Right Bol Loop 16.3.2.111 of order 16

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

Centre: 0 2

Centrum: 0 2

Nucleus: 0 2

Left Nucleus: 0 2 5 6

Middle Nucleus: 0 1 2 3

Right Nucleus: 0 1 2 3

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

3 Elements of order 2: 2 4 7

8 Elements of order 4: 1 3 5 6 12 13 14 15

4 Elements of order 8: 8 9 10 11

Commutator Subloop: 0 1 2 3

Associator Subloop: 0 1 2 3

2 Conjugacy Classes of size 1:

3 Conjugacy Classes of size 2:

2 Conjugacy Classes of size 4:

8 9 10 11
12 13 14 15

Automorphic Inverse Property: FAILS. (1^-1)(13^-1) neq (1*13)^-1

A_l Property: FAILS. The left inner mapping L_4,8 = (8,10,11,9)(12,15,14,13) is not an automorphism. L_4,8(4*8) neq L_4,8(4)*L_4,8(8)

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

Right (Left, Full) Mult Group Orders: 32 (512, 2048)

/ revised October, 2001

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

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

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