Right Bol Loop 16.5.4.4 of order 16

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

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

5 Elements of order 2: 3 5 9 12 15

10 Elements of order 4: 1 2 4 6 7 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:

9 12

3 Conjugacy Classes of size 4:

1 2 7 8
3 4 5 6
10 11 13 14

Automorphic Inverse Property: FAILS. (1^-1)(4^-1) neq (1*4)^-1

A_l Property: FAILS. The left inner mapping L_1,1 = (3,4,5,6)(10,11,13,14) is not an automorphism. L_1,1(1*3) neq L_1,1(1)*L_1,1(3)

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

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

/ revised October, 2001

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

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

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