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G = Dic3order 12 = 22·3

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic3, C3⋊C4, C6.C2, C2.S3, SmallGroup(12,1)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3
C1C3C6 — Dic3
C3 — Dic3
C1C2

Generators and relations for Dic3
 G = < a,b | a6=1, b2=a3, bab-1=a-1 >

3C4

Character table of Dic3

 class 1234A4B6
 size 112332
ρ1111111    trivial
ρ2111-1-11    linear of order 2
ρ31-11i-i-1    linear of order 4
ρ41-11-ii-1    linear of order 4
ρ522-100-1    orthogonal lifted from S3
ρ62-2-1001    symplectic faithful, Schur index 2

Permutation representations of Dic3
Regular action on 12 points - transitive group 12T5
Generators in S12
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 8 4 11)(2 7 5 10)(3 12 6 9)

G:=sub<Sym(12)| (1,2,3,4,5,6)(7,8,9,10,11,12), (1,8,4,11)(2,7,5,10)(3,12,6,9)>;

G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12), (1,8,4,11)(2,7,5,10)(3,12,6,9) );

G=PermutationGroup([(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,8,4,11),(2,7,5,10),(3,12,6,9)])

G:=TransitiveGroup(12,5);

Polynomial with Galois group Dic3 over ℚ
actionf(x)Disc(f)
12T5x12-4x11-46x10+152x9+638x8-1789x7-3048x6+6884x5+6512x4-9622x3-4913x2+4165x-17118·179·592·4432·5572·5772·55632·614932

Matrix representation of Dic3 in GL2(𝔽5) generated by

01
41
,
32
02
G:=sub<GL(2,GF(5))| [0,4,1,1],[3,0,2,2] >;

Dic3 in GAP, Magma, Sage, TeX

{\rm Dic}_3
% in TeX

G:=Group("Dic3");
// GroupNames label

G:=SmallGroup(12,1);
// by ID

G=gap.SmallGroup(12,1);
# by ID

G:=PCGroup([3,-2,-2,-3,6,74]);
// Polycyclic

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

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