Quality report on orthogonal/seminormal models N = dimension nz = average number of nonzero entries in each row of s[n] MN = largest absolute numerator in s[n] MD = largest denominator in s[n] LCD = least common denominator for all entries in s[n] * indicates a representation that is not totally free r5 = sqrt(5), r7=sqrt(7), ... F4 N nz ortho: MN,MD,LCD seminormal: MN,MD,LCD 1 1 1.00 [1, 1, 1] [1, 1, 1] 2 1 1.00 [1, 1, 1] [1, 1, 1] 3 1 1.00 [1, 1, 1] [1, 1, 1] 4 1 1.00 [1, 1, 1] [1, 1, 1] 5 2 2.00 [r3, 2, 2] [3, 2, 2] 6 2 2.00 [r3, 2, 2] [3, 2, 2] 7 2 1.00 [1, 1, 1] [1, 1, 1] 8 2 1.00 [1, 1, 1] [1, 1, 1] 9 4 1.50 [r3, 2, 2] [3, 2, 2] 10 4 1.50 [r3, 2, 2] [3, 2, 2] 11 4 1.50 [r3, 2, 2] [3, 2, 2] 12 4 1.50 [r3, 2, 2] [3, 2, 2] 13 4 2.00 [r3, 2, 2] [3, 2, 2] 14 6 1.67 [r3, 2, 2] [3, 2, 2] 15 6 1.67 [r3, 2, 2] [3, 2, 2] 16 8 3.00 [3, 4, 4] [9, 4, 4] 17 8 3.00 [3, 4, 4] [9, 4, 4] 18 8 2.50 [3, 4, 4] [9, 4, 4] 19 8 2.50 [3, 4, 4] [3, 4, 4] 20 9 2.67 [1, 2, 2] [1, 2, 2] 21 9 2.67 [1, 2, 2] [1, 2, 2] 22 9 2.67 [1, 2, 2] [1, 2, 2] 23 9 2.67 [1, 2, 2] [1, 2, 2] 24 12 3.33 [3, 4, 4] [9, 4, 4] 25 16 5.00 [3*r3, 8, 8] [9, 8, 8] E6 1 1 1.00 [1, 1, 1] [1, 1, 1] 2 1 1.00 [1, 1, 1] [1, 1, 1] 3 6 1.33 [r15, 4, 4] [15, 4, 4] 4 6 1.33 [r15, 4, 4] [15, 4, 4] 5 10 1.00 [1, 1, 1] [1, 1, 1] 6 15 1.67 [r3, 2, 2] [3, 2, 2] 7 15 1.67 [r3, 2, 2] [3, 2, 2] 8 15 1.53 [r15, 4, 4] [15, 4, 4] 9 15 1.53 [r15, 4, 4] [15, 4, 4] 10 20 2.50 [7, 8, 8] [9, 8, 8] 11 20 2.50 [7, 8, 8] [9, 8, 8] 12 20 1.60 [r15, 4, 4] [15, 4, 4] 13 24 1.33 [r15, 4, 4] [15, 4, 4] 14 24 1.33 [r15, 4, 4] [15, 4, 4] 15 30 1.87 [3*r5, 8, 8] [15, 8, 8] 16 30 1.87 [3*r5, 8, 8] [15, 8, 8] 17 60 3.27 [3*r3, 8, 8] [9, 8, 8] 18 60 3.27 [3*r3, 8, 8] [9, 8, 8] 19 60 2.87 [r15, 4, 4] [9, 4, 4] 20 64 3.86 [7*r15, 64, 64] [105, 64, 64] 21 64 3.86 [7*r15, 64, 64] [135, 64, 64] 22 80 3.40 [3*r15, 16, 16] [15, 16, 16] 23 81 3.67 [5, 12, 12] [15, 12, 12] 24 81 3.67 [5, 12, 12] [5, 12, 12] 25 90 3.87 [r15, 8, 8*r3] [15, 24, 24] E7 1 1 1.00 [1, 1, 1] [1, 1, 1] 2 1 1.00 [1, 1, 1] [1, 1, 1] 3 7 1.29 [2*r2, 3, 3] [4, 3, 3] 4 7 1.29 [2*r2, 3, 3] [4, 3, 3] 5 15 1.00 [1, 1, 1] [1, 1, 1] 6 15 1.00 [1, 1, 1] [1, 1, 1] 7 21 1.10 [4*r5, 9, 9] [20, 9, 9] 8 21 1.10 [4*r5, 