C2xS3wrC2 - GroupNames

G = C₂xS₃wrC₂ order 144 = 2⁴·3²

Direct product of C₂ and S₃wrC₂

direct product, non-abelian, soluble, monomial, rational

Aliases: C₂xS₃wrC₂, C₃:S₃:D₄, (C₃xC₆):D₄, S₃²:C₂², C₃²:(C₂xD₄), C₃²:C₄:C₂², C₃:S₃.₁C₂³, (C₂xS₃²):₅C₂, (C₂xC₃²:C₄):₃C₂, (C₂xC₃:S₃).₆C₂², SmallGroup(144,186)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃:S₃ — C₂xS₃wrC₂

Chief series

C₁ — C₃² — C₃:S₃ — S₃² — S₃wrC₂ — C₂xS₃wrC₂

Lower central

C₃² — C₃:S₃ — C₂xS₃wrC₂

Upper central

C₁ — C₂

Generators and relations for C₂xS₃wrC₂
G = < a,b,c,d,e | a²=b³=c³=d⁴=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=c, ebe=dcd^-1=b^-1, ce=ec, ede=d^-1 >

Subgroups: 414 in 86 conjugacy classes, 21 normal (9 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, D₄, C₂³, C₃², D₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃:S₃, C₃xC₆, C₂²xS₃, C₃²:C₄, S₃², S₃², S₃xC₆, C₂xC₃:S₃, S₃wrC₂, C₂xC₃²:C₄, C₂xS₃², C₂xS₃wrC₂
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, S₃wrC₂, C₂xS₃wrC₂

Character table of C₂xS₃wrC₂

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	6A	6B	6C	6D	6E	6F
size	1	1	6	6	6	6	9	9	4	4	18	18	4	4	12	12	12	12
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	-1	-1	1	1	-1	1	1	1	-1	-1	-1	-1	1	-1	1	linear of order 2
ρ₃	1	1	-1	1	-1	1	1	1	1	1	-1	-1	1	1	1	-1	-1	1	linear of order 2
ρ₄	1	-1	-1	-1	1	1	1	-1	1	1	-1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₅	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₆	1	-1	-1	1	1	-1	1	-1	1	1	1	-1	-1	-1	1	-1	1	-1	linear of order 2
ρ₇	1	1	1	-1	1	-1	1	1	1	1	-1	-1	1	1	-1	1	1	-1	linear of order 2
ρ₈	1	-1	1	1	-1	-1	1	-1	1	1	-1	1	-1	-1	1	1	-1	-1	linear of order 2
ρ₉	2	2	0	0	0	0	-2	-2	2	2	0	0	2	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	0	0	0	0	-2	2	2	2	0	0	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	4	4	0	2	0	2	0	0	-2	1	0	0	1	-2	-1	0	0	-1	orthogonal lifted from S₃wrC₂
ρ₁₂	4	4	-2	0	-2	0	0	0	1	-2	0	0	-2	1	0	1	1	0	orthogonal lifted from S₃wrC₂
ρ₁₃	4	-4	0	2	0	-2	0	0	-2	1	0	0	-1	2	-1	0	0	1	orthogonal faithful
ρ₁₄	4	-4	-2	0	2	0	0	0	1	-2	0	0	2	-1	0	1	-1	0	orthogonal faithful
ρ₁₅	4	-4	0	-2	0	2	0	0	-2	1	0	0	-1	2	1	0	0	-1	orthogonal faithful
ρ₁₆	4	-4	2	0	-2	0	0	0	1	-2	0	0	2	-1	0	-1	1	0	orthogonal faithful
ρ₁₇	4	4	2	0	2	0	0	0	1	-2	0	0	-2	1	0	-1	-1	0	orthogonal lifted from S₃wrC₂
ρ₁₈	4	4	0	-2	0	-2	0	0	-2	1	0	0	1	-2	1	0	0	1	orthogonal lifted from S₃wrC₂

Permutation representations of C₂xS₃wrC₂
►On 12 points - transitive group 12T77

Generators in S₁₂

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

(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)| (1,4)(2,3)(5,12)(6,9)(7,10)(8,11), (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( (1,4)(2,3)(5,12)(6,9)(7,10)(8,11), (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([[(1,4),(2,3),(5,12),(6,9),(7,10),(8,11)], [(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(12,77);

►On 12 points - transitive group 12T78

Generators in S₁₂

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

(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)| (1,3)(2,4)(5,12)(6,9)(7,10)(8,11), (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( (1,3)(2,4)(5,12)(6,9)(7,10)(8,11), (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([[(1,3),(2,4),(5,12),(6,9),(7,10),(8,11)], [(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(12,78);

►On 18 points - transitive group 18T63

Generators in S₁₈

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

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

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

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

(4 6)(7 8)(9 10)(12 14)(15 16)(17 18)

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

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

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

G:=TransitiveGroup(18,63);

►On 24 points - transitive group 24T261

Generators in S₂₄

(1 17)(2 18)(3 19)(4 20)(5 15)(6 16)(7 13)(8 14)(9 21)(10 22)(11 23)(12 24)

