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## G = C2×C12order 24 = 23·3

### Abelian group of type [2,12]

Aliases: C2×C12, SmallGroup(24,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12
 Chief series C1 — C2 — C6 — C12 — C2×C12
 Lower central C1 — C2×C12
 Upper central C1 — C2×C12

Generators and relations for C2×C12
G = < a,b | a2=b12=1, ab=ba >

Character table of C2×C12

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 -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 1 1 1 1 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 -1 -1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ5 1 -1 1 -1 1 1 i -i i -i 1 -1 -1 -1 1 -1 -i -i i -i i -i i i linear of order 4 ρ6 1 1 -1 -1 1 1 i -i -i i -1 1 -1 -1 -1 1 i -i i -i -i i -i i linear of order 4 ρ7 1 -1 1 -1 1 1 -i i -i i 1 -1 -1 -1 1 -1 i i -i i -i i -i -i linear of order 4 ρ8 1 1 -1 -1 1 1 -i i i -i -1 1 -1 -1 -1 1 -i i -i i i -i i -i linear of order 4 ρ9 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ10 1 -1 -1 1 ζ3 ζ32 1 1 -1 -1 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 ζ6 ζ32 ζ3 ζ3 ζ65 ζ65 ζ6 ζ32 linear of order 6 ρ11 1 -1 1 -1 ζ3 ζ32 i -i i -i ζ32 ζ6 ζ65 ζ6 ζ3 ζ65 ζ43ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ4ζ32 linear of order 12 ρ12 1 1 -1 -1 ζ3 ζ32 i -i -i i ζ6 ζ32 ζ65 ζ6 ζ65 ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 linear of order 12 ρ13 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ6 ζ6 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ14 1 -1 -1 1 ζ3 ζ32 -1 -1 1 1 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 ζ32 ζ6 ζ65 ζ65 ζ3 ζ3 ζ32 ζ6 linear of order 6 ρ15 1 -1 1 -1 ζ3 ζ32 -i i -i i ζ32 ζ6 ζ65 ζ6 ζ3 ζ65 ζ4ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ43ζ32 linear of order 12 ρ16 1 1 -1 -1 ζ3 ζ32 -i i i -i ζ6 ζ32 ζ65 ζ6 ζ65 ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 linear of order 12 ρ17 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ18 1 -1 -1 1 ζ32 ζ3 1 1 -1 -1 ζ65 ζ65 ζ32 ζ3 ζ6 ζ6 ζ65 ζ3 ζ32 ζ32 ζ6 ζ6 ζ65 ζ3 linear of order 6 ρ19 1 -1 1 -1 ζ32 ζ3 i -i i -i ζ3 ζ65 ζ6 ζ65 ζ32 ζ6 ζ43ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ4ζ3 linear of order 12 ρ20 1 1 -1 -1 ζ32 ζ3 i -i -i i ζ65 ζ3 ζ6 ζ65 ζ6 ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 linear of order 12 ρ21 1 1 1 1 ζ32 ζ3 -1 -1 -1 -1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ65 ζ65 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ22 1 -1 -1 1 ζ32 ζ3 -1 -1 1 1 ζ65 ζ65 ζ32 ζ3 ζ6 ζ6 ζ3 ζ65 ζ6 ζ6 ζ32 ζ32 ζ3 ζ65 linear of order 6 ρ23 1 -1 1 -1 ζ32 ζ3 -i i -i i ζ3 ζ65 ζ6 ζ65 ζ32 ζ6 ζ4ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ43ζ3 linear of order 12 ρ24 1 1 -1 -1 ζ32 ζ3 -i i i -i ζ65 ζ3 ζ6 ζ65 ζ6 ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 linear of order 12

Permutation representations of C2×C12
Regular action on 24 points - transitive group 24T2
Generators in S24
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 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)

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

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

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

G:=TransitiveGroup(24,2);

C2×C12 is a maximal subgroup of   C4.Dic3  Dic3⋊C4  C4⋊Dic3  D6⋊C4  C4○D12

Polynomial with Galois group C2×C12 over ℚ
actionf(x)Disc(f)
24T2x24-x23+x19-x18+x17-x16+x14-x13+x12-x11+x10-x8+x7-x6+x5-x+1518·720

Matrix representation of C2×C12 in GL2(𝔽13) generated by

 12 0 0 12
,
 9 0 0 5
G:=sub<GL(2,GF(13))| [12,0,0,12],[9,0,0,5] >;

C2×C12 in GAP, Magma, Sage, TeX

C_2\times C_{12}
% in TeX

G:=Group("C2xC12");
// GroupNames label

G:=SmallGroup(24,9);
// by ID

G=gap.SmallGroup(24,9);
# by ID

G:=PCGroup([4,-2,-2,-3,-2,48]);
// Polycyclic

G:=Group<a,b|a^2=b^12=1,a*b=b*a>;
// generators/relations

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