9, 9] [20, 9, 9] 9 21 1.48 [2*r2, 3, 3] [4, 3, 3] 10 21 1.48 [2*r2, 3, 3] [4, 3, 3] 11 27 1.59 [4*r35, 27, 27] [280, 27, 27] 12 27 1.59 [4*r35, 27, 27] [280, 27, 27] 13 35 1.57 [2*r2, 3, 3] [4, 3, 3] 14 35 1.57 [2*r2, 3, 3] [4, 3, 3] 15 35 1.57 [2*r2, 3, 3] [4, 3, 3] 16 35 1.57 [2*r2, 3, 3] [4, 3, 3] 17 56 1.48 [r5, 3, 3] [5, 3, 3] 18 56 1.48 [r5, 3, 3] [5, 3, 3] 19 70 1.29 [2*r2, 3, 3] [4, 3, 3] 20 70 1.29 [2*r2, 3, 3] [4, 3, 3] 21 84 1.48 [2*r2, 3, 3] [4, 3, 3] 22 84 1.48 [2*r2, 3, 3] [4, 3, 3] 23 105 2.05 [16*r2, 45, 45] [64, 45, 45] 24 105 2.05 [16*r2, 45, 45] [32, 45, 45] 25 105 1.38 [2*r6, 5, 5] [12, 5, 5] 26 105 1.38 [2*r6, 5, 5] [12, 5, 5] 27 105 1.95 [2*r2, 3, 3] [4, 3, 3] 28 105 1.95 [2*r2, 3, 3] [4, 3, 3] 29 120 2.32 [8*r7, 45, 45] [140, 45, 45] 30 120 2.32 [8*r7, 45, 45] [140, 135, 135] 31 168 2.45 [7, 9, 9] [10, 9, 9] 32 168 2.45 [7, 9, 9] [20, 9, 9] 33 189 2.40 [41, 45, 135] [80, 45, 135] 34 189 2.40 [41, 45, 135] [41, 45, 135] 35 189 3.00 [4*r10, 27, 27] [40, 27, 27] 36 189 3.00 [4*r10, 27, 27] [40, 27, 27] 37 189 2.16 [16*r2, 27, 27] [32, 27, 27] 38 189 2.16 [16*r2, 27, 27] [32, 27, 27] 39 210 2.33 [4*r2, 9, 9] [8, 9, 9] 40 210 2.33 [4*r2, 9, 9] [8, 9, 9] 41 210 2.66 [13, 45, 45] [20, 45, 45] 42 210 2.66 [13, 45, 45] [20, 45, 135] 43 216 2.25 [r70, 9, 18] [10, 9, 18] 44 216 2.25 [r70, 9, 18] [70, 18, 18] 45 280 2.66 [4*r2, 15, 30] [12, 15, 30] 46 280 2.66 [4*r2, 15, 30] [8, 15, 30] 47 280 2.55 [7, 15, 30] [10, 30, 30] 48 280 2.55 [7, 15, 30] [10, 30, 30] 49 315 3.25 [32, 135, 135] [64, 135, 135] 50 315 3.25 [32, 135, 135] [32, 135, 135] 51 336 2.91 [9, 45, 360] [15, 45, 360] 52 336 2.91 [9, 45, 360] [12, 45, 360] 53 378 3.72 [16*r2, 27, 27] [32, 27, 27] 54 378 3.72 [16*r2, 27, 27] [32, 27, 27] 55 405 3.79 [121, 405, 405] [224, 405, 405] 56 405 3.79 [121, 405, 405] [121, 405, 405] 57 420 3.64 [16, 45, 180] [18, 45, 180] 58 420 3.64 [16, 45, 180] [18, 45, 180] 59 512 4.03 [3*r21, 32, 96] [63, 32, 96] 60 512 4.03 [3*r21, 32, 96] [63, 96, 96] E8 1 1 1.00 [1, 1, 1] [1, 1, 1] 2 1 1.00 [1, 1, 1] [1, 1, 1] 3 8 1.25 [r3, 2, 2] [3, 2, 2] 4 8 1.25 [r3, 2, 2] [3, 2, 2] 5 28 1.43 [r3, 2, 2] [3, 2, 2] 6 28 1.43 [r3, 2, 2] [3, 2, 2] 7 35 1.51 [3*r3, 14, 14] [27, 14, 14] 8 35 1.51 [3*r3, 14, 14] [27, 14, 14] 9 50 1.60 [r3, 2, 2] [3, 2, 2] 10 50 1.60 [r3, 2, 2] [3, 2, 2] 11 56 1.54 [r3, 2, 2] [3, 2, 2] 12 56 1.54 [r3, 2, 2] [3, 2, 2] 13 70 1.57 [r3, 2, 2] [3, 2, 2] 14 84 2.50 [3*r5, 14, 28] [45, 14, 28] 15 84 2.50 [3*r5, 14, 28] [45, 14, 28] 16 112 2.32 [9*r5, 28, 84] [45, 28, 84] 17 112 2.32 [9*r5, 28, 84] [45, 28, 84] 18 160 1.90 [r35, 8, 8] [35, 8, 8] 19 160 1.90 [r35, 8, 8] [35, 8, 8] 20 168 1.71 [r3, 2, 2] [3, 2, 2] 21 175 1.69 [r3, 2, 2] [3, 2, 2] 22 175 1.69 [r3, 2, 2] [3, 2, 2] 23 210 2.18 [3*r5, 14, 14] [15, 14, 70] 24 210 2.18 [3*r5, 14, 14] [15, 14, 14] 25 300 1.87 [3*r3, 10, 10] [9, 10, 10] 26 300 1.87 [3*r3, 10, 10] [9, 10, 10] 27 350 1.99 [3*r3, 10, 10] [9, 10, 10] 28 350 1.99 [3*r3, 10, 10] [9, 10, 10] 29 400 2.50 [7, 10, 20] [9, 20, 20] 30 400 2.50 [7, 10, 20] [9, 10, 20] 31 420 1.71 [r3, 2, 2] [3, 2, 2] 32 448 2.07 [3*r15, 16, 16] [45, 16, 16] 33 448 2.36 [3*r3, 8, 8] [9, 8, 8] 34 448 2.36 [3*r3, 8, 8] [9, 8, 8] 35 525 2.40 [3*r3, 10, 30] [10, 10, 60] 36 525 2.40 [3*r3, 10, 30] [9, 10, 30] 37 560 3.69 [27, 56, 840] [27, 60, 840] 38 560 3.69 [27, 56, 840] [45, 140, 840] 39 567 3.41 [13, 42, 126] [30, 105, 630] 40 567 3.41 [13, 42, 126] [15, 315, 630] 41 700 3.32 [27, 140, 140] [27, 140, 420] 42 700 3.32 [27, 140, 140] [27, 140, 140] 43 700 2.58 [7, 14, 210] [9, 14, 210] 44 700 2.58 [7, 14, 210] [9, 14, 210] 45 840 2.41 [3*r15, 16, 16] [15, 16, 16] 46 840 2.41 [3*r15, 16, 16] [15, 16, 16] 47 840 2.20 [r3, 2, 2] [3, 6, 12] 48 840 2.20 [r3, 2, 2] [3, 6, 6] 49 972 3.00 [11, 18, 18] [70, 18, 90] 50 972 3.00 [11, 18, 18] [70, 18, 36] 51 1008 3.64 [11, 28, 168] [45, 168, 1680] 52 1008 3.64 [11, 28, 168] [15, 140, 840] 53 1050 3.16 [23, 70, 210] [23, 70, 210] 54 1050 3.16 [23, 70, 210] [23, 70, 210] 55 1134 2.30 [r3, 2*r3, 2*r3] [3, 6, 6] 56 1296 3.97 [5*r7, 48, 144] [35, 48, 144] 57 1296 3.97 [5*r7, 48, 144] [35, 120, 720] 58 1344 4.91 [27, 112, 672] [135, 560, 3360] 59 1344 4.91 [27, 112, 672] [45, 112, 3360] 60 1344 3.45 [3*r3, 8, 8] [9, 8, 8] 61 1400 3.99 [19, 70, 280] [45, 70, 1680] 62 1400 3.99 [19, 70, 280] [19, 70, 840] 63 1400 3.78 [5*r3, 28, 140] [21, 210, 840] 64 1400 3.78 [5*r3, 28, 140] [75, 70, 280] 65 1400 4.91 [27, 140, 1120] [45, 280, 