - transitive group 6T13
Generators in S6
(1 4 6)(2 3 5)
(1 6 4)(2 3 5)
(1 2)(3 4 5 6)
(1 2)(3 6)(4 5)

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

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

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

G:=TransitiveGroup(6,13);

On 9 points: primitive - transitive group 9T16
Generators in S9
(1 5 3)(2 8 9)(4 7 6)
(1 4 2)(3 6 9)(5 7 8)
(2 3 4 5)(6 7 8 9)
(3 5)(6 7)(8 9)

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

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

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

G:=TransitiveGroup(9,16);

On 12 points - transitive group 12T34
Generators in S12
(1 7 5)(2 8 6)(3 11 9)(4 10 12)
(1 7 5)(2 6 8)(3 9 11)(4 10 12)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 3)(2 4)(5 11)(6 10)(7 9)(8 12)

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

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

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

G:=TransitiveGroup(12,34);

On 12 points - transitive group 12T35
Generators in S12
(1 9 8)(2 10 5)(3 6 11)(4 7 12)
(1 9 8)(2 5 10)(3 6 11)(4 12 7)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)

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

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

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

G:=TransitiveGroup(12,35);

On 12 points - transitive group 12T36
Generators in S12
(1 9 8)(3 6 11)
(2 5 10)(4 12 7)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 12)(6 11)(7 10)(8 9)

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

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

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

G:=TransitiveGroup(12,36);

On 18 points - transitive group 18T34
Generators in S18
(1 11 13)(2 3 5)(4 15 7)(6 9 17)(8 14 18)(10 16 12)
(1 6 4)(2 14 12)(3 18 10)(5 8 16)(7 13 17)(9 15 11)
(1 2)(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 2)(3 13)(4 12)(5 11)(6 14)(7 10)(8 9)(15 16)(17 18)

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

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

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

G:=TransitiveGroup(18,34);

On 18 points - transitive group 18T36
Generators in S18
(1 6 4)(2 17 15)(3 10 7)(5 9 8)(11 16 12)(13 14 18)
(1 5 3)(2 16 18)(4 8 7)(6 9 10)(11 14 15)(12 13 17)
(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 2)(3 18)(4 17)(5 16)(6 15)(7 13)(8 12)(9 11)(10 14)

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

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

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

G:=TransitiveGroup(18,36);

On 24 points - transitive group 24T72
Generators in S24
(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)| (1,21,20)(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( (1,21,20)(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([(1,21,20),(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(24,72);

S3≀C2 is a maximal subgroup of
AΓL1(𝔽9)  C33⋊D4  C322D12  S32⋊D5  C32⋊D20
S3≀C2 is a maximal quotient of
S32⋊C4  C3⋊S3.Q8  C32⋊D8  C322SD16  C32⋊Q16  He3⋊D4  C33⋊D4  C322D12  S32⋊D5  C32⋊D20

Polynomial with Galois group S3≀C2 over ℚ
actionf(x)Disc(f)
6T13x6-2x5+x4+1-26·5·149
9T16x9-10x7-3x6+25x5+5x4-21x3+5x-126·193·1373
12T34x12+12x10+54x8+104x6+57x4-36x2+16256·320
12T35x12-4x10+12x8-51x6+164x4-8x2+433212·36·194·314·4333·268214
12T36x12-x9-x6-x3+1-315·136

Matrix representation of S3≀C2 in GL4(ℤ) generated by

1000
0100
0001
00-1-1
,
0100
-1-100
0010
0001
,
00-10
0011
-1000
0-100
,
-1000
0-100
00-10
0011
G:=sub<GL(4,Integers())| [1,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,-1],[0,-1,0,0,1,-1,0,0,0,0,1,0,0,0,0,1],[0,0,-1,0,0,0,0,-1,-1,1,0,0,0,1,0,0],[-1,0,0,0,0,-1,0,0,0,0,-1,1,0,0,0,1] >;

S3≀C2 in GAP, Magma, Sage, TeX

S_3\wr C_2
% in TeX

G:=Group("S3wrC2");
// GroupNames label

G:=SmallGroup(72,40);
// by ID

G=gap.SmallGroup(72,40);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,3,61,323,248,93,109,314]);
// Polycyclic

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

Export

Subgroup lattice of S3≀C2 in TeX
Character table of S3≀C2 in TeX

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