S3wrC2 - GroupNames

class	1	2A	2B	2C	3A	3B	4	6A	6B
size	1	6	6	9	4	4	18	12	12
ρ₁	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	-1	1	-1	linear of order 2
ρ₃	1	1	-1	1	1	1	-1	-1	1	linear of order 2
ρ₄	1	-1	-1	1	1	1	1	-1	-1	linear of order 2
ρ₅	2	0	0	-2	2	2	0	0	0	orthogonal lifted from D₄
ρ₆	4	0	-2	0	-2	1	0	1	0	orthogonal faithful
ρ₇	4	-2	0	0	1	-2	0	0	1	orthogonal faithful
ρ₈	4	0	2	0	-2	1	0	-1	0	orthogonal faithful
ρ₉	4	2	0	0	1	-2	0	0	-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

action	f(x)	Disc(f)
6T13	x⁶-2x⁵+x⁴+1	-2⁶·5·149
9T16	x⁹-10x⁷-3x⁶+25x⁵+5x⁴-21x³+5x-1	2⁶·19³·137³
12T34	x¹²+12x¹⁰+54x⁸+104x⁶+57x⁴-36x²+16	2⁵⁶·3²⁰
12T35	x¹²-4x¹⁰+12x⁸-51x⁶+164x⁴-8x²+433	2¹²·3⁶·19⁴·31⁴·433³·26821⁴
12T36	x¹²-x⁹-x⁶-x³+1	-3¹⁵·13⁶

Matrix representation of S₃wrC₂ ►in GL₄(Z) generated by

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	1	1
-1	0	0	0
0	-1	0	0

-1	0	0	0
0	-1	0	0
0	0	-1	0
0	0	1	1

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

G = S3wrC2 order 72 = 23·32

Wreath product of S3 by C2

G = S₃wrC₂ order 72 = 2³·3²

Wreath product of S₃ by C₂