Moufang Loop 28.21.1.0 of order 28

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27
1	0	3	2	8	7	17	5	4	13	12	18	10	9	27	26	20	6	11	22	16	23	19	21	25	24	15	14
2	3	0	1	9	6	5	17	13	4	16	14	20	8	11	21	10	7	27	24	12	15	25	26	19	22	23	18
3	2	1	0	5	4	9	8	7	6	11	10	18	17	16	19	14	13	12	15	27	24	26	25	21	23	22	20
4	8	7	5	0	3	10	2	1	11	6	9	17	18	19	16	15	12	13	14	26	25	27	24	23	21	20	22
5	9	6	4	3	0	2	10	11	1	7	8	15	14	13	12	17	16	19	18	21	20	23	22	27	26	25	24
6	17	5	7	10	2	0	3	12	16	4	15	8	20	21	11	9	1	26	25	13	14	24	27	22	19	18	23
7	13	4	6	11	1	16	12	3	0	5	19	26	27	9	10	21	2	8	23	17	22	18	20	14	15	24	25
8	4	13	9	1	11	14	18	0	3	19	5	22	2	6	23	24	27	7	10	25	26	12	15	16	20	21	17
9	5	17	8	2	10	3	0	18	14	15	4	7	1	24	25	6	20	22	11	23	16	21	19	26	27	12	13
10	14	16	11	6	9	4	15	19	5	0	3	25	24	1	7	2	21	23	8	22	17	20	18	13	12	27	26
11	18	12	10	7	8	15	4	5	19	3	0	2	22	23	6	25	26	1	9	24	27	13	14	20	16	17	21
12	27	11	16	19	13	21	26	6	7	1	23	24	25	0	5	22	4	3	20	2	18	8	17	9	10	14	15
13	7	8	17	18	12	20	1	2	27	26	22	5	0	25	24	23	3	4	21	6	19	11	16	15	14	10	9
14	10	20	18	17	15	8	9	22	24	25	2	0	5	26	27	3	23	21	4	19	6	16	11	12	13	7	1
15	24	21	19	16	14	11	25	23	10	9	6	27	26	5	0	4	22	20	3	18	2	17	8	1	7	13	12
16	20	10	12	15	17	7	6	26	21	2	25	3	23	22	4	0	5	24	27	1	9	14	13	18	11	8	19
17	6	9	13	14	16	1	20	27	2	21	24	23	3	4	22	5	0	25	26	7	10	15	12	11	18	19	8
18	11	27	14	13	19	24	22	9	8	23	1	21	4	3	20	26	25	0	5	15	12	7	10	6	17	16	2
19	22	26	15	12	18	25	11	10	23	8	7	4	21	20	3	27	24	5	0	14	13	1	9	17	6	2	16
20	16	14	27	24	21	13	23	25	17	22	26	19	6	2	18	1	9	15	12	0	5	10	7	4	8	11	3
21	23	15	26	25	20	12	16	24	22	17	27	6	19	18	2	7	10	14	13	5	0	9	1	8	4	3	11
22	19	25	24	27	23	26	21	14	18	20	13	16	11	8	17	12	15	9	1	10	7	0	5	3	2	6	4
23	21	24	25	26	22	27	19	15	20	18	12	11	16	17	8	13	14	10	7	9	1	5	0	2	3	4	6
24	15	23	22	20	25	18	14	21	26	27	17	9	10	12	13	8	19	16	2	11	3	6	4	7	1	0	5
25	26	22	23	21	24	19	27	20	15	14	16	13	12	10	9	11	18	17	6	8	4	2	3	5	0	1	7
26	25	19	21	23	27	22	24	16	12	13	20	14	15	7	1	18	11	6	17	4	8	3	2	0	5	9	10
27	12	18	20	22	26	23	13	17	25	24	21	1	7	15	14	19	8	2	16	3	11	4	6	10	9	5	0

