### Table 1: Binary codes of the 24 24 Hadamard matrices Code Hadamard Matrices

"... In PAGE 24: ... All of the six possible [24; 12] codes appeared. The results are given in Table1 , where the equivalence classes of matrices are as listed in [21], with 1; 2; ; 59 representing H1; H2; ; H59 and K representing the 60th class (found by Kimura). Also we use the notation of [33] for the six codes that occur.... ..."

### Table 1. The 6 codes associated with the 60 Hadamard matrices of order 24.

2007

"... In PAGE 17: ...2, Assmus and Key classified the 60 H-classes according to the doubly-even binary codes associated to the columns of the matrices. (See Table1 in [3] or Table 7.1 in [2], but beware that 4232 D , listed with the code D, should be listed with the code C, and that 3242 D in line 3 of the table should be changed to 3242 C .... ..."

### Table 1. Type II codes from the 16 skew Hadamard matrices of order 24

"... In PAGE 6: ... There exist at least three optimal [20; 10; 8] self-dual codes over GF(5), all of which are from Hadamard matrices of order 20, two being from skew Hadamard matrices of order 20. For more detail about these codes, see [22, Table1 , Remark 3]. 4.... In PAGE 6: ... We have checked that there are exactly six Type II codes from the 16 SH matrices, one of them is the extended Golay code G24 of length 24 and the other have minimum weight 4. For detail, see Table1 . Here the flrst column refers to the binary Type II codes from [15] and the second column refers to the indices of the skew Hadamard matrices in [21].... ..."

### 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 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 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