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G = C2×S4order 48 = 24·3

Direct product of C2 and S4

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

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

C1C22A4 — C2×S4
C1C22A4S4 — C2×S4
A4 — C2×S4
C1C2

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
3C2×C4
3D4
4D6
3C2×D4

Character table of C2×S4

 class 12A2B2C2D2E34A4B6
 size 1133668668
ρ11111111111    trivial
ρ21-11-11-11-11-1    linear of order 2
ρ31111-1-11-1-11    linear of order 2
ρ41-11-1-1111-1-1    linear of order 2
ρ52-22-200-1001    orthogonal lifted from D6
ρ6222200-100-1    orthogonal lifted from S3
ρ733-1-1110-1-10    orthogonal lifted from S4
ρ83-3-11-110-110    orthogonal faithful
ρ93-3-111-101-10    orthogonal faithful
ρ1033-1-1-1-10110    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 2)(3 4)(5 16)(6 14)(7 15)(8 13)(9 11)(10 12)
(1 16)(2 5)(3 8)(4 13)(6 7)(9 10)(11 12)(14 15)
(1 14)(2 6)(3 9)(4 11)(5 7)(8 10)(12 13)(15 16)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
(1 3)(2 4)(5 12)(6 11)(7 13)(8 15)(9 14)(10 16)

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

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

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

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

-100
0-10
00-1
,
-100
0-10
001
,
100
0-10
00-1
,
001
100
010
,
-100
00-1
0-10
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

׿
×
𝔽