Equation 677 Database

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Magma 29114da6b210…

magma 29114da6b210
Size
841
Isomorphism class hash
29114da6b2104ef81c020a93ef8aa1bc3b71ec816d4186c3cb104944af226769
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
yes
Submitted by
b-reinke
Submitted at
2026-05-14 05:11:58
Display reorder
0,2,793,314,572,269,540,627,487,332,335,731,714,466,206,363,787,565,82,268,651,663,179,145,195,448,623,112,614,1,522,823,657,520,411,654,538,203,665,348,427,551,547,768,215,489,36,412,356,367,434,723,679,610,375,19,739,835,227,174,775,831,583,473,125,672,172,480,569,509,271,113,69,218,539,231,491,101,774,33,640,277,305,834,45,369,53,634,502,355,382,230,328,391,16,625,371,748,680,500,815,694,392,527,183,637,754,703,51,531,553,838,34,449,469,442,239,707,293,302,84,103,339,67,404,95,146,109,421,626,683,410,705,837,561,508,316,745,676,177,435,163,106,306,746,323,497,444,526,632,689,743,386,350,464,548,749,387,675,11,377,437,833,733,557,495,620,225,664,29,46,178,810,366,692,159,290,511,256,110,212,262,216,494,458,38,660,559,589,568,354,18,238,465,816,65,794,54,157,795,819,86,133,182,712,409,415,681,244,37,688,750,751,344,596,321,311,73,426,513,111,89,114,566,168,307,151,807,440,298,100,710,758,512,61,510,357,528,669,338,9,537,26,479,647,381,644,713,690,365,541,825,240,813,402,401,424,193,600,452,764,555,17,428,631,378,315,249,15,175,248,186,347,587,629,160,97,481,467,294,127,194,580,729,607,773,603,643,802,711,673,732,827,74,341,57,202,171,131,616,229,730,292,498,329,666,85,153,446,736,400,283,586,93,735,394,425,134,296,558,507,595,397,536,639,21,699,56,447,396,820,708,474,333,505,759,454,235,685,642,4,352,753,188,808,360,376,532,98,702,136,671,591,678,141,334,288,301,207,503,405,139,62,297,741,430,346,700,451,399,79,234,158,740,832,621,90,264,32,618,209,77,201,330,535,788,107,140,263,358,726,622,309,609,190,646,782,709,482,567,30,443,658,560,461,327,319,722,143,331,390,161,83,219,450,523,698,75,99,761,192,576,606,419,282,286,691,817,436,47,308,121,760,720,124,247,144,142,223,645,232,677,39,3,58,130,118,779,374,478,276,71,50,588,514,496,839,836,574,236,310,780,485,544,612,22,180,102,734,525,20,165,585,716,254,599,199,772,320,592,132,648,383,317,299,608,636,191,472,778,433,253,486,196,27,590,562,641,617,170,325,613,456,25,777,304,258,530,204,322,438,704,604,353,783,72,259,653,477,185,721,137,345,12,638,724,289,155,181,279,423,797,602,122,92,342,10,661,706,803,189,476,582,205,373,550,598,575,462,624,278,684,492,208,220,728,6,543,738,439,805,659,552,372,744,416,556,361,24,490,432,417,380,662,318,41,615,828,670,459,406,517,14,718,407,763,343,649,818,769,674,351,210,422,533,652,830,605,362,484,546,198,515,31,370,429,66,542,695,579,149,8,246,806,252,44,483,809,549,682,379,501,455,237,784,519,785,650,123,593,280,228,135,55,241,147,63,76,408,840,757,791,801,413,393,388,752,250,398,265,781,260,418,786,275,767,771,270,776,747,796,403,737,742,762,255,245,150,7,457,368,184,5,597,176,633,619,545,471,656,117,274,719,273,628,577,701,87,453,284,340,570,200,798,337,792,156,717,755,431,312,187,693,78,173,715,42,281,571,70,214,326,119,303,686,601,94,445,385,349,116,812,414,756,518,52,727,420,581,441,324,224,197,822,129,88,766,696,725,166,104,291,765,395,80,287,126,336,668,389,313,148,493,611,105,154,814,789,233,584,488,242,285,81,49,460,563,68,257,506,128,152,811,384,655,211,251,687,790,60,13,554,524,468,169,40,635,521,64,564,120,770,364,272,96,213,504,667,826,48,300,804,534,266,167,578,222,829,28,226,108,475,59,164,221,573,799,697,267,594,35,138,217,529,821,91,115,630,470,824,261,43,243,800,516,463,23,162,295,499,359 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

Size-841 = 29² idempotent right-cancellative magma satisfying Eq 677 and Eq 255 (but NOT Eq 3345). Constructed using the exceptional Zassenhaus near-field of order 29². The multiplicative group of this near-field has order 840 = 29² − 1 with structure Z₅ × Z₇ × SL(2, 3) (the binary tetrahedral group of order 24, extended by cyclic factors Z₅ and Z₇ — a non-cyclic group of order 840). The automorphism group of the magma is the sharply 2-transitive (Z/29)² ⋊ (Z₅ × Z₇ × SL(2, 3)) of order 706,440 = 841 × 840. Operation: x ◇ y = x + f(y − x) on the additive group (Z/29)² ≅ GF(841), where f is left-multiplication by a fixed near-field element c of order 10 in the mult group. f has cycle structure (1, 10⁸⁴) — fixed at 0, with 84 orbits of length 10 partitioning F_841* (the 84 cosets of ⟨c⟩ in the order-840 multiplicative group). f is NOT F_29-linear (so this is NOT the simple linear F_841 magma of the form x ◇ y = x + c(y − x) over GF(841) with commutative ·): in the basis used by the display reorder, f(1, 0) + f(0, 1) = (8, 23) ≠ (7, 3) = f(1, 1) in (Z/29)² coordinates. The axis-additivity of f together with its global non-additivity is the structural fingerprint of the exceptional near-field's non-distributive multiplication. Direct analog of magma#5ebfbb80 (size 121 = 11², exceptional near-field of order 11²), which has the same template but with Z₅ × SL(2, 3) replacing Z₅ × Z₇ × SL(2, 3), and which additionally satisfies Eq 3345. Display reorder presents elements as (i, j) ↦ τ₁^i τ₂^j(0) at index 29i + j, where τ₁, τ₂ are commuting order-29 fix-free magma automorphisms (the hidden additive translations by '1' and '2' in (Z/29)²). Under this reorder the Cayley table is fully (Z/29)²-translation-invariant; the rendered image shows diagonal banding across the 841×841 grid. [text written by Claude]

last edited by dwrensha at 2026-05-14 20:46:15 · history