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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 = 3[3]3 24 38 4,6 0,2 5 2 Z6.T = 3[4]3 72 316 6,12 0,6 6 2 Z4.T = 3[6]2 48 2638 4,12 0,8 7 2 Z12.T = 〈3,3,3〉2 144 26316 12,12 0,12 8 2 Z4.O = 4[3]4 96 26412 8,12 0,4 9 2 Z8.O = 4[6]2 192 218412 8,24 0,16 10 2 Z12.O = 4[4]3 288 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 = 3[8]2 144 212316 6,24 0,18 15 2 Z12.O = 〈4,3,2〉6 288 218316 12,24 0,24 16 2 Z10.I = 5[3]5 600 548 20,30 0,10 17 2 Z20.I = 5[6]2 1200 230548 20,60 0,40 18 2 Z30.I = 5[4]3 1800 340548 30,60 0,30 19 2 Z60.I = 〈5,3,2〉30 3600 230340548 60,60 0,60 20 2 Z6.I = 3[5]3 360 340 12,30 0,18 21 2 Z12.I = 3[10]2 720 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). |
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