Right Bol Loop 16.9.4.1 of order 16


0123456789101112131415
1012101113141592438657
2915111014131201347568
3101409151213114786125
4111390121514103875216
5131115120910146124783
6141012159011135213874
7159131410110128652431
8120141311109157561342
9214365870111015141312
1034127856111215149013
1143218765101512130914
1287654321151413011109
1356781234149011121510
1465872143130910151211
1578563412121314910110

Centre:   0   15

Centrum:   0   9   12   15

Nucleus:   0   15

Left Nucleus:   0   9   12   15

Middle Nucleus:   0   15

Right Nucleus:   0   15


Comm(L):   This graph has as its 7 vertices the nontrivial cosets of the centre. Edges represent non-commuting cosets. Here we print (in reverse video) the complementary graph, in which edges represent commuting cosets.


1 Element of order 1:   0

9 Elements of order 2:   1   3   4   5   6   7   9   12   15

6 Elements of order 4:   2   8   10   11   13   14

Commutator Subloop:   0   9   12   15

Associator Subloop:   0   9   12   15

2 Conjugacy Classes of size 1:

1 Conjugacy Class of size 2:

3 Conjugacy Classes of size 4:

Automorphic Inverse Property:   FAILS.   (1-1)(3-1) neq (1*3)-1

Al Property:   FAILS. The left inner mapping L1,1 = (2,8)(3,4)(5,6)(9,12)(10,11)(13,14) is not an automorphism.   L1,1(2*3) neq L1,1(2)*L1,1(3)

Ar Property:   FAILS. The right inner mapping R1,3 = (1,2)(3,5)(4,6)(7,8)(10,14)(11,13) is not an automorphism.   R1,3(1*1) neq R1,3(1)*R1,3(1)

Right (Left, Full) Mult Group Orders:   64 (4096, 16384)


/ revised October, 2001