Right Bol Loop 16.9.2.312 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	7	3	4	8	5	6	2
2	9	0	11	10	14	13	12	15	8	4	3	7	6	5	1
3	10	13	0	15	9	12	11	14	5	8	7	6	2	1	4
4	11	14	9	12	0	15	10	13	6	7	8	5	1	2	3
5	13	10	15	0	12	9	14	11	3	2	1	4	8	7	6
6	14	11	12	9	15	0	13	10	4	1	2	3	7	8	5
7	15	12	13	14	10	11	0	9	1	5	6	2	3	4	8
8	12	15	14	13	11	10	9	0	2	6	5	1	4	3	7
9	2	1	4	3	6	5	8	7	0	11	10	15	14	13	12
10	3	5	1	7	2	8	4	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	4	8	2	7	1	5	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 12

Centrum: 0 12

Nucleus: 0 12

Left Nucleus: 0 9 12 15

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

9 Elements of order 2: 1 2 3 6 7 8 9 12 15

6 Elements of order 4: 4 5 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)(4^-1) neq (1*4)^-1

A_l Property: FAILS. The left inner mapping L_1,1 = (2,7)(9,15) 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	7	3	4	8	5	6	2
2	9	0	11	10	14	13	12	15	8	4	3	7	6	5	1
3	10	13	0	15	9	12	11	14	5	8	7	6	2	1	4
4	11	14	9	12	0	15	10	13	6	7	8	5	1	2	3
5	13	10	15	0	12	9	14	11	3	2	1	4	8	7	6
6	14	11	12	9	15	0	13	10	4	1	2	3	7	8	5
7	15	12	13	14	10	11	0	9	1	5	6	2	3	4	8
8	12	15	14	13	11	10	9	0	2	6	5	1	4	3	7
9	2	1	4	3	6	5	8	7	0	11	10	15	14	13	12
10	3	5	1	7	2	8	4	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	4	8	2	7	1	5	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	0	9	10	11	13	14	15	12	7	3	4	8	5	6	2
2	9	0	11	10	14	13	12	15	8	4	3	7	6	5	1
3	10	13	0	15	9	12	11	14	5	8	7	6	2	1	4
4	11	14	9	12	0	15	10	13	6	7	8	5	1	2	3
5	13	10	15	0	12	9	14	11	3	2	1	4	8	7	6
6	14	11	12	9	15	0	13	10	4	1	2	3	7	8	5
7	15	12	13	14	10	11	0	9	1	5	6	2	3	4	8
8	12	15	14	13	11	10	9	0	2	6	5	1	4	3	7
9	2	1	4	3	6	5	8	7	0	11	10	15	14	13	12
10	3	5	1	7	2	8	4	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	4	8	2	7	1	5	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	0	9	10	11	13	14	15	12	7	3	4	8	5	6	2
2	9	0	11	10	14	13	12	15	8	4	3	7	6	5	1
3	10	13	0	15	9	12	11	14	5	8	7	6	2	1	4
4	11	14	9	12	0	15	10	13	6	7	8	5	1	2	3
5	13	10	15	0	12	9	14	11	3	2	1	4	8	7	6
6	14	11	12	9	15	0	13	10	4	1	2	3	7	8	5
7	15	12	13	14	10	11	0	9	1	5	6	2	3	4	8
8	12	15	14	13	11	10	9	0	2	6	5	1	4	3	7
9	2	1	4	3	6	5	8	7	0	11	10	15	14	13	12
10	3	5	1	7	2	8	4	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	4	8	2	7	1	5	3	13	0	9	10	15	12	11
15	7	8	5	6	3	4	1	2	12	13	14	9	10	11	0