List of Irreducible Complex Reflection Groups
There are a few duplicates in the first 3 lines of this list; see the previous section for details.
- ST is the Shephard–Todd number of the reflection group.
- Rank is the dimension of the complex vector space the group acts on.
- Structure describes the structure of the group. The symbol * stands for a central product of two groups. For rank 2, the quotient by the (cyclic) center is the group of rotations of a tetrahedron, octahedron, or icosahedron (T = Alt(4), O = Sym(4), I = Alt(5), of orders 12, 24, 60), as stated in the table. For the notation 21+4, see extra special group.
- Order is the number of elements of the group.
- Reflections describes the number of reflections: 26412 means that there are 6 reflections of order 2 and 12 of order 4.
- Degrees gives the degrees of the fundamental invariants of the ring of polynomial invariants. For example, the invariants of group number 4 form a polynomial ring with 2 generators of degrees 4 and 6.
ST | Rank | Structure and names | Order | Reflections | Degrees | Codegrees |
---|---|---|---|---|---|---|
1 | n−1 | Symmetric group G(1,1,n) = Sym(n) | n! | 2n(n − 1)/2 | 2, 3, ...,n | 0,1,...,n − 2 |
2 | n | G(m,p,n) m > 1, n > 1, p|m (G(2,2,2) is reducible) | mnn!/p | 2mn(n−1)/2,dnφ(d) (d|m/p, d > 1) | m,2m,..,(n − 1)m; mn/p | 0,m,..., (n − 1)m if p < m; 0,m,...,(n − 2)m, (n − 1)m − n if p = m |
3 | 1 | Cyclic group G(m,1,1) = Zm | m | dφ(d) (d|m, d > 1) | m | 0 |
4 | 2 | Z2.T = 33 | 24 | 38 | 4,6 | 0,2 |
5 | 2 | Z6.T = 33 | 72 | 316 | 6,12 | 0,6 |
6 | 2 | Z4.T = 32 | 48 | 2638 | 4,12 | 0,8 |
7 | 2 | Z12.T = 〈3,3,3〉2 | 144 | 26316 | 12,12 | 0,12 |
8 | 2 | Z4.O = 44 | 96 | 26412 | 8,12 | 0,4 |
9 | 2 | Z8.O = 42 | 192 | 218412 | 8,24 | 0,16 |
10 | 2 | Z12.O = 43 | 288 | 26316412 | 12,24 | 0,12 |
11 | 2 | Z24.O = 〈4,3,2〉12 | 576 | 218316412 | 24,24 | 0,24 |
12 | 2 | Z2.O= GL2(F3) | 48 | 212 | 6,8 | 0,10 |
13 | 2 | Z4.O = 〈4,3,2〉2 | 96 | 218 | 8,12 | 0,16 |
14 | 2 | Z6.O = 32 | 144 | 212316 | 6,24 | 0,18 |
15 | 2 | Z12.O = 〈4,3,2〉6 | 288 | 218316 | 12,24 | 0,24 |
16 | 2 | Z10.I = 55 | 600 | 548 | 20,30 | 0,10 |
17 | 2 | Z20.I = 52 | 1200 | 230548 | 20,60 | 0,40 |
18 | 2 | Z30.I = 53 | 1800 | 340548 | 30,60 | 0,30 |
19 | 2 | Z60.I = 〈5,3,2〉30 | 3600 | 230340548 | 60,60 | 0,60 |
20 | 2 | Z6.I = 33 | 360 | 340 | 12,30 | 0,18 |
21 | 2 | Z12.I = 32 | 720 | 230340 | 12,60 | 0,48 |
22 | 2 | Z4.I = 〈5,3,2〉2 | 240 | 230 | 12,20 | 0,28 |
23 | 3 | W(H3) = Z2 × PSL2(5), Coxeter | 120 | 215 | 2,6,10 | 0,4,8 |
24 | 3 | W(J3(4)) = Z2 × PSL2(7), Klein | 336 | 221 | 4,6,14 | 0,8,10 |
25 | 3 | W(L3) = W(P3) = 31+2.SL2(3), Hessian | 648 | 324 | 6,9,12 | 0,3,6 |
26 | 3 | W(M3) =Z2 ×31+2.SL2(3), Hessian | 1296 | 29 324 | 6,12,18 | 0,6,12 |
27 | 3 | W(J3(5)) = Z2 ×(Z3.Alt(6)), Valentiner | 2160 | 245 | 6,12,30 | 0,18,24 |
28 | 4 | W(F4) = (SL2(3)* SL2(3)).(Z2 × Z2) Weyl | 1152 | 212+12 | 2,6,8,12 | 0,4,6,10 |
29 | 4 | W(N4) = (Z4*21 + 4).Sym(5) | 7680 | 240 | 4,8,12,20 | 0,8,12,16 |
30 | 4 | W(H4) = (SL2(5)*SL2(5)).Z2 Coxeter | 14400 | 260 | 2, 12, 20,30 | 0,10,18,28 |
31 | 4 | W(EN4) = W(O4) = (Z4*21 + 4).Sp4(2) | 46080 | 260 | 8,12,20,24 | 0,12,16,28 |
32 | 4 | W(L4) = Z3 × Sp4(3) | 155520 | 380 | 12,18,24,30 | 0,6,12,18 |
33 | 5 | W(K5) = Z2 ×Ω5(3) = Z2 × PSp4(3) = Z2 × PSU4(2) | 51840 | 245 | 4,6,10,12,18 | 0,6,8,12,14 |
34 | 6 | W(K6)= Z3.Ω− 6(3).Z2, Mitchell's group |
39191040 | 2126 | 6,12,18,24,30,42 | 0,12,18,24,30,36 |
35 | 6 | W(E6) = SO5(3) = O− 6(2) = PSp4(3).Z2 = PSU4(2).Z2, Weyl |
51840 | 236 | 2,5,6,8,9,12 | 0,3,4,6,7,10 |
36 | 7 | W(E7) = Z2 ×Sp6(2), Weyl | 2903040 | 263 | 2,6,8,10,12,14,18 | 0,4,6,8,10,12,16 |
37 | 8 | W(E8)= Z2.O+ 8(2), Weyl |
696729600 | 2120 | 2,8,12,14,18,20,24,30 | 0,6,10,12,16,18,22,28 |
For more information, including diagrams, presentations, and codegrees of complex reflection groups, see the tables in (Michel Broué, Gunter Malle & Raphaël Rouquier 1998).
Read more about this topic: Complex Reflection Group
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