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G = C2×C12  order 24 = 23·3

Abelian group of type [2,12]

direct product, abelian, monomial, 2-elementary

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 >

C1
C2
C2
C2
C3
C4
C22
C4
C6
C6
C6
C2×C4
C12
C12
C2×C6
C2×C12

Character table of C2×C12

 class 12A2B2C3A3B4A4B4C4D6A6B6C6D6E6F12A12B12C12D12E12F12G12H
 size 111111111111111111111111
ρ1111111111111111111111111    trivial
ρ21-1-111111-1-1-1-111-1-1-1111-1-1-11    linear of order 2
ρ3111111-1-1-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41-1-1111-1-111-1-111-1-11-1-1-1111-1    linear of order 2
ρ51-11-111i-ii-i1-1-1-11-1-i-ii-ii-iii    linear of order 4
ρ611-1-111i-i-ii-11-1-1-11i-ii-i-ii-ii    linear of order 4
ρ71-11-111-ii-ii1-1-1-11-1ii-ii-ii-i-i    linear of order 4
ρ811-1-111-iii-i-11-1-1-11-ii-iii-ii-i    linear of order 4
ρ91111ζ3ζ321111ζ32ζ32ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ101-1-11ζ3ζ3211-1-1ζ6ζ6ζ3ζ32ζ65ζ65ζ6ζ32ζ3ζ3ζ65ζ65ζ6ζ32    linear of order 6
ρ111-11-1ζ3ζ32i-ii-iζ32ζ6ζ65ζ6ζ3ζ65ζ43ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ4ζ32    linear of order 12
ρ1211-1-1ζ3ζ32i-i-iiζ6ζ32ζ65ζ6ζ65ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ131111ζ3ζ32-1-1-1-1ζ32ζ32ζ3ζ32ζ3ζ3ζ6ζ6ζ65ζ65ζ65ζ65ζ6ζ6    linear of order 6
ρ141-1-11ζ3ζ32-1-111ζ6ζ6ζ3ζ32ζ65ζ65ζ32ζ6ζ65ζ65ζ3ζ3ζ32ζ6    linear of order 6
ρ151-11-1ζ3ζ32-ii-iiζ32ζ6ζ65ζ6ζ3ζ65ζ4ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ43ζ32    linear of order 12
ρ1611-1-1ζ3ζ32-iii-iζ6ζ32ζ65ζ6ζ65ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ171111ζ32ζ31111ζ3ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ181-1-11ζ32ζ311-1-1ζ65ζ65ζ32ζ3ζ6ζ6ζ65ζ3ζ32ζ32ζ6ζ6ζ65ζ3    linear of order 6
ρ191-11-1ζ32ζ3i-ii-iζ3ζ65ζ6ζ65ζ32ζ6ζ43ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ4ζ3    linear of order 12
ρ2011-1-1ζ32ζ3i-i-iiζ65ζ3ζ6ζ65ζ6ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    linear of order 12
ρ211111ζ32ζ3-1-1-1-1ζ3ζ3ζ32ζ3ζ32ζ32ζ65ζ65ζ6ζ6ζ6ζ6ζ65ζ65    linear of order 6
ρ221-1-11ζ32ζ3-1-111ζ65ζ65ζ32ζ3ζ6ζ6ζ3ζ65ζ6ζ6ζ32ζ32ζ3ζ65    linear of order 6
ρ231-11-1ζ32ζ3-ii-iiζ3ζ65ζ6ζ65ζ32ζ6ζ4ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ43ζ3    linear of order 12
ρ2411-1-1ζ32ζ3-iii-iζ65ζ3ζ6ζ65ζ6ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12

Permutation representations of C2×C12
►Regular action on 24 points - transitive group 24T2
Generators in S24
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)

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

120
012
,
90
05
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
 

Export

Subgroup lattice of C2×C12 in TeX
Character table of C2×C12 in TeX

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