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
| C3 — Dic3 |
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 |
►Regular action on 12 points - transitive group 12T5
(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) C52⋊2Dic3 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 C52⋊2Dic3 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 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 12T5 | x12-4x11-46x10+152x9+638x8-1789x7-3048x6+6884x5+6512x4-9622x3-4913x2+4165x-17 | 118·179·592·4432·5572·5772·55632·614932 |
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
Export
Subgroup lattice of Dic3 in TeX
Character table of Dic3 in TeX