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## G = C2×S4order 48 = 24·3

### Direct product of C2 and S4

Aliases: C2×S4, C2S3, O3(𝔽3), CO3(𝔽3), C23⋊S3, C22⋊D6, A4⋊C22, (C2×A4)⋊C2, group of symmetries of a cube (and its dual - regular octahedron), SmallGroup(48,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C2×S4
 Chief series C1 — C22 — A4 — S4 — C2×S4
 Lower central A4 — C2×S4
 Upper central C1 — C2

Generators and relations for C2×S4
G = < a,b,c,d,e | a2=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

3C2
3C2
6C2
6C2
4C3
3C4
3C22
3C22
3C4
3C22
3C22
6C22
6C22
4S3
4S3
4C6
3D4
3C23
3D4
3D4
3D4
4D6

Character table of C2×S4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 6 size 1 1 3 3 6 6 8 6 6 8 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 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 linear of order 2 ρ4 1 -1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ5 2 -2 2 -2 0 0 -1 0 0 1 orthogonal lifted from D6 ρ6 2 2 2 2 0 0 -1 0 0 -1 orthogonal lifted from S3 ρ7 3 3 -1 -1 1 1 0 -1 -1 0 orthogonal lifted from S4 ρ8 3 -3 -1 1 -1 1 0 -1 1 0 orthogonal faithful ρ9 3 -3 -1 1 1 -1 0 1 -1 0 orthogonal faithful ρ10 3 3 -1 -1 -1 -1 0 1 1 0 orthogonal lifted from S4

Permutation representations of C2×S4
On 6 points - transitive group 6T11
Generators in S6
(1 6)(2 4)(3 5)
(1 6)(2 4)
(2 4)(3 5)
(1 2 3)(4 5 6)
(1 6)(2 5)(3 4)

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

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

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

G:=TransitiveGroup(6,11);

On 8 points - transitive group 8T24
Generators in S8
(1 2)(3 7)(4 8)(5 6)
(1 7)(2 3)(4 5)(6 8)
(1 8)(2 4)(3 5)(6 7)
(3 4 5)(6 7 8)
(1 2)(3 6)(4 8)(5 7)

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

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

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

G:=TransitiveGroup(8,24);

On 12 points - transitive group 12T21
Generators in S12
(1 9)(2 7)(3 8)(4 10)(5 11)(6 12)
(2 7)(3 8)(4 10)(6 12)
(1 9)(3 8)(4 10)(5 11)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 10)(2 12)(3 11)(4 9)(5 8)(6 7)

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

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

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

G:=TransitiveGroup(12,21);

On 12 points - transitive group 12T22
Generators in S12
(1 6)(2 4)(3 5)(7 11)(8 12)(9 10)
(1 11)(2 8)(3 5)(4 12)(6 7)(9 10)
(1 6)(2 12)(3 9)(4 8)(5 10)(7 11)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(2 3)(4 5)(7 11)(8 10)(9 12)

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

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

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

G:=TransitiveGroup(12,22);

On 12 points - transitive group 12T23
Generators in S12
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)
(1 9)(2 7)(4 12)(6 11)
(2 7)(3 8)(4 12)(5 10)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(2 3)(4 5)(7 8)(10 12)

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

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

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

G:=TransitiveGroup(12,23);

On 12 points - transitive group 12T24
Generators in S12
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)
(1 9)(2 7)(4 12)(6 11)
(2 7)(3 8)(4 12)(5 10)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 9)(2 8)(3 7)(4 10)(5 12)(6 11)

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

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

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

G:=TransitiveGroup(12,24);

On 16 points - transitive group 16T61
Generators in S16
(1 3)(2 4)(5 14)(6 15)(7 16)(8 11)(9 12)(10 13)
(1 5)(2 9)(3 14)(4 12)(6 7)(8 10)(11 13)(15 16)
(1 6)(2 10)(3 15)(4 13)(5 7)(8 9)(11 12)(14 16)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
(1 2)(3 4)(5 8)(6 10)(7 9)(11 14)(12 16)(13 15)

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

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

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

G:=TransitiveGroup(16,61);

On 24 points - transitive group 24T46
Generators in S24
(1 11)(2 12)(3 10)(4 22)(5 23)(6 24)(7 19)(8 20)(9 21)(13 16)(14 17)(15 18)
(1 11)(2 15)(3 16)(4 20)(5 23)(6 7)(8 22)(9 21)(10 13)(12 18)(14 17)(19 24)
(1 17)(2 12)(3 13)(4 8)(5 21)(6 24)(7 19)(9 23)(10 16)(11 14)(15 18)(20 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 8)(2 7)(3 9)(4 17)(5 16)(6 18)(10 21)(11 20)(12 19)(13 23)(14 22)(15 24)

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

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

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

G:=TransitiveGroup(24,46);

On 24 points - transitive group 24T47
Generators in S24
(1 11)(2 12)(3 10)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(2 7)(3 8)(4 22)(6 24)(10 14)(12 13)(16 19)(18 21)
(1 9)(3 8)(4 22)(5 23)(10 14)(11 15)(16 19)(17 20)
(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 16)(2 18)(3 17)(4 15)(5 14)(6 13)(7 21)(8 20)(9 19)(10 23)(11 22)(12 24)

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

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

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

G:=TransitiveGroup(24,47);

On 24 points - transitive group 24T48
Generators in S24
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 22)(11 23)(12 24)(13 18)(14 16)(15 17)
(1 14)(2 12)(3 20)(4 17)(5 9)(6 23)(7 16)(8 24)(10 13)(11 21)(15 19)(18 22)
(1 21)(2 15)(3 10)(4 24)(5 18)(6 7)(8 17)(9 22)(11 14)(12 19)(13 20)(16 23)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(2 3)(4 5)(8 9)(10 15)(11 14)(12 13)(16 23)(17 22)(18 24)(19 20)

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

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

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

G:=TransitiveGroup(24,48);

Polynomial with Galois group C2×S4 over ℚ
actionf(x)Disc(f)
6T11x6-x4+1-26·232
8T24x8-x7+x6+x2+x+126·54·372
12T21x12-15x10+84x8-215x6+243x4-90x2+4222·314·236
12T22x12-9x9-5x8+7x6-18x5-125x4+9x3+23x2-57x+1224·36·232·372·532·2295·44632
12T23x12-15x10+75x8-148x6+120x4-35x2+1212·56·78·1974
12T24x12-4x11+8x10-8x9+3x8+2x7-2x5+3x4-2x3+2x2-2x+1218·116·1672

Matrix representation of C2×S4 in GL3(ℤ) generated by

 -1 0 0 0 -1 0 0 0 -1
,
 -1 0 0 0 -1 0 0 0 1
,
 1 0 0 0 -1 0 0 0 -1
,
 0 0 1 1 0 0 0 1 0
,
 -1 0 0 0 0 -1 0 -1 0
G:=sub<GL(3,Integers())| [-1,0,0,0,-1,0,0,0,-1],[-1,0,0,0,-1,0,0,0,1],[1,0,0,0,-1,0,0,0,-1],[0,1,0,0,0,1,1,0,0],[-1,0,0,0,0,-1,0,-1,0] >;

C2×S4 in GAP, Magma, Sage, TeX

C_2\times S_4
% in TeX

G:=Group("C2xS4");
// GroupNames label

G:=SmallGroup(48,48);
// by ID

G=gap.SmallGroup(48,48);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,2,122,483,133,304,239]);
// Polycyclic

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

׿
×
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