Right Bol Loop 16.5.2.0 of order 16

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

Centre: 0 9

Centrum: 0 9

Nucleus: 0 9

Left Nucleus: 0 9 14 15

Middle Nucleus: 0 9 10 11

Right Nucleus: 0 9 10 11

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

5 Elements of order 2: 9 12 13 14 15

2 Elements of order 4: 10 11

8 Elements of order 8: 1 2 3 4 5 6 7 8

Commutator Subloop: 0 9 10 11

Associator Subloop: 0 9 10 11

2 Conjugacy Classes of size 1:

1 Conjugacy Class of size 2:

10 11

3 Conjugacy Classes of size 4:

1 2 3 4
5 6 7 8
12 13 14 15

Automorphic Inverse Property: FAILS. (1^-1)(8^-1) neq (1*8)^-1

A_l Property: FAILS. The left inner mapping L_1,1 = (5,8,6,7)(12,15,13,14) is not an automorphism. L_1,1(1*5) neq L_1,1(1)*L_1,1(5)

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

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

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