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## G = Dic3order 12 = 22·3

### Dicyclic group

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

Series: Derived Chief Lower central Upper central

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

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

Character table of Dic3

 class 1 2 3 4A 4B 6 size 1 1 2 3 3 2 ρ1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 1 linear of order 2 ρ3 1 -1 1 i -i -1 linear of order 4 ρ4 1 -1 1 -i i -1 linear of order 4 ρ5 2 2 -1 0 0 -1 orthogonal lifted from S3 ρ6 2 -2 -1 0 0 1 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);

Dic3 is a maximal subgroup of
C4×S3  C3⋊D4  C3⋊Dic3  CSU2(𝔽3)  A4⋊C4  C33⋊C4  SL2(𝔽5)  C522Dic3  C52⋊Dic3
Dic3p: Dic6  Dic9  Dic15  Dic21  Dic33  Dic39  Dic51  Dic57 ...
C3p⋊C4, p=1 mod 4: C3⋊F5  C39⋊C4  C51⋊C4  C87⋊C4  C37⋊Dic3  C41⋊Dic3 ...
Dic3 is a maximal quotient of
A4⋊C4  C33⋊C4  C522Dic3  C52⋊Dic3
C2p.S3: C3⋊C8  Dic9  C3⋊Dic3  Dic15  Dic21  Dic33  Dic39  Dic51 ...
C3p⋊C4, p=1 mod 4: C3⋊F5  C39⋊C4  C51⋊C4  C87⋊C4  C37⋊Dic3  C41⋊Dic3 ...

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

 0 1 4 1
,
 3 2 0 2
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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