Complex Reflection Group - List of Irreducible Complex Reflection Groups

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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