### Table 1 lists these groups. We let Zn denote the cyclic group of order n, and Dn denote the dihedral group of order 2n. 3n

1990

"... In PAGE 7: ...Table 1 The groups in Table1 were found in the following way. Let L be the linear form LA, LB, or LC for Theorems 3A, 3B, and 3C respectively.... In PAGE 9: ...the xed point set of g in Zt?1, then 1 X n=0 p(tn+r)qn 1 (q)t 1 X (m;n3;:::;nt)2FP qQ(m;n3;:::;nt)+LA(m;n3;:::;nt) mod p: For the rst application of Proposition 4 and Table1 we take p(5n + 4) 0 mod 5. Table 1 shows that the symmetry group for 5n + 4 contains an element g of order ve.... In PAGE 9: ...the xed point set of g in Zt?1, then 1 X n=0 p(tn+r)qn 1 (q)t 1 X (m;n3;:::;nt)2FP qQ(m;n3;:::;nt)+LA(m;n3;:::;nt) mod p: For the rst application of Proposition 4 and Table 1 we take p(5n + 4) 0 mod 5. Table1 shows that the symmetry group for 5n + 4 contains an element g of order ve. Thus we need F P (g) = ?.... In PAGE 13: ... We could not nd appropriate residue classes for the three variable quadratic functions which occur for t = 6. However, for t = 7, the eight cycle for r = 0; 2, or 6 (see Table1 ) has a one dimensional xed point set, and we nd a one variable quadratic function. Lemma 1, with modulus 169, gives the Subbarao conjecture in these cases.... ..."

Cited by 8

### TABLE 1. For each of the ten smallest known closed hyperbolic three-manifolds, all of which are orientable, we give: a surgery description that embodies the manifold apos;s full symmetry group; the volume; the Chern{Simons invariant (mod 1 2); the rst homology; the symmetry group; and the length spectrum to = 2:0, with entries corresponding to core curves marked with asterisks. Dn = dihedral group of order 2n; S16 = semidihedral group of order 16, with presentation hx; y j x8 = y2 = 1; y?1xy = x3i.

### Table 2: The group D 4 of dihedral symmetry of a square.

"... In PAGE 2: ... DIHEDRAL SYMMETRY OPERATIONS ON JPEG IMAGES The operationsdefined by compositionsof flips aboutthe di- agonal and about the Y-axis form the group of dihedral sym- metry of the square, referred to as D 4 . These operations are listed and described in Table2 . The second column defines the result of of an operation o on pixel block f;; along with the equivalent DCT-domain relationshipbetween oF and F.... In PAGE 3: ... Let F k denote the 8 8 block numbered k in raster or- der of blocks of quantized DCT coefficients for I. Let I o de- note the result of applyingoperation o on the image (where o is one of the D 4 operations from Table2 ). From the preced- ing section, it is apparent that the quantized DCT coefficient blocks of I o will essentially be the same as those in I, with possible block reordering, transposition, and sign changes, and the quantization table will also be the same, with possi- ble transposition.... ..."

### Table 2: The group D 4 of dihedral symmetry of a square.

"... In PAGE 2: ... DIHEDRAL SYMMETRY OPERATIONS ON JPEG IMAGES The operationsdefined by compositionsof flips aboutthe di- agonal and about the Y-axis form the group of dihedral sym- metry of the square, referred to as D 4 . These operations are listed and described in Table2 . The second column defines the result of of an operation o on pixel block f;; along with the equivalent DCT-domain relationshipbetween oF and F.... In PAGE 3: ... Let F k denote the 8 8 block numbered k in raster or- der of blocks of quantized DCT coefficients for I. Let I o de- note the result of applyingoperation o on the image (where o is one of the D 4 operations from Table2 ). From the preced- ing section, it is apparent that the quantized DCT coefficient blocks of I o will essentially be the same as those in I, with possible block reordering, transposition, and sign changes, and the quantization table will also be the same, with possi- ble transposition.... ..."

