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G = C2×C12order 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

C1 — C2×C12
C1C2C6C12 — C2×C12
C1 — C2×C12
C1 — C2×C12

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


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
ρ111-11-1ζ62ζ32ζ2ζ2ζ2ζ2ζ32ζ6ζ65ζ6ζ62ζ65ζ43ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ4ζ32    linear of order 12
ρ1211-1-1ζ62ζ32ζ2ζ2ζ2ζ2ζ6ζ32ζ65ζ6ζ65ζ62ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    linear of order 12
ρ131111ζ3ζ32-1-1-1-1ζ32ζ32ζ3ζ32ζ3ζ3ζ6ζ6ζ65ζ65ζ65ζ65ζ6ζ6    linear of order 6
ρ141-1-11ζ3ζ32-1-111ζ6ζ6ζ3ζ32ζ65ζ65ζ32ζ6ζ65ζ65ζ3ζ3ζ32ζ6    linear of order 6
ρ151-11-1ζ62ζ32ζ2ζ2ζ2ζ2ζ32ζ6ζ65ζ6ζ62ζ65ζ4ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ43ζ32    linear of order 12
ρ1611-1-1ζ62ζ32ζ2ζ2ζ2ζ2ζ6ζ32ζ65ζ6ζ65ζ62ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    linear of order 12
ρ171111ζ32ζ31111ζ3ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ181-1-11ζ32ζ311-1-1ζ65ζ65ζ32ζ3ζ6ζ6ζ65ζ3ζ32ζ32ζ6ζ6ζ65ζ3    linear of order 6
ρ191-11-1ζ32ζ62ζ2ζ2ζ2ζ2ζ62ζ65ζ6ζ65ζ32ζ6ζ43ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ4ζ3    linear of order 12
ρ2011-1-1ζ32ζ62ζ2ζ2ζ2ζ2ζ65ζ62ζ6ζ65ζ6ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    linear of order 12
ρ211111ζ32ζ3-1-1-1-1ζ3ζ3ζ32ζ3ζ32ζ32ζ65ζ65ζ6ζ6ζ6ζ6ζ65ζ65    linear of order 6
ρ221-1-11ζ32ζ3-1-111ζ65ζ65ζ32ζ3ζ6ζ6ζ3ζ65ζ6ζ6ζ32ζ32ζ3ζ65    linear of order 6
ρ231-11-1ζ32ζ62ζ2ζ2ζ2ζ2ζ62ζ65ζ6ζ65ζ32ζ6ζ4ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ43ζ3    linear of order 12
ρ2411-1-1ζ32ζ62ζ2ζ2ζ2ζ2ζ65ζ62ζ6ζ65ζ6ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    linear of order 12

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

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

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