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G = C2×A4order 24 = 23·3

Direct product of C2 and A4

direct product, metabelian, soluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C22 — C2×A4
C1C22A4 — C2×A4
C22 — C2×A4
C1C2

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 >

3C2
3C2
4C3
3C22
3C22
4C6

Character table of C2×A4

 class 12A2B2C3A3B6A6B
 size 11334444
ρ111111111    trivial
ρ21-1-1111-1-1    linear of order 2
ρ31111ζ32ζ3ζ32ζ3    linear of order 3
ρ41-1-11ζ32ζ3ζ6ζ65    linear of order 6
ρ51-1-11ζ3ζ32ζ65ζ6    linear of order 6
ρ61111ζ3ζ32ζ3ζ32    linear of order 3
ρ733-1-10000    orthogonal lifted from A4
ρ83-31-10000    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);

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

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