Moufang Loop 16.1.2.31 of order 16

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

Centre: 0 1

Centrum: 0 1

Nucleus: 0 1

Left Nucleus: 0 1

Middle Nucleus: 0 1

Right Nucleus: 0 1

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: 1

14 Elements of order 4: 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Commutator Subloop: 0 1

Associator Subloop: 0 1

2 Conjugacy Classes of size 1:

7 Conjugacy Classes of size 2:

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

Automorphic Inverse Property: FAILS. (2^-1)(5^-1) neq (2*5)^-1

A_l Property: HOLDS (i.e. every left inner mapping L_a,b is an automorphism)

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

Right (Left, Full) Mult Group Orders: 128 (128, 1024)

/ revised October, 2001

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

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

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