(1 5 9)(2 6 10)(3 11 7)(4 12 8)(13 19 23)(14 20 24)(15 21 17)(16 22 18)

(1 5 9)(2 10 6)(3 11 7)(4 8 12)(13 19 23)(14 24 20)(15 21 17)(16 18 22)

(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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)

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

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

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

G:=TransitiveGroup(24,261);

►On 24 points - transitive group 24T262

Generators in S₂₄

(1 18)(2 19)(3 20)(4 17)(5 13)(6 14)(7 15)(8 16)(9 22)(10 23)(11 24)(12 21)

(1 7 12)(2 9 8)(3 10 5)(4 6 11)(13 20 23)(14 24 17)(15 21 18)(16 19 22)

(1 12 7)(2 9 8)(3 5 10)(4 6 11)(13 23 20)(14 24 17)(15 18 21)(16 19 22)

(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 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 22)(14 21)(15 24)(16 23)(17 18)(19 20)

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

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

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

G:=TransitiveGroup(24,262);

►On 24 points - transitive group 24T263

Generators in S₂₄

(1 5)(2 6)(3 8)(4 7)(9 14)(10 15)(11 16)(12 13)(17 23)(18 24)(19 21)(20 22)

(1 22 24)(2 23 21)(3 9 11)(4 10 12)(5 20 18)(6 17 19)(7 15 13)(8 14 16)

(1 22 24)(2 21 23)(3 9 11)(4 12 10)(5 20 18)(6 19 17)(7 13 15)(8 14 16)

(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 4)(2 3)(5 7)(6 8)(9 21)(10 24)(11 23)(12 22)(13 20)(14 19)(15 18)(16 17)

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

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

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

G:=TransitiveGroup(24,263);

►On 24 points - transitive group 24T264

Generators in S₂₄

(1 3)(2 4)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)(17 24)(18 21)(19 22)(20 23)

(2 22 17)(4 19 24)(5 14 10)(7 12 16)

(1 21 20)(3 18 23)(6 11 15)(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)| (1,3)(2,4)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14)(17,24)(18,21)(19,22)(20,23), (2,22,17)(4,19,24)(5,14,10)(7,12,16), (1,21,20)(3,18,23)(6,11,15)(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( (1,3)(2,4)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14)(17,24)(18,21)(19,22)(20,23), (2,22,17)(4,19,24)(5,14,10)(7,12,16), (1,21,20)(3,18,23)(6,11,15)(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([[(1,3),(2,4),(5,7),(6,8),(9,15),(10,16),(11,13),(12,14),(17,24),(18,21),(19,22),(20,23)], [(2,22,17),(4,19,24),(5,14,10),(7,12,16)], [(1,21,20),(3,18,23),(6,11,15),(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(24,264);

C₂xS₃wrC₂ is a maximal subgroup of
C₂.AΓL₁(F₉) S₃²:D₄ C₄:S₃wrC₂ D₆wrC₂ C₆²:D₄
C₂xS₃wrC₂ is a maximal quotient of
S₃²:Q₈ C₄.₄S₃wrC₂ C₃²:C₄:Q₈ C₃²:D₈:₅C₂ C₃²:D₈:C₂ C₃:S₃:D₈ C₃²:Q₁₆:C₂ C₃:S₃:₂SD₁₆ C₃:S₃:Q₁₆ S₃²:D₄ C₄:S₃wrC₂ C₆².₉D₄ C₆².₁₂D₄ C₆².₁₃D₄ C₆².₁₅D₄ D₆wrC₂ C₆²:D₄

Polynomial with Galois group C₂xS₃wrC₂ over Q

action	f(x)	Disc(f)
12T77	x¹²-2x¹⁰-x⁸-2x⁶+5x⁴-6x²+4	2²⁶·5⁶·11⁴·509²
12T78	x¹²-12x¹⁰+54x⁸-104x⁶+57x⁴+36x²-16	-2⁵⁶·3¹²·5⁶

Matrix representation of C₂xS₃wrC₂ ►in GL₄(Z) generated by

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

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

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

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

0	0	1	0
0	0	0	1
1	0	0	0
0	1	0	0

G:=sub<GL(4,Integers())| [-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,-1,0,0,1,-1,0,0,0,0,-1,1,0,0,-1,0],[-1,1,0,0,-1,0,0,0,0,0,-1,1,0,0,-1,0],[0,0,-1,0,0,0,0,-1,1,-1,0,0,0,-1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

C₂xS₃wrC₂ in GAP, Magma, Sage, TeX

C_2\times S_3\wr C_2

% in TeX

G:=Group("C2xS3wrC2");

// GroupNames label

G:=SmallGroup(144,186);

// by ID

G=gap.SmallGroup(144,186);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,121,964,730,142,299,455]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂xS₃wrC₂ in TeX

G = C2xS3wrC2 order 144 = 24·32

Direct product of C2 and S3wrC2

G = C₂xS₃wrC₂ order 144 = 2⁴·3²

Direct product of C₂ and S₃wrC₂