Centre: 0

Centrum: 0

Nucleus: 0

Left Nucleus: 0

Middle Nucleus: 0

Right Nucleus: 0

1 Element of order 1: 0

21 Elements of order 2: 1 2 3 4 5 6 8 10 11 13 15 16 17 18 19 20 21 22 23 25 27

6 Elements of order 7: 7 9 12 14 24 26

Commutator Subloop: 0 7 9 12 14 24 26

Associator Subloop: 0 7 9 12 14 24 26

1 Conjugacy Class of size 1:

3 Conjugacy Classes of size 2:

7 9
12 14
24 26

3 Conjugacy Classes of size 7:

1 5 10 13 15 25 27
2 4 11 17 19 20 23
3 6 8 16 18 21 22

Automorphic Inverse Property: FAILS. (1^-1)(6^-1) neq (1*6)^-1

A_l Property: FAILS. The left inner mapping L_1,5 = (2,11,23,17,4,19,20)(3,16,22,8,6,21,18) is not an automorphism. L_1,5(1*2) neq L_1,5(1)*L_1,5(2)

A_r Property: FAILS. The right inner mapping R_1,5 = (2,11,23,17,4,19,20)(3,16,22,8,6,21,18) is not an automorphism. R_1,5(1*2) neq R_1,5(1)*R_1,5(2)

Right (Left, Full) Mult Group Orders: 19208 (19208, 76832)

/ revised November, 2001

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27
1	0	3	2	8	7	17	5	4	13	12	18	10	9	27	26	20	6	11	22	16	23	19	21	25	24	15	14
2	3	0	1	9	6	5	17	13	4	16	14	20	8	11	21	10	7	27	24	12	15	25	26	19	22	23	18
3	2	1	0	5	4	9	8	7	6	11	10	18	17	16	19	14	13	12	15	27	24	26	25	21	23	22	20
4	8	7	5	0	3	10	2	1	11	6	9	17	18	19	16	15	12	13	14	26	25	27	24	23	21	20	22
5	9	6	4	3	0	2	10	11	1	7	8	15	14	13	12	17	16	19	18	21	20	23	22	27	26	25	24
6	17	5	7	10	2	0	3	12	16	4	15	8	20	21	11	9	1	26	25	13	14	24	27	22	19	18	23
7	13	4	6	11	1	16	12	3	0	5	19	26	27	9	10	21	2	8	23	17	22	18	20	14	15	24	25
8	4	13	9	1	11	14	18	0	3	19	5	22	2	6	23	24	27	7	10	25	26	12	15	16	20	21	17
9	5	17	8	2	10	3	0	18	14	15	4	7	1	24	25	6	20	22	11	23	16	21	19	26	27	12	13
10	14	16	11	6	9	4	15	19	5	0	3	25	24	1	7	2	21	23	8	22	17	20	18	13	12	27	26
11	18	12	10	7	8	15	4	5	19	3	0	2	22	23	6	25	26	1	9	24	27	13	14	20	16	17	21
12	27	11	16	19	13	21	26	6	7	1	23	24	25	0	5	22	4	3	20	2	18	8	17	9	10	14	15
13	7	8	17	18	12	20	1	2	27	26	22	5	0	25	24	23	3	4	21	6	19	11	16	15	14	10	9
14	10	20	18	17	15	8	9	22	24	25	2	0	5	26	27	3	23	21	4	19	6	16	11	12	13	7	1
15	24	21	19	16	14	11	25	23	10	9	6	27	26	5	0	4	22	20	3	18	2	17	8	1	7	13	12
16	20	10	12	15	17	7	6	26	21	2	25	3	23	22	4	0	5	24	27	1	9	14	13	18	11	8	19
17	6	9	13	14	16	1	20	27	2	21	24	23	3	4	22	5	0	25	26	7	10	15	12	11	18	19	8
18	11	27	14	13	19	24	22	9	8	23	1	21	4	3	20	26	25	0	5	15	12	7	10	6	17	16	2
19	22	26	15	12	18	25	11	10	23	8	7	4	21	20	3	27	24	5	0	14	13	1	9	17	6	2	16
20	16	14	27	24	21	13	23	25	17	22	26	19	6	2	18	1	9	15	12	0	5	10	7	4	8	11	3
21	23	15	26	25	20	12	16	24	22	17	27	6	19	18	2	7	10	14	13	5	0	9	1	8	4	3	11
22	19	25	24	27	23	26	21	14	18	20	13	16	11	8	17	12	15	9	1	10	7	0	5	3	2	6	4
23	21	24	25	26	22	27	19	15	20	18	12	11	16	17	8	13	14	10	7	9	1	5	0	2	3	4	6
24	15	23	22	20	25	18	14	21	26	27	17	9	10	12	13	8	19	16	2	11	3	6	4	7	1	0	5
25	26	22	23	21	24	19	27	20	15	14	16	13	12	10	9	11	18	17	6	8	4	2	3	5	0	1	7
26	25	19	21	23	27	22	24	16	12	13	20	14	15	7	1	18	11	6	17	4	8	3	2	0	5	9	10
27	12	18	20	22	26	23	13	17	25	24	21	1	7	15	14	19	8	2	16	3	11	4	6	10	9	5	0

