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## G = C2×Dic3order 24 = 23·3

### Direct product of C2 and Dic3

Aliases: C2×Dic3, C6⋊C4, C2.2D6, C22.S3, C6.4C22, C32(C2×C4), (C2×C6).C2, SmallGroup(24,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×Dic3
 Chief series C1 — C3 — C6 — Dic3 — C2×Dic3
 Lower central C3 — C2×Dic3
 Upper central C1 — C22

Generators and relations for C2×Dic3
G = < a,b,c | a2=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C2×Dic3

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C size 1 1 1 1 2 3 3 3 3 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 1 -1 1 i i -i -i -1 -1 1 linear of order 4 ρ6 1 1 -1 -1 1 -i i i -i -1 1 -1 linear of order 4 ρ7 1 1 -1 -1 1 i -i -i i -1 1 -1 linear of order 4 ρ8 1 -1 1 -1 1 -i -i i i -1 -1 1 linear of order 4 ρ9 2 -2 -2 2 -1 0 0 0 0 -1 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 -1 0 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 -2 -1 0 0 0 0 1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ12 2 -2 2 -2 -1 0 0 0 0 1 1 -1 symplectic lifted from Dic3, Schur index 2

Permutation representations of C2×Dic3
Regular action on 24 points - transitive group 24T6
Generators in S24
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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)
(1 16 4 13)(2 15 5 18)(3 14 6 17)(7 20 10 23)(8 19 11 22)(9 24 12 21)

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

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

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

G:=TransitiveGroup(24,6);

Matrix representation of C2×Dic3 in GL3(𝔽13) generated by

 12 0 0 0 12 0 0 0 12
,
 1 0 0 0 1 12 0 1 0
,
 1 0 0 0 5 8 0 0 8
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[1,0,0,0,1,1,0,12,0],[1,0,0,0,5,0,0,8,8] >;

C2×Dic3 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3
% in TeX

G:=Group("C2xDic3");
// GroupNames label

G:=SmallGroup(24,7);
// by ID

G=gap.SmallGroup(24,7);
# by ID

G:=PCGroup([4,-2,-2,-2,-3,16,259]);
// Polycyclic

G:=Group<a,b,c|a^2=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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