Finite Group T

The finite group T is one of the three non-Abelian groups of order 12 (out of a total of fives groups of order 12), the other two being the alternating group A_4 and the dihedral group D_6. However, it is highly unfortunate that the symbol T is used to refer this particular group, since the symbol T is also used to denote the point group T that constitutes the pure rotational subgroup of the full tetrahedral group T_h and is isomorphic to A_4. Thus, of the three distinct non-Abelian groups of order 12, two different ones are each known as T under some circumstances. Extreme caution is therefore needed.


T is the semidirect product of C_3 by C_4 by the map g:C_4->Aut(C_3) given by g(k)=a^k, where a is the automorphism a(x)=-x. The group can be constructed from the generators

x=[0 i; i 0]
y=[omega 0; 0 omega^2],

where omega=e^(2pii/3) as the group elements 1, y, y^2, x, xy, xy^2, x^2, x^2y, x^2y^2, x^3, x^3y, and x^3y^2. The multiplication table is illustrated above.

T has conjugacy classes {1}, {x^2}, {y,y^2}, {x^2y,x^2y^2}, {x^3,x^3y,x^3y^2}, and {x,xy,xy^2}. There are 8 subgroups, and their lengths are 1, 2, 3, 4, 4, 4, 6, and 12. Of these, the following five are normal: {1}, {1,x^2}, {1,y,y^2}, {1,y,y^2,x^2,x^2y,x^2y^2}, and the entire group.


The cycle graph of T is illustrated above. The numbers of elements for with A^k=1 for k=1, 2, ... are 1, 2, 3, 8, 1, 6, 1, 8, 3, 2, 1, and 12.

The finite group T has the presentations




See also

Finite Group, Tetrahedral Group

Explore with Wolfram|Alpha


Pedersen, J. "Groups of Small Order."

Referenced on Wolfram|Alpha

Finite Group T

Cite this as:

Weisstein, Eric W. "Finite Group T." From MathWorld--A Wolfram Web Resource.

Subject classifications