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27
1	0	3	2	8	7	17	5	4	13	12	18	10	9	27	26	20	6	11	22	16	23	19	21	25	24	15	14
2	3	0	1	9	6	5	17	13	4	16	14	20	8	11	21	10	7	27	24	12	15	25	26	19	22	23	18
3	2	1	0	5	4	9	8	7	6	11	10	18	17	16	19	14	13	12	15	27	24	26	25	21	23	22	20
4	8	7	5	0	3	10	2	1	11	6	9	17	18	19	16	15	12	13	14	26	25	27	24	23	21	20	22
5	9	6	4	3	0	2	10	11	1	7	8	15	14	13	12	17	16	19	18	21	20	23	22	27	26	25	24
6	17	5	7	10	2	0	3	12	16	4	15	8	20	21	11	9	1	26	25	13	14	24	27	22	19	18	23
7	13	4	6	11	1	16	12	3	0	5	19	26	27	9	10	21	2	8	23	17	22	18	20	14	15	24	25
8	4	13	9	1	11	14	18	0	3	19	5	22	2	6	23	24	27	7	10	25	26	12	15	16	20	21	17
9	5	17	8	2	10	3	0	18	14	15	4	7	1	24	25	6	20	22	11	23	16	21	19	26	27	12	13
10	14	16	11	6	9	4	15	19	5	0	3	25	24	1	7	2	21	23	8	22	17	20	18	13	12	27	26
11	18	12	10	7	8	15	4	5	19	3	0	2	22	23	6	25	26	1	9	24	27	13	14	20	16	17	21
12	27	11	16	19	13	21	26	6	7	1	23	24	25	0	5	22	4	3	20	2	18	8	17	9	10	14	15
13	7	8	17	18	12	20	1	2	27	26	22	5	0	25	24	23	3	4	21	6	19	11	16	15	14	10	9
14	10	20	18	17	15	8	9	22	24	25	2	0	5	26	27	3	23	21	4	19	6	16	11	12	13	7	1
15	24	21	19	16	14	11	25	23	10	9	6	27	26	5	0	4	22	20	3	18	2	17	8	1	7	13	12
16	20	10	12	15	17	7	6	26	21	2	25	3	23	22	4	0	5	24	27	1	9	14	13	18	11	8	19
17	6	9	13	14	16	1	20	27	2	21	24	23	3	4	22	5	0	25	26	7	10	15	12	11	18	19	8
18	11	27	14	13	19	24	22	9	8	23	1	21	4	3	20	26	25	0	5	15	12	7	10	6	17	16	2
19	22	26	15	12	18	25	11	10	23	8	7	4	21	20	3	27	24	5	0	14	13	1	9	17	6	2	16
20	16	14	27	24	21	13	23	25	17	22	26	19	6	2	18	1	9	15	12	0	5	10	7	4	8	11	3
21	23	15	26	25	20	12	16	24	22	17	27	6	19	18	2	7	10	14	13	5	0	9	1	8	4	3	11
22	19	25	24	27	23	26	21	14	18	20	13	16	11	8	17	12	15	9	1	10	7	0	5	3	2	6	4
23	21	24	25	26	22	27	19	15	20	18	12	11	16	17	8	13	14	10	7	9	1	5	0	2	3	4	6
24	15	23	22	20	25	18	14	21	26	27	17	9	10	12	13	8	19	16	2	11	3	6	4	7	1	0	5
25	26	22	23	21	24	19	27	20	15	14	16	13	12	10	9	11	18	17	6	8	4	2	3	5	0	1	7
26	25	19	21	23	27	22	24	16	12	13	20	14	15	7	1	18	11	6	17	4	8	3	2	0	5	9	10
27	12	18	20	22	26	23	13	17	25	24	21	1	7	15	14	19	8	2	16	3	11	4	6	10	9	5	0

