ABSTRACT
G = 〈a, b, c, d : a2 = b2 = c2 = d2 = [a, b] = [a, c] = [a, d] = [b, c] = [b, d] = [c, d] = 1〉. D8 = 1 + a + b+ c+ d+ abcd
G = 〈a, b, c : a4 = b2 = c2 = baba = [a, c] = [b, c] = 1〉. D9 = 1 + a + a2 + b+ ac+ a2bc D10 = 1 + a+ b+ a2b+ c+ a3c
G = 〈a, b, c : a4 = c2 = b2a2 = a3ba−1b−1 = [a, c] = [b, c] = 1〉. D11 = 1 + a+ a2 + b+ c+ abc D12 = 1 + a+ a2 + b+ ac+ a2bc
G = 〈a, b, c : a4 = c4 = b4 = a2b2 = b2c2 = a3ba−1b−1 = [a, c] = [b, c] = 1〉. D13 = 1 + a+ a2 + b+ ac+ bc D14 = 1 + a+ b+ ab+ c+ a2c
G = 〈a, b, c : a4 = b2 = bc2 = a3bca−1c−1 = [a, b] = 1〉. D15 = 1 + a+ a2 + ab+ c + a2bc D16 = 1 + a+ a2 + ab+ ac+ a3bc D17 = 1 + a+ b+ a3b+ ac+ a3c D18 = 1 + a+ a2 + a3b+ ac + abc
G = 〈a, b : a4 = b4 = a3ba−1b−1 = 1〉. D19 = 1 + a+ a2 + b+ ab2 + a2b3 D20 = 1 + a+ a2 + b+ b3 + a3b2
D21 = 1 + a+ b+ a2b+ b2 + a3b2
G = 〈a, b : a8 = b2 = a5ba−1b−1 = 1〉. D22 = 1 + a+ a2 + a5 + a4b+ a2b D23 = 1 + a+ a3 + a4 + a3b+ a5b
G = 〈a, b : a8 = b4 = a4b2 = a3ba−1b−1 = 1〉. D24 = 1 + a+ a2 + a5 + b+ a2b D25 = 1 + a+ a3 + a4 + ab+ a3b
G = 〈a, b : a8 = b4 = a4b2 = a7ba−1b−1 = 1〉. D26 = 1 + a+ a2 + a5 + b+ a2b D27 = 1 + a+ a3 + a4 + b+ a2b
18.78 Table (Kibler [1295]) (36,15,6) difference sets. There are 14 groups of order 36. Five do not have difference sets. Two abelian groups have seven difference sets giving seven nonisomorphic designs. Seven nonabelian groups have 28 difference sets but 26 of these arise as automorphism groups of the seven designs constructible by abelian groups. (These seven designs occur as follows. There are three designs with 3-rank equal to 12, given by the families of difference sets from the table: {D5, D8, D17, D20, D34}, {D6, D18}, {D7, D19, D33}. There are four designs with 3-rank equal to 14, given by the four families: {D3, D12, D24, D32, D35}, {D1, D13, D21, D25, D27, D29}, {D2, D14}, {D4, D11, D15, D16, D22, D23, D26, D28, D30, D31}.) Two difference sets (D9 and D10) are genuinely nonabelian; the designs constucted from D9 and D10 are dual to each other and have 3-rank 16. All 35 difference sets are given. (D35 does not occur in Kibler’s work.) G = 〈a3 = b3 = c4 = [a, b] = [a, c] = [b, c] = 1〉 (abelian)
D1 = (1 + a+ a2)(1 + b) + (1 + b+ b2)c+ (1 + ab+ a2b2)c2 + (1 + a2b+ ab2)c3
D2 = (1 + a+ a2)(1 + b) + (1 + b+ b2)c+ (1 + ab+ a2b2)c2 + (a + b+ a2b2)c3
D3 = (1 + a+ a2)(1 + b) + (1 + b+ b2)c+ (1 + ab+ a2b2)c2 + (a2 + ab+ b2)c3
D4 = (1 + a+ a2)(1 + b) + (1 + b+ b2)c+ (a+ a2b+ b2)c2 + (a2 + ab+ b2)c3
G = 〈a, b, c : a3 = b3 = c4 = [a, b] = [a, c] = b2cb−1c−1 = 1〉. D5 = (1+a+a2)+(1+a+a2)b+(1+ab+a2b2)c+(1+b+b2)c2+(1+a2b+ab2)c3
D6 = (1+a+a2)+(1+a+a2)b+(1+ab+a2b2)c+(1+b+b2)c2+a(1+a2b+ab2)c3
