Right Bol Loop 16.5.2.37 of order 16

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

Centre: 0 12

Centrum: 0 12

Nucleus: 0 12

Left Nucleus: 0 10 12 14

Middle Nucleus: 0 12

Right Nucleus: 0 12

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: 1 8 9 12 15

10 Elements of order 4: 2 3 4 5 6 7 10 11 13 14

Commutator Subloop: 0 12

Associator Subloop: 0 12

2 Conjugacy Classes of size 1:

7 Conjugacy Classes of size 2:

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

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

A_l Property: FAILS. The left inner mapping L_1,1 = (3,6)(10,14) is not an automorphism. L_1,1(2*3) neq L_1,1(2)*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 (1024, 2048)

/ revised October, 2001

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