118 
ON THE SUBSTITUTION GROUPS 
[918 
1 
3 
1 
6 
to I 
1 
2 
1 
5 
1 
2 
2 
6 
1 
2 
3 
4 
2 
6 
1 
3 
1 
7 
8 
4 
1 
8 
1 
10 
4 
4 
2 
10 
4 
6 
1 
12 
1 
12 
3 
6 
9 
12 
25 
8 
1 
24 
1 
20 
60 
6 
8 
9 
1 
14 
20 
6 
1 
12 
15 
1 
1 
2 
i 
1 
120 
1 
12 
16 
1 
21 
24 
22 
3 
16 
18 
1 
7 
3 
18 
1 
36 
4 
24 
4 
24 
1 
40 
3 
30 
2 
36 
1 
42 
12 
32 
2 
48 
1 
48 
5 
36 
1 
60 
3 
72 
14 
48 
1 
72 
2 
120 
3 
60 
1 
120 
1 
144 
3 
64 
1 
360 
1 
240 
3 
72 
1 
720 
1 
2520 
5040 
10 
96 
120 
144 
34 
1 
3 
2 
38 
1 
168 
1 
180 
3 
192 
1 
240 
3 
288 
1 
336 
3 
360 
1 
384 
2 
576 
3 
720 
1 
1152 
1 
1440 
1 
20160 
1 
40320 
155 
Here the top line shows the numbers of letters; each second column shows the 
order of the groups, and each first column the number of groups of the several 
orders: the sums at the foot of the first columns show therefore the whole number 
of groups, viz. 
No. of letters = 2, 3, 4, 5, 6, 7, 8 
No. of groups = 1, 2, 7, 8, 34, 38, 155’ 
In the enumeration of the groups, I use some notations which must be explained. 
For greater simplicity I omit parentheses, and write ab, abc, ab. cd, abc. def, &c., to 
denote substitutions, viz. ab is the interchange of a and b; abc the cyclical change 
a into b, b into c, c into a; ab .cd the combined interchange of a and b and of 
c and d; abc. def the combined cyclical changes a into b, b into c, c into a, and 
d into e, e into f f into d; and so in other cases. 
Again, (abc) all, means the complete group of all the substitutions 
(1, abc, acb, be, ca, ab)