Right Bol Loop 16.7.2.270 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	14	12	15	13
2	3	0	1	7	6	5	4	11	10	9	8	15	14	13	12
3	0	1	2	6	4	7	5	10	8	11	9	13	15	12	14
4	5	7	6	2	3	1	0	12	14	13	15	11	9	10	8
5	7	6	4	3	0	2	1	14	15	12	13	10	11	8	9
6	4	5	7	1	2	0	3	13	12	15	14	9	8	11	10
7	6	4	5	0	1	3	2	15	13	14	12	8	10	9	11
8	10	11	9	15	13	14	12	2	3	1	0	4	6	5	7
9	8	10	11	13	12	15	14	3	0	2	1	5	4	7	6
10	11	9	8	14	15	12	13	1	2	0	3	6	7	4	5
11	9	8	10	12	14	13	15	0	1	3	2	7	5	6	4
12	13	15	14	8	9	10	11	7	6	5	4	2	1	3	0
13	15	14	12	10	8	11	9	5	7	4	6	1	0	2	3
14	12	13	15	9	11	8	10	6	4	7	5	3	2	0	1
15	14	12	13	11	10	9	8	4	5	6	7	0	3	1	2

Centre: 0 2

Centrum: 0 2

Nucleus: 0 2

Left Nucleus: 0 1 2 3 4 5 6 7

Middle Nucleus: 0 2

Right Nucleus: 0 2

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

7 Elements of order 2: 2 5 6 9 10 13 14

8 Elements of order 4: 1 3 4 7 8 11 12 15

Commutator Subloop: 0 2

Associator Subloop: 0 2

2 Conjugacy Classes of size 1:

7 Conjugacy Classes of size 2:

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

Automorphic Inverse Property: FAILS. (4^-1)(9^-1) neq (4*9)^-1

A_l Property: FAILS. The left inner mapping L_1,8 = (4,7)(5,6)(8,11)(9,10)(12,15)(13,14) is not an automorphism. L_1,8(4*8) neq L_1,8(4)*L_1,8(8)

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

Right (Left, Full) Mult Group Orders: 32 (1024, 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	14	12	15	13
2	3	0	1	7	6	5	4	11	10	9	8	15	14	13	12
3	0	1	2	6	4	7	5	10	8	11	9	13	15	12	14
4	5	7	6	2	3	1	0	12	14	13	15	11	9	10	8
5	7	6	4	3	0	2	1	14	15	12	13	10	11	8	9
6	4	5	7	1	2	0	3	13	12	15	14	9	8	11	10
7	6	4	5	0	1	3	2	15	13	14	12	8	10	9	11
8	10	11	9	15	13	14	12	2	3	1	0	4	6	5	7
9	8	10	11	13	12	15	14	3	0	2	1	5	4	7	6
10	11	9	8	14	15	12	13	1	2	0	3	6	7	4	5
11	9	8	10	12	14	13	15	0	1	3	2	7	5	6	4
12	13	15	14	8	9	10	11	7	6	5	4	2	1	3	0
13	15	14	12	10	8	11	9	5	7	4	6	1	0	2	3
14	12	13	15	9	11	8	10	6	4	7	5	3	2	0	1
15	14	12	13	11	10	9	8	4	5	6	7	0	3	1	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	14	12	15	13
2	3	0	1	7	6	5	4	11	10	9	8	15	14	13	12
3	0	1	2	6	4	7	5	10	8	11	9	13	15	12	14
4	5	7	6	2	3	1	0	12	14	13	15	11	9	10	8
5	7	6	4	3	0	2	1	14	15	12	13	10	11	8	9
6	4	5	7	1	2	0	3	13	12	15	14	9	8	11	10
7	6	4	5	0	1	3	2	15	13	14	12	8	10	9	11
8	10	11	9	15	13	14	12	2	3	1	0	4	6	5	7
9	8	10	11	13	12	15	14	3	0	2	1	5	4	7	6
10	11	9	8	14	15	12	13	1	2	0	3	6	7	4	5
11	9	8	10	12	14	13	15	0	1	3	2	7	5	6	4
12	13	15	14	8	9	10	11	7	6	5	4	2	1	3	0
13	15	14	12	10	8	11	9	5	7	4	6	1	0	2	3
14	12	13	15	9	11	8	10	6	4	7	5	3	2	0	1
15	14	12	13	11	10	9	8	4	5	6	7	0	3	1	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	14	12	15	13
2	3	0	1	7	6	5	4	11	10	9	8	15	14	13	12
3	0	1	2	6	4	7	5	10	8	11	9	13	15	12	14
4	5	7	6	2	3	1	0	12	14	13	15	11	9	10	8
5	7	6	4	3	0	2	1	14	15	12	13	10	11	8	9
6	4	5	7	1	2	0	3	13	12	15	14	9	8	11	10
7	6	4	5	0	1	3	2	15	13	14	12	8	10	9	11
8	10	11	9	15	13	14	12	2	3	1	0	4	6	5	7
9	8	10	11	13	12	15	14	3	0	2	1	5	4	7	6
10	11	9	8	14	15	12	13	1	2	0	3	6	7	4	5
11	9	8	10	12	14	13	15	0	1	3	2	7	5	6	4
12	13	15	14	8	9	10	11	7	6	5	4	2	1	3	0
13	15	14	12	10	8	11	9	5	7	4	6	1	0	2	3
14	12	13	15	9	11	8	10	6	4	7	5	3	2	0	1
15	14	12	13	11	10	9	8	4	5	6	7	0	3	1	2