Copied to
clipboard

G = C2×C6order 12 = 22·3

Abelian group of type [2,6]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C6, SmallGroup(12,5)

Series: Derived Chief Lower central Upper central

C1 — C2×C6
C1C3C6 — C2×C6
C1 — C2×C6
C1 — C2×C6

Generators and relations for C2×C6
 G = < a,b | a2=b6=1, ab=ba >


Character table of C2×C6

 class 12A2B2C3A3B6A6B6C6D6E6F
 size 111111111111
ρ1111111111111    trivial
ρ21-11-1111-1-11-1-1    linear of order 2
ρ311-1-111-111-1-1-1    linear of order 2
ρ41-1-1111-1-1-1-111    linear of order 2
ρ51111ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ32    linear of order 3
ρ61-11-1ζ3ζ32ζ3ζ65ζ6ζ32ζ65ζ6    linear of order 6
ρ711-1-1ζ3ζ32ζ65ζ3ζ32ζ6ζ65ζ6    linear of order 6
ρ81-1-11ζ3ζ32ζ65ζ65ζ6ζ6ζ3ζ32    linear of order 6
ρ91111ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ3    linear of order 3
ρ101-11-1ζ32ζ3ζ32ζ6ζ65ζ3ζ6ζ65    linear of order 6
ρ1111-1-1ζ32ζ3ζ6ζ32ζ3ζ65ζ6ζ65    linear of order 6
ρ121-1-11ζ32ζ3ζ6ζ6ζ65ζ65ζ32ζ3    linear of order 6

Permutation representations of C2×C6
Regular action on 12 points - transitive group 12T2
Generators in S12
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)
(1 2 3 4 5 6)(7 8 9 10 11 12)

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

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

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

G:=TransitiveGroup(12,2);

Polynomial with Galois group C2×C6 over ℚ
actionf(x)Disc(f)
12T2x12-x6+1212·318

Matrix representation of C2×C6 in GL3(𝔽7) generated by

100
060
006
,
200
060
001
G:=sub<GL(3,GF(7))| [1,0,0,0,6,0,0,0,6],[2,0,0,0,6,0,0,0,1] >;

C2×C6 in GAP, Magma, Sage, TeX

C_2\times C_6
% in TeX

G:=Group("C2xC6");
// GroupNames label

G:=SmallGroup(12,5);
// by ID

G=gap.SmallGroup(12,5);
# by ID

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

G:=Group<a,b|a^2=b^6=1,a*b=b*a>;
// generators/relations

׿
×
𝔽