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27
1	0	3	2	8	7	17	5	4	13	12	18	10	9	27	26	20	6	11	22	16	23	19	21	25	24	15	14
2	3	0	1	9	6	5	17	13	4	16	14	20	8	11	21	10	7	27	24	12	15	25	26	19	22	23	18
3	2	1	0	5	4	9	8	7	6	11	10	18	17	16	19	14	13	12	15	27	24	26	25	21	23	22	20
4	8	7	5	0	3	10	2	1	11	6	9	17	18	19	16	15	12	13	14	26	25	27	24	23	21	20	22
5	9	6	4	3	0	2	10	11	1	7	8	15	14	13	12	17	16	19	18	21	20	23	22	27	26	25	24
6	17	5	7	10	2	0	3	12	16	4	15	8	20	21	11	9	1	26	25	13	14	24	27	22	19	18	23
7	13	4	6	11	1	16	12	3	0	5	19	26	27	9	10	21	2	8	23	17	22	18	20	14	15	24	25
8	4	13	9	1	11	14	18	0	3	19	5	22	2	6	23	24	27	7	10	25	26	12	15	16	20	21	17
9	5	17	8	2	10	3	0	18	14	15	4	7	1	24	25	6	20	22	11	23	16	21	19	26	27	12	13
10	14	16	11	6	9	4	15	19	5	0	3	25	24	1	7	2	21	23	8	22	17	20	18	13	12	27	26
11	18	12	10	7	8	15	4	5	19	3	0	2	22	23	6	25	26	1	9	24	27	13	14	20	16	17	21
12	27	11	16	19	13	21	26	6	7	1	23	24	25	0	5	22	4	3	20	2	18	8	17	9	10	14	15
13	7	8	17	18	12	20	1	2	27	26	22	5	0	25	24	23	3	4	21	6	19	11	16	15	14	10	9
14	10	20	18	17	15	8	9	22	24	25	2	0	5	26	27	3	23	21	4	19	6	16	11	12	13	7	1
15	24	21	19	16	14	11	25	23	10	9	6	27	26	5	0	4	22	20	3	18	2	17	8	1	7	13	12
16	20	10	12	15	17	7	6	26	21	2	25	3	23	22	4	0	5	24	27	1	9	14	13	18	11	8	19
17	6	9	13	14	16	1	20	27	2	21	24	23	3	4	22	5	0	25	26	7	10	15	12	11	18	19	8
18	11	27	14	13	19	24	22	9	8	23	1	21	4	3	20	26	25	0	5	15	12	7	10	6	17	16	2
19	22	26	15	12	18	25	11	10	23	8	7	4	21	20	3	27	24	5	0	14	13	1	9	17	6	2	16
20	16	14	27	24	21	13	23	25	17	22	26	19	6	2	18	1	9	15	12	0	5	10	7	4	8	11	3
21	23	15	26	25	20	12	16	24	22	17	27	6	19	18	2	7	10	14	13	5	0	9	1	8	4	3	11
22	19	25	24	27	23	26	21	14	18	20	13	16	11	8	17	12	15	9	1	10	7	0	5	3	2	6	4
23	21	24	25	26	22	27	19	15	20	18	12	11	16	17	8	13	14	10	7	9	1	5	0	2	3	4	6
24	15	23	22	20	25	18	14	21	26	27	17	9	10	12	13	8	19	16	2	11	3	6	4	7	1	0	5
25	26	22	23	21	24	19	27	20	15	14	16	13	12	10	9	11	18	17	6	8	4	2	3	5	0	1	7
26	25	19	21	23	27	22	24	16	12	13	20	14	15	7	1	18	11	6	17	4	8	3	2	0	5	9	10
27	12	18	20	22	26	23	13	17	25	24	21	1	7	15	14	19	8	2	16	3	11	4	6	10	9	5	0