Right Bol Loop 28.21.1.1 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	5	4	3	2	19	18	21	24	27	26	23	25	22	20	17	16	7	6	15	8	14	12	9	13	11	10
2	3	0	19	6	1	4	5	7	11	8	9	10	15	12	13	14	22	21	24	16	18	17	27	26	20	25	23
3	2	7	0	1	18	24	21	27	19	23	25	22	26	17	16	15	20	5	4	13	10	12	14	6	11	9	8
4	5	18	1	0	7	9	8	10	6	12	13	14	11	16	17	20	15	2	3	25	27	23	22	19	26	24	21
5	4	1	6	19	0	3	2	18	26	21	24	27	20	23	25	22	14	8	9	17	7	16	10	11	15	13	12
6	7	21	5	2	8	11	10	12	4	14	15	16	9	20	22	25	13	0	1	26	23	27	17	3	24	19	18
7	6	3	9	24	4	1	0	2	25	18	19	21	16	27	26	23	12	10	11	22	5	20	8	13	17	15	14
8	9	19	11	26	6	5	4	0	20	2	3	18	14	21	24	27	10	12	13	23	1	25	7	15	22	17	16
9	8	27	7	18	10	13	12	14	0	16	17	20	6	25	23	26	11	4	5	24	22	21	15	1	19	3	2
10	11	24	13	25	9	7	6	4	16	0	1	2	12	18	19	21	8	14	15	27	3	26	5	17	23	22	20
11	10	23	8	21	12	15	14	16	2	20	22	25	4	26	27	24	9	6	7	19	17	18	13	5	3	1	0
12	13	26	15	20	11	8	9	6	14	4	5	0	10	2	3	18	7	16	17	21	19	24	1	22	27	23	25
13	12	22	10	27	14	17	16	20	18	25	23	26	0	24	21	19	6	9	8	3	15	2	11	7	1	5	4
14	15	25	17	16	13	10	11	9	12	6	7	4	8	0	1	2	5	20	22	18	24	19	3	23	21	27	26
15	14	17	12	23	16	22	20	25	21	26	27	24	2	19	18	3	4	11	10	1	13	0	9	8	5	7	6
16	17	20	22	14	15	12	13	11	10	9	8	6	7	4	5	0	1	25	23	2	26	3	19	27	18	21	24
17	16	15	14	22	20	23	25	26	27	24	21	19	18	3	2	1	0	13	12	5	11	4	6	10	7	8	9
18	19	4	24	9	3	0	1	5	13	7	6	8	17	10	11	12	23	27	26	14	2	15	21	25	16	20	22
19	18	8	2	5	21	26	27	23	3	22	20	17	24	15	14	13	25	1	0	11	12	10	16	4	9	6	7
20	22	16	23	12	17	14	15	13	8	11	10	9	5	6	7	4	3	26	27	0	25	1	24	21	2	18	19
21	24	6	26	11	19	2	3	1	15	5	4	7	22	8	9	10	27	23	25	12	0	13	18	20	14	16	17
22	20	13	16	17	25	27	26	24	23	19	18	3	21	1	0	5	2	15	14	7	9	6	4	12	8	10	11
23	25	11	20	15	26	21	24	19	22	3	2	1	27	5	4	7	18	17	16	8	6	9	0	14	10	12	13
24	21	10	18	7	27	25	23	22	1	17	16	15	19	13	12	11	26	3	2	9	14	8	20	0	6	4	5
25	23	14	27	10	22	16	17	15	7	13	12	11	1	9	8	6	19	24	21	4	20	5	26	18	0	2	3
26	27	12	21	8	23	20	22	17	5	15	14	13	3	11	10	9	24	19	18	6	16	7	25	2	4	0	1
27	26	9	25	13	24	18	19	3	17	1	0	5	23	7	6	8	21	22	20	10	4	11	2	16	12	14	15

Centre: 0

Centrum: 0

Nucleus: 0

Left Nucleus: 0 1 16 17

Middle Nucleus: 0 6 11 15 18 22 27

Right Nucleus: 0 6 11 15 18 22 27

1 Element of order 1: 0

21 Elements of order 2: 1 2 3 4 5 7 8 9 10 12 13 14 16 17 19 20 21 23 24 25 26

6 Elements of order 7: 6 11 15 18 22 27

Commutator Subloop: 0 6 11 15 18 22 27

Associator Subloop: 0 6 11 15 18 22 27

1 Conjugacy Class of size 1:

3 Conjugacy Classes of size 2:

6 18
11 27
15 22

3 Conjugacy Classes of size 7:

1 7 10 14 19 20 26
2 4 9 13 17 21 23
3 5 8 12 16 24 25

Automorphic Inverse Property: FAILS. (1^-1)(7^-1) neq (1*7)^-1

A_l Property: FAILS. The left inner mapping L_1,2 = (2,4,9,13,17,23,21)(3,24,25,16,12,8,5) is not an automorphism. L_1,2(2*1) neq L_1,2(2)*L_1,2(1)

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

Right (Left, Full) Mult Group Orders: 56 (9604, 19208)

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

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

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