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## G = C2×A4order 24 = 23·3

### Direct product of C2 and A4

Aliases: C2×A4, C2C3, AΣL1(𝔽8), C23⋊C3, C22⋊C6, SmallGroup(24,13)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×A4
G = < a,b,c,d | a2=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Character table of C2×A4

 class 1 2A 2B 2C 3A 3B 6A 6B size 1 1 3 3 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 -1 -1 1 ζ32 ζ3 ζ6 ζ65 linear of order 6 ρ5 1 -1 -1 1 ζ3 ζ32 ζ65 ζ6 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ7 3 3 -1 -1 0 0 0 0 orthogonal lifted from A4 ρ8 3 -3 1 -1 0 0 0 0 orthogonal faithful

Permutation representations of C2×A4
On 6 points - transitive group 6T6
Generators in S6
(1 4)(2 5)(3 6)
(2 5)(3 6)
(1 4)(3 6)
(1 2 3)(4 5 6)

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

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

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

G:=TransitiveGroup(6,6);

On 8 points - transitive group 8T13
Generators in S8
(1 2)(3 6)(4 7)(5 8)
(1 6)(2 3)(4 5)(7 8)
(1 7)(2 4)(3 5)(6 8)
(3 4 5)(6 7 8)

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

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

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

G:=TransitiveGroup(8,13);

On 12 points - transitive group 12T6
Generators in S12
(1 4)(2 5)(3 6)(7 12)(8 10)(9 11)
(1 4)(2 9)(3 12)(5 11)(6 7)(8 10)
(1 10)(2 5)(3 7)(4 8)(6 12)(9 11)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)

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

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

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

G:=TransitiveGroup(12,6);

On 12 points - transitive group 12T7
Generators in S12
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)
(2 8)(3 9)(5 11)(6 12)
(1 7)(3 9)(4 10)(6 12)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)

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

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

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

G:=TransitiveGroup(12,7);

Regular action on 24 points - transitive group 24T9
Generators in S24
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)
(1 15)(2 16)(3 12)(4 20)(5 9)(6 24)(7 19)(8 22)(10 18)(11 13)(14 17)(21 23)
(1 10)(2 13)(3 17)(4 22)(5 21)(6 7)(8 20)(9 23)(11 16)(12 14)(15 18)(19 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)

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

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

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

G:=TransitiveGroup(24,9);

C2×A4 is a maximal subgroup of
A4⋊C4  C23.3A4  C24⋊C6  C42⋊C6  C23.A4  Q8⋊A4  C23⋊A4  AΓL1(𝔽8)  D7⋊A4  D13⋊A4  D19⋊A4
C2×A4 is a maximal quotient of
C4.A4  C24⋊C6  C42⋊C6  C23.A4  D7⋊A4  D13⋊A4  D19⋊A4

Polynomial with Galois group C2×A4 over ℚ
actionf(x)Disc(f)
6T6x6-x4-2x2+1-26·74
8T13x8-x7+x6-x5-x4+x3+x2+x+174·134
12T6x12-20x10+148x8-503x6+800x4-512x2+64242·114·438
12T7x12-21x10+135x8-345x6+321x4-39x2+1212·316·176·12794

Matrix representation of C2×A4 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 1 0 0 0 1 1 0 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],[0,0,1,1,0,0,0,1,0] >;

C2×A4 in GAP, Magma, Sage, TeX

C_2\times A_4
% in TeX

G:=Group("C2xA4");
// GroupNames label

G:=SmallGroup(24,13);
// by ID

G=gap.SmallGroup(24,13);
# by ID

G:=PCGroup([4,-2,-3,-2,2,78,151]);
// Polycyclic

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

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