Right Bol Loop 16.15.2.0 of order 16

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	0	3	2	5	4	7	6	9	8	11	10	15	14	13	12
2	3	0	1	6	7	4	5	10	11	8	9	13	12	15	14
3	2	1	0	7	6	5	4	11	10	9	8	14	15	12	13
4	5	6	7	0	1	2	3	12	15	13	14	8	10	11	9
5	4	7	6	1	0	3	2	15	12	14	13	9	11	10	8
6	7	4	5	2	3	0	1	13	14	12	15	10	8	9	11
7	6	5	4	3	2	1	0	14	13	15	12	11	9	8	10
8	9	11	10	14	13	15	12	0	1	2	3	4	6	7	5
9	8	10	11	13	14	12	15	1	0	3	2	5	7	6	4
10	11	9	8	15	12	14	13	2	3	0	1	6	4	5	7
11	10	8	9	12	15	13	14	3	2	1	0	7	5	4	6
12	15	14	13	11	10	9	8	4	5	6	7	0	2	3	1
13	14	15	12	9	8	11	10	6	7	4	5	2	0	1	3
14	13	12	15	8	9	10	11	7	6	5	4	3	1	0	2
15	12	13	14	10	11	8	9	5	4	7	6	1	3	2	0

Centre: 0 1

Centrum: 0 1

Nucleus: 0 1

Left Nucleus: 0 1 2 3 4 5 6 7

Middle Nucleus: 0 1

Right Nucleus: 0 1

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

15 Elements of order 2: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Commutator Subloop: 0 1 2 3

Associator Subloop: 0 1

2 Conjugacy Classes of size 1:

1 Conjugacy Class of size 2:

3 Conjugacy Classes of size 4:

4 5 6 7
8 9 10 11
12 13 14 15

Automorphic Inverse Property: HOLDS

A_l Property: FAILS. The left inner mapping L_2,8 = (4,5)(6,7)(8,9)(10,11)(12,15)(13,14) is not an automorphism. L_2,8(4*8) neq L_2,8(4)*L_2,8(8)

A_r Property: HOLDS (i.e. every right inner mapping R_a,b is an automorphism)

Right (Left, Full) Mult Group Orders: 64 (1024, 4096)

/ revised October, 2001

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	0	3	2	5	4	7	6	9	8	11	10	15	14	13	12
2	3	0	1	6	7	4	5	10	11	8	9	13	12	15	14
3	2	1	0	7	6	5	4	11	10	9	8	14	15	12	13
4	5	6	7	0	1	2	3	12	15	13	14	8	10	11	9
5	4	7	6	1	0	3	2	15	12	14	13	9	11	10	8
6	7	4	5	2	3	0	1	13	14	12	15	10	8	9	11
7	6	5	4	3	2	1	0	14	13	15	12	11	9	8	10
8	9	11	10	14	13	15	12	0	1	2	3	4	6	7	5
9	8	10	11	13	14	12	15	1	0	3	2	5	7	6	4
10	11	9	8	15	12	14	13	2	3	0	1	6	4	5	7
11	10	8	9	12	15	13	14	3	2	1	0	7	5	4	6
12	15	14	13	11	10	9	8	4	5	6	7	0	2	3	1
13	14	15	12	9	8	11	10	6	7	4	5	2	0	1	3
14	13	12	15	8	9	10	11	7	6	5	4	3	1	0	2
15	12	13	14	10	11	8	9	5	4	7	6	1	3	2	0

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	0	3	2	5	4	7	6	9	8	11	10	15	14	13	12
2	3	0	1	6	7	4	5	10	11	8	9	13	12	15	14
3	2	1	0	7	6	5	4	11	10	9	8	14	15	12	13
4	5	6	7	0	1	2	3	12	15	13	14	8	10	11	9
5	4	7	6	1	0	3	2	15	12	14	13	9	11	10	8
6	7	4	5	2	3	0	1	13	14	12	15	10	8	9	11
7	6	5	4	3	2	1	0	14	13	15	12	11	9	8	10
8	9	11	10	14	13	15	12	0	1	2	3	4	6	7	5
9	8	10	11	13	14	12	15	1	0	3	2	5	7	6	4
10	11	9	8	15	12	14	13	2	3	0	1	6	4	5	7
11	10	8	9	12	15	13	14	3	2	1	0	7	5	4	6
12	15	14	13	11	10	9	8	4	5	6	7	0	2	3	1
13	14	15	12	9	8	11	10	6	7	4	5	2	0	1	3
14	13	12	15	8	9	10	11	7	6	5	4	3	1	0	2
15	12	13	14	10	11	8	9	5	4	7	6	1	3	2	0

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	0	3	2	5	4	7	6	9	8	11	10	15	14	13	12
2	3	0	1	6	7	4	5	10	11	8	9	13	12	15	14
3	2	1	0	7	6	5	4	11	10	9	8	14	15	12	13
4	5	6	7	0	1	2	3	12	15	13	14	8	10	11	9
5	4	7	6	1	0	3	2	15	12	14	13	9	11	10	8
6	7	4	5	2	3	0	1	13	14	12	15	10	8	9	11
7	6	5	4	3	2	1	0	14	13	15	12	11	9	8	10
8	9	11	10	14	13	15	12	0	1	2	3	4	6	7	5
9	8	10	11	13	14	12	15	1	0	3	2	5	7	6	4
10	11	9	8	15	12	14	13	2	3	0	1	6	4	5	7
11	10	8	9	12	15	13	14	3	2	1	0	7	5	4	6
12	15	14	13	11	10	9	8	4	5	6	7	0	2	3	1
13	14	15	12	9	8	11	10	6	7	4	5	2	0	1	3
14	13	12	15	8	9	10	11	7	6	5	4	3	1	0	2
15	12	13	14	10	11	8	9	5	4	7	6	1	3	2	0