Binary Cubic Groups

On this page:

  • Q8 - the quaternion group
  • 2T - the binary tetrahedral group
  • 2O - the binary octahedral group
  • 2I - the binary icosahedral group (unfinished)

See also the binary dihedral groups, or dicyclic groups.

The binary cubic groups are the preimages under the double cover SU(2) → SO(3) of the cubic groups T, O, and I. We also include the quaternion group Q8, which is in fact the double cover of the group of roations by π about each of the Cartesian axes. If G is the cubic group in question, and 2G the binary version, then there is a short exact sequence,

$$\mathbb{Z}_2^c \longrightarrow 2G \longrightarrow G $$

where \(\mathbb{Z}_2^c =\{± I\}\) is the centre of SU(2). In particular, any representation of G gives rise to a representation of 2G, via the homomorphism 2G → G. Note that SU(2) ≅ Sp(1) (unit quaternions).

Q8

Q8 is the quaternion group and has order 8. It consists of the unit quaternions ±1, ±i, ±j, ±k. It is the double cover of the group D2, which is the subgoup of SO(3) consisting of the identity and the rotation by π about each of the x- y- and z-axes, so could be called the binary Klein group. The character table is:

Q8 1 -1 {±i} {±j} {±k} notes
# 1 1 2 2 2 |Q8|=8
A0 1 1 1 1 1 trivial rep
A1 1 1 1 -1 -1
A2 1 1 -1 1 -1
A3 1 1 -1 -1 1
H 4 -4 0 0 0 quaternionic type
  • Notice that jij-1=-i, so i and -i are conjugate, as are j and -j, and k and -k.
  • The 1-dimensional reps are all induced from the reps of D2 = Z2×Z2.
  • The irreducible representation H is the natural action of Q8 on the space of all quaternions (hence its name). It is therefore of quaternionic type. It is also the underlying representation of the 2-dimensional complex irreducible representation given by Q8 ⊂ SU(2)

2T

Here T ≅ A4 is the group of all rotational symmetries of the tetrahedron. The binary group 2T is therefore denoted 2A4. It is (isomorphic to) SL(2; Z3). See Tony Phillips' pages and for details about this group. It contains Q8 as a normal subgroup. As unit quaternions, the elements are of the form

±1, ±i, ±j, ±k,   ½ (±1 ±i ±j ±k).

The elements of the form ½(1 ±i ±j ±k) are of order 6, while those of the form ½(-1 ±i ±j ±k) are of order 3. And of course ±i etc are of order 4, and -1 is of order 2.

Conjugation within the group permutes the coefficients of the 3 "imaginary" parts and changes an even number of signs, but does not change the real part of a quaternion. For example,

½(1 + i + j + k) is conjugate to ½(1 - i - j + k), and +i is conjugate to -i and to -k.

The conjugacy classes are therefore represented by:

1, -1, i, a = ½(1 + i + j + k), b = ½(1 + i + j - k), c = ½(-1 + i + j + k), d = ½(-1 + i + j - k), so a and b are of order 6 and c, d are of order 3.

Note that a² = d² = c, and b² = c² = d. Finally, Q8 is a normal subgroup.

2T 1 -1 i a b c d notes
# 1 1 6 4 4 4 4 |2T|=24
A0 1 1 1 1 1 1 1 trivial rep
E 2 2 2 -1 -1 -1 -1 complex type
T 3 3 -1 0 0 0 0  
G 4 -4 0 -1 -1 1 1 complex type
H 4 -4 0 2 2 -2 -2 quaternionic type
  • A0 E and T arise from reps of T (see A4) via the double covering.
  • A0 and T are absolutely irreducible (ie, of real type), E and G are of complex type and H is of quaternionic type
  • E, G and H split over C, and have characters:
    E = (1, 1, 1, ω, ω², ω², ω) ⊕ (1, 1, 1, ω², ω, ω, ω²), where ω = ½(-1+√3 i),
    G = (2, -2, 0, ω, ω², -ω², -ω) ⊕ (2, -2, 0, ω², ω, -ω, -ω²), and
    H = 2 (2, -2, 0, 1, 1, -1, -1) (2 copies of the same complex rep)
  • The elements of this group considered as points in the unit sphere in R4 (space of quaternions) are the vertices of the convex regular 24-cell. The permutation representation on this set of points is the regular representation, equal to A0 + E + 3T +2G + H.
  • H is the natural representation of this group acting on the quaternions.

2O

where O is the group of all rotational symmetries of the cube. O is isomorphic to S4, so 2O is often denoted 2S4, it has order 48. See wikipedia for details.

The binary tetrahedral group 2T is a normal subgroup, as is Q8. The remaining elements are all of the form (1/√2)(±1±i) etc.

The conjugacy classes are represented by 1, -1, i, and

a = ½(1 +i + j + k), c = ½(-1 +i + j + k),
e = (1/√2)(1+i), f = (1/√2)(-1+i), g = (1/√2)(i+j).
2O 1 -1 i a c e f g notes
# 1 1 6 8 8 6 6 12 |2O|=48
A0 1 1 1 1 1 1 1 1 trivial rep
A1 1 1 1 1 1 -1 -1 -1  
E 2 2 2 -1 -1 0 0 0  
T1 3 3 -1 0 0 1 1 -1  
T2 3 3 -1 0 0 -1 -1 1  
H1 4 -4 0 2 -2 2√2 -2√2 0 action on quaternions
H2 4 -4 0 2 -2 -2√2 2√2 0
L 8 -8 0 -2 2 0 0 0
  • The reps A0, A1, E, T1 and T2 arise from the corresponding reps of the octahedral group; they are all absolutely irreducible (since they are for S4).
  • The reps H1, H2 and L are all irreducible of quaternionic type. The corresponding complex irreducible reps have simply half the character.

2I

where I is the group of all rotational symmetries of the icosahedron.

I is isomorphic to A5, so 2I is often denoted 2A5 (a double cover of A5); it is isomorphic to SL(3,F5). See wikipedia for details.

The elements of 2I are ...

... Unfinished ...