### Table 5: Decomposition Table for the Dihedral Group of Order 8

1992

"... In PAGE 17: ...pproximately 0:082, i.e., about twice the computational work. The advantage of the decomposition of Table5 stems mainly from the fact that all characters have real values. Let us now turn to our main point of interest, namely to the flnite groups of rigid motions in R3.... ..."

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### Table 2: Quaternion group Q and dihedral group D4 e E i I j J k K

"... In PAGE 6: ... The contents of its columns are the following: group { name of the group; n { order of the group; # { number of ordered partitions, (0) = feg; a; a0 2 ; #w { number of interval norms; wmax { range of maximal interval norm; #wmax { number of maximal interval norms; #u { number of ultrametric interval norms; umax { range of maximal ultrametric interval norm; # { number of bad partitions. For the quaternion group (its Cayley table is given on the left part of Table2 ) we obtained 3 maximal ultrametric interval norms... In PAGE 8: ...and 12 maximal interval norms e E i I j J k K 0 1 2 2 3 3 4 4 0 1 2 2 4 4 3 3 0 1 3 3 2 2 4 4 0 1 3 3 4 4 2 2 0 1 4 4 2 2 3 3 0 1 4 4 3 3 2 2 e E i I j J k K 0 2 1 1 3 3 4 4 0 2 1 1 4 4 3 3 0 2 3 3 1 1 4 4 0 2 3 3 4 4 1 1 0 2 4 4 1 1 3 3 0 2 4 4 3 3 1 1 For the dihedral group D4 (its Cayley table is given on the right part of Table2 ) we list only the 7 maximal ultrametric interval norms e a b c E A B C 0 2 1 2 3 3 3 3 0 3 1 3 2 3 2 3 0 3 1 3 3 2 3 2 0 3 2 3 1 3 2 3 0 3 2 3 2 3 1 3 0 3 2 3 3 1 3 2 0 3 2 3 3 2 3 1... ..."

### Table 2: An example for grouping 8 points into 3 clusters. ! 2( + Dn) (44)

"... In PAGE 25: ... The third condition speci es the clustering criteria whereby the maximum diameter among all the clusters is minimized. For example, Table2 shows the output from an 8 3 neural network, where 8 points are grouped into 3 clusters. The neural network solution to this problem is approached in a manner similar to the point-congruence problem described in Section 4.... ..."

### Table 4: Expected block structure for Dn-symmetric primary and higher order bifurcation branches o and including a D128-symmetric solution path. Axially compressed cylindrical shell with: N=128, M=41. Average bandwidth for each block is bw 3jV ij 100 .

"... In PAGE 25: ... Of course, which symmetries actually occur in the analysis is dependent upon the physics in the problem. Table4 displays the block structure for all generic solution branches with dihedral symmetry expected with a 41 128 mesh. As is evident in Table 4, the less symmetry a solution branch has the less bene t a group theoretic approach o ers as far as allowing computation to be done in dimensionally-reduced subspaces.... In PAGE 25: ... Table 4 displays the block structure for all generic solution branches with dihedral symmetry expected with a 41 128 mesh. As is evident in Table4 , the less symmetry a solution branch has the less bene t a group theoretic approach o ers as far as allowing computation to be done in dimensionally-reduced subspaces. However, the group theoretic approach still allows one to avoid numerical ill-conditioning due to closely space bifurcation points which is crucial in some problems [32].... ..."

### Table 1 Irreducible unitary representations of the dihedral group D4. Example 5.4 In this example we analyze a variation of the Robinson form, described in [26]. This instance has an interesting dihedral symmetry, and it is given by:

2002

"... In PAGE 16: ... They satisfy the commutation relation- ship s = dsd. The group D4 has five irreducible representations, shown in Table1 , of degrees 1, 1, 1, 1, and 2 [27,7]. All of them are absolutely irreducible.... ..."

Cited by 22