3360] 66 1400 4.91 [27, 140, 1120] [135, 280, 1120] 67 1400 2.63 [r5, 6, 6] [5, 6, 30] 68 1575 4.23 [9*r5, 224, 3360] [45, 224, 3360] 69 1575 4.23 [9*r5, 224, 3360] [45, 224, 3360] 70 1680 3.92 [9*r3, 40, 120] [27, 60, 120] 71 2016 3.99 [19, 42, 84] [25, 63, 252] 72 2100 4.64 [29, 80, 240] [29, 80, 240] 73 2100 4.64 [29, 80, 240] [45, 80, 240] 74 2100 4.19 [13, 30, 60] [18, 30, 60] 75 2240 5.18 [15*r3, 56, 224] [105, 480, 3360] 76 2240 5.18 [15*r3, 56, 224] [75, 56, 672] 77 2268 4.90 [29, 112, 1008] [75, 112, 5040] 78 2268 4.90 [29, 112, 1008] [225, 560, 5040] 79 2400 4.74 [17, 40, 720] [63, 60, 720] 80 2400 4.74 [17, 40, 720] [35, 40, 720] 81 2688 4.38 [9*r3, 42, 252] [135, 140, 1260] 82 2800 5.70 [37, 112, 5040] [45, 560, 5040] 83 2800 5.70 [37, 112, 5040] [45, 280, 5040] 84 2835 5.31 [47, 126, 252] [47, 126, 3780] 85 2835 5.31 [47, 126, 252] [60, 126, 252] 86 3150 5.29 [33, 70, 420] [126, 315, 1260] 87 3200 4.96 [9*r7, 80, 10080] [189, 160, 10080] 88 3200 4.96 [9*r7, 80, 10080] [315, 210, 10080] 89 *3240 7.14 [21, 96, 20160] [135, 280, 20160] 90 *3240 7.14 [21, 96, 20160] [225, 840, 20160] 91 3360 5.92 [20*r5, 378, 15120] [105, 756, 15120] 92 3360 5.92 [20*r5, 378, 15120] [100, 378, 15120] 93 4096 7.65 [243, 512, 32256] [1575, 3584, 161280] 94 4096 7.65 [243, 512, 32256] [6075, 17920, 161280] 95 4096 7.82 [135*r15, 3584, 483840] [2835, 17920, 483840] 96 4096 7.82 [135*r15, 3584, 483840] [2025, 17920, 483840] 97 4200 6.86 [47, 210, 5040] [47, 315, 5040] 98 4200 6.86 [47, 210, 5040] [63, 210, 5040] 99 4200 6.65 [127, 378, 7560] [144, 378, 7560] 100 4200 6.65 [127, 378, 7560] [127, 378, 7560] 101 4200 6.55 [23, 70, 2520] [189, 280, 2520] 102 4480 7.13 [135*r3, 448, 6720] [405, 2240, 6720] 103 *4536 7.69 [121, 630, 5040] [121, 630, 15120] 104 *4536 7.69 [121, 630, 5040] [121, 630, 5040] 105 4536 7.09 [79, 378, 9072] [675, 1134, 45360] 106 5600 8.24 [43, 210, 10080] [135, 315, 10080] 107 *5600 8.32 [83, 630, 30240] [140, 945, 30240] 108 *5600 8.32 [83, 630, 30240] [135, 630, 30240] 109 5670 8.42 [121, 1890, 90720] [350, 2268, 90720] 110 *6075 9.14 [79, 1134, 90720] [140, 1134, 90720] 111 *6075 9.14 [79, 1134, 90720] [400, 1134, 90720] 112 *7168 10.23 [167, 896, 40320] [405, 4480, 40320]