Right Bol Loop 16.1.2.3 of order 16

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

Centre: 0 6

Centrum: 0 6

Nucleus: 0 6

Left Nucleus: 0 6 8 15

Middle Nucleus: 0 6

Right Nucleus: 0 6

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

1 Element of order 2: 6

6 Elements of order 4: 2 4 8 13 14 15

8 Elements of order 8: 1 3 5 7 9 10 11 12

Commutator Subloop: 0 2 4 6

Associator Subloop: 0 2 4 6

2 Conjugacy Classes of size 1:

1 Conjugacy Class of size 2:

3 Conjugacy Classes of size 4:

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

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

A_l Property: FAILS. The left inner mapping L_1,1 = (8,13)(9,10)(11,12)(14,15) is not an automorphism. L_1,1(1*9) neq L_1,1(1)*L_1,1(9)

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

Right (Left, Full) Mult Group Orders: 64 (1024, 4096)

/ revised October, 2001

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

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

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