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G = S3wrC2order 72 = 23·32

Wreath product of S3 by C2

non-abelian, soluble, monomial, rational

Aliases: S3wrC2, 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 — S3wrC2
C1C32C3:S3S32 — S3wrC2
C32C3:S3 — S3wrC2
C1

Generators and relations for S3wrC2
 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 >

Subgroups: 112 in 26 conjugacy classes, 7 normal (5 characteristic)
Quotients: C1, C2, C22, D4, S3wrC2
6C2
6C2
9C2
2C3
2C3
9C4
9C22
9C22
2S3
2S3
6S3
6S3
6C6
6C6
9D4
6D6
6D6
2C3xS3
2C3xS3

Character table of S3wrC2

 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 S3wrC2
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 9 11)(4 10 12)
(1 7 5)(2 6 8)(3 9 11)(4 12 10)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 10)(6 9)(7 12)(8 11)

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

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

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

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

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

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

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

G:=TransitiveGroup(18,34);

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

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

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

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

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);

S3wrC2 is a maximal subgroup of
AΓL1(F9)  C33:D4  C32:2D12  S32:D5  C32:D20
S3wrC2 is a maximal quotient of
S32:C4  C3:S3.Q8  C32:D8  C32:2SD16  C32:Q16  He3:D4  C33:D4  C32:2D12  S32:D5  C32:D20

Polynomial with Galois group S3wrC2 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 S3wrC2 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] >;

S3wrC2 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 S3wrC2 in TeX
Character table of S3wrC2 in TeX

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