D7 = (1+a+b)+(ab+b2+ab2)+(1+ab+a2b2)c+(1+a+a2)c2+(1+a2b+ab2)c3
D8 = (1+a+b)+(ab+b2+ab2)+(1+ab+a2b2)c+(1+a+a2 )c2+a(1+a2b+ab2)c3
G = 〈a, b, c : a3 = b3 = c4 = [a, b] = [a, c] = b2cb−1c−1 = 1〉. D9 = (1+ a+ b)+ (a+ b+ ab)ab+(1+ a+ a2)c+(1+ ab+ a2b2)c2+(1+ b+ b2)c3
D10 = (1+a+ b)+(a+ b+ab)ab+(1+a+a2)c+(1+ab+a2b2)c2+(1+ b+ b2)ac3
G = 〈a, b, c : a3 = b3 = c4 = [a, b] = a2ca−1c−1 = b2cb−1c−1 = 1〉. D11 = (1+a+a2)+(1+a+a2)b+(a2+b+ab2)c+(1+b+b2)c2+(1+a2b+ab2)c3
G = 〈a, b, c : a3 = b3 = c4 = [a, b] = bca−1c−1 = a2cb−1c−1 = 1〉. D12 = (1+a+a2)+(1+a+a2)b+(a2+b+ab2)c+(1+b+b2)c2+(1+a2b+ab2)c3
D13 = (1+a+a2)+(1+a+a2)b+(a2+b+ab2)c+(1+b+b2)c2+a(1+a2b+ab2)c3
D14 = (1+a+a2)+(1+a+a2)b+(a2+b+ab2)c+(1+b+b2)c2+a2(1+a2b+ab2)c3
D15 = (1+a+a2)+(1+a+a2)b+(a2+b+ab2)c+a(1+b+b2)c2+(1+a2b+ab2)c3
D16 = (1+a+a2)+(1+a+a2)b+(a2+ab+b2)c+(1+b+b2)c2+(1+ab+a2b2)c3
G = 〈a3 = b3 = c2 = d2 = [a, b] = [c, d] = [a, c] = [a, d] = [b, c] = [b, d] = 1〉 (abelian) D17 = (1+a+a2)+(1+a2+b+a2b+b2+a2b2)c+(a+a2b+b2)d+(a+b+a2b2)cd D18 = (1+a+a2)+(1+a2+b+a2b+b2+a2b2)c+(a2+ab+b2)d+(1+ab+a2b2)cd D19 = (1+a+a2)+(1+a2+ab+a2b+b2+ab2)c+(1+b+b2)d+(1+ab+a2b2)cd
G = 〈a, b, c, d : a3=b3=c2=d2=[a, b]=[c, d]=[a, d]=[b, d]=1, a2c= ca, b2c=cb〉. D20 = (1+a+a2)+(1+a+a2)b+(1+b+b2)c+(1+ab+a2b2)d+(1+a2b+ab2)cd
G = 〈a, b, c, d : a3=b3=c2=d2=[a, b]=[c, d]=[a, d]=[b, c]=[b, d]=a2ca−1c−1 = 1〉. D21 = (1+a+a2)+(1+a+a2)b+(1+ab+a2b2)c+(1+b+b2)d+(1+a2b+ab2)cd D22 = (1+a+a2)+(1+a+a2)b+(1+ab+a2b2)c+(1+b+b2)d+a(1+a2b+ab2)cd D23 = (1+a+b)+(ab+b2+ab2)+(1+ab+a2b2)c+(1+a+a2)d+(1+a2b+ab2)cd D24 = (1+a+b)+(ab+b2+ab2)+(1+ab+a2b2)c+(1+a+a2)d+a(1+a2b+ab2)cd D25 = (1+a+b)+a(ab+ b2+ab2)+(1+a+a2)c+(1+ab+a2b2)d+(1+b+b2)cd D26 = (1+a+b)+a(ab+b2+ab2)+(1+a+a2)c+a2(1+ab+a2b2)d+(1+b+b2)cd
G = 〈a, b, c, d : a3=b3=c2=d2=[a, b]=[c, d]=[a, d]=[b, c]=1, b2d=db, a2c=ca〉. D27 = (1+a+a2)+(b+ab+a2b)+(c+abc+a2b2c)+(d+a2bd+ab2d)+(cd+bcd+b2cd) D28 = (1+a+a2)+(1+a+a2)b+(1+ab+a2b2)c+(1+a2b+ab2)d+a2(1+b+b2)cd D29 = (1+a+b)+(a2b+ab2+a2b2)+(c+ac+a2c)+(d+bd+b2d)+(cd+abcd+a2b2cd)
D30 = (1+a+b)+(a2b+ab2+a2b2)+(1+a+a2)c+a2(1+b+b2)d+(1+ab+a2b2)cd D31 = (1+a+b)+(a2b+ab2+a2b2)+(1+b+b2)c+(1+a+a2)d+(1+ab+a2b2)cd D32 = (1+a+b)+(a2b+ab2+a2b2)+(1+b+b2)c+(1+a+a2)d+a(1+ab+a2b2)cd
G = 〈a, b, c, d : a3=b3=c2=d2=[a, b]=[c, d]=[b, c]= [d, b]=1, da=ac, cda=ad〉. D33 = (1+ c+ d+ cd+ a+ ca+ da2)+ (c+ ca+ cda+ cda2)b+(c+ ca+ da+ a2)b2
D34 = (1+c+d+cda+a+ca2+cda2)+(a+da+ca2+da2)b+(a+ca+ca2+a2)b2
D35 = 1+a+ b+d+ c+a2+ac+ b2+abd+abc+a2bc+ab2c+abdc+ b2dc+a2bdc
18.79 Remarks
1. The multipliers of the difference sets with k < 20 are also known, see [1295]. 2. Tables 18.75, 18.77, and 18.78 contain all difference sets, up to equivalence, with k < 20.
