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## G = Dic6order 24 = 23·3

### Dicyclic group

Aliases: Dic6, C3⋊Q8, C4.S3, C2.3D6, C12.1C2, Dic3.C2, C6.1C22, SmallGroup(24,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — Dic6
 Chief series C1 — C3 — C6 — Dic3 — Dic6
 Lower central C3 — C6 — Dic6
 Upper central C1 — C2 — C4

Generators and relations for Dic6
G = < a,b | a12=1, b2=a6, bab-1=a-1 >

Character table of Dic6

 class 1 2 3 4A 4B 4C 6 12A 12B size 1 1 2 2 6 6 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 linear of order 2 ρ3 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ5 2 2 -1 2 0 0 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 -1 -2 0 0 -1 1 1 orthogonal lifted from D6 ρ7 2 -2 2 0 0 0 -2 0 0 symplectic lifted from Q8, Schur index 2 ρ8 2 -2 -1 0 0 0 1 √3 -√3 symplectic faithful, Schur index 2 ρ9 2 -2 -1 0 0 0 1 -√3 √3 symplectic faithful, Schur index 2

Permutation representations of Dic6
Regular action on 24 points - transitive group 24T5
Generators in S24
(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 13 7 19)(2 24 8 18)(3 23 9 17)(4 22 10 16)(5 21 11 15)(6 20 12 14)

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

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

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

G:=TransitiveGroup(24,5);

Matrix representation of Dic6 in GL2(𝔽11) generated by

 2 7 7 3
,
 0 10 1 0
G:=sub<GL(2,GF(11))| [2,7,7,3],[0,1,10,0] >;

Dic6 in GAP, Magma, Sage, TeX

{\rm Dic}_6
% in TeX

G:=Group("Dic6");
// GroupNames label

G:=SmallGroup(24,4);
// by ID

G=gap.SmallGroup(24,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-3,16,49,21,259]);
// Polycyclic

G:=Group<a,b|a^12=1,b^2=a^6,b*a*b^-1=a^-1>;
// generators/relations

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