M22 and Steiner S(3,6,22)

I thought it would be fun to map hexads in a 22-tET system. I found a
listing of a good Steiner S(3,6,22) system on the Internet, and have
been crunching the numbers. First, the Polya formula for hexads in 22
generate 3,399 hexads (22C6 + 11C3 / 22). Now reducing for mirrors gives
1782 Tn/TnI types. Reducingfor Z-relations (there are 5 triples and 55
pairs) gives -10 and -55 -> 1,717. (Assuming the triples are all
symmetrical sets, I think they are).
Now an S(3,6,22) Steiner system is a BIBD (Block Design) such that all
triples (triads), are contained in exactly one hexad. There are 77
hexads here,and of course 20 possible triad subsets (6C3) so 77 * 20 =
1,540 which is the right count. Now I found that there are 64 different
Tn/TnI types used,the reduction is purely for Tn and TnI, so no
reduction a la Z-relation here. However, some of the Steiner sets here
may be a Z-relation, there are 60 types that are Z-related which out of
1,782 isn't a big percentage, so there are probably just 2 or 3. So this
will not be a focus of this post. Here are the 77 Steiner Sets:

1

2

3

4

5

6

1

2

7

8

9

10

1

2

11

12

13

14

1

2

15

16

17

18

1

2

19

20

21

22

1

3

8

12

16

20

1

3

9

13

17

21

1

3

10

14

18

22

1

3

7

11

15

19

1

4

7

12

17

22

1

4

8

11

18

21

1

4

9

14

15

20

1

4

10

13

16

19

1

5

7

13

18

20

1

5

8

14

17

19

1

5

9

11

16

22

1

5

10

12

15

21

1

6

7

14

16

21

1

6

8

13

15

22

1

6

9

12

18

19

1

6

10

11

17

20

2

3

7

12

18

21

2

3

8

11

17

22

2

3

9

14

16

19

2

3

10

13

15

20

2

4

7

11

16

20

2

4

8

12

15

19

2

4

9

13

18

22

2

4

10

14

17

21

2

5

7

14

15

22

2

5

8

13

16

21

2

5

9

12

17

20

2

5

10

11

18

19

2

6

7

13

17

19

2

6

8

14

18

20

2

6

9

11

15

21

2

6

10

12

16

22

3

4

7

8

13

14

3

4

9

10

11

12

3

4

15

16

21

22

3

4

17

18

19

20

3

5

7

10

16

17

3

5

8

9

15

18

3

5

11

14

20

21

3

5

12

13

19

22

3

6

7

9

20

22

3

6

8

10

19

21

3

6

11

13

16

18

3

6

12

14

15

17

4

5

7

9

19

21

4

5

8

10

20

22

4

5

11

13

15

17

4

5

12

14

16

18

4

6

7

10

15

18

4

6

8

9

16

17

4

6

11

14

19

22

4

6

12

13

20

21

5

6

7

8

11

12

5

6

9

10

13

14

5

6

15

16

19

20

5

6

17

18

21

22

7

8

15

17

20

21

7

8

16

18

19

22

7

9

11

14

17

18

7

9

12

13

15

16

7

10

11

13

21

22

7

10

12

14

19

20

8

9

11

13

19

20

8

9

12

14

21

22

8

10

11

14

15

16

8

10

12

13

17

18

9

10

15

17

19

22

9

10

16

18

20

21

11

12

15

18

20

22

11

12

16

17

19

21

13

14

15

18

19

22

13

14

16

17

20

21

Now, I played with the Interval Vectors, and also looked at even and odd
slots (1,3,5,7,9 and 2,4,6,8,10) so see about any Affine actions here,
but that doesn't seem to be very key here. I know that the two poles in
22,6 are not used (Vectors <1,2,1,2,1,2,1,2,1,2,0> and
<3,0,3,0,3,0,3,0,3,0,0>) but I thought there might be some scrambling in
the intevecs of Steiners here. The behaviour here just isn't that
interesting. Of course, this isn't the only Steiner system possible for
M22, (The automorphism group of a S(3,6,22) system). In fact, you can
swap any elements at all and get another one. Fun fact is that the
intvecs of a Steiner system always add up to a flat line interval vector
sum (tallying each slot) and of course the whole set C(22,6) does also.
(There are 969 * 77 total hexads, so the Methusaleh number comes into
play here.) in M12, there are 7 times (132 * 7 = 924). So anyway here
are the vectors, reduced down, for a few that have transpositions and
TnI sets used, and then showing taking the odds and taking the evens. I
have sorted from based on left to right. There are just 64 types used. I
have listed the sets, the main intevec, odd slots, and even slots.

2

5

9

12

17

20

0

0

3

2

1

0

4

2

0

2

1

0

3

1

4

0

0

2

0

2

2

2

5

8

13

16

21

0

0

4

0

2

2

0

4

1

0

2

0

4

2

0

1

0

0

2

4

0

1

4

10

13

16

19

0

0

4

1

0

3

2

0

3

2

0

0

4

0

2

3

0

1

3

0

2

1

3

10

14

18

22

0

1

0

5

0

2

0

4

0

3

0

0

0

0

0

0

1

5

2

4

3

2

4

8

12

15

19

0

1

1

3

1

1

3

1

1

1

2

0

1

1

3

1

1

3

1

1

1

1

3

7

11

15

19

0

1

1

3

2

0

2

2

2

1

1

0

1

2

2

2

1

3

0

2

1

2

4

7

11

16

20

0

1

1

3

2

1

1

1

4

1

0

0

1

2

1

4

1

3

1

1

1

4

6

11

14

19

22

0

1

2

1

2

1

2

3

1

1

1

0

2

2

2

1

1

1

1

3

1

2

4

10

14

17

21

0

1

2

2

1

1

3

1

1

2

1

0

2

1

3

1

1

2

1

1

2

1

5

8

14

17

19

0

1

2

2

1

2

1

1

3

1

1

0

2

1

1

3

1

2

2

1

1

2

6

9

11

15

21

0

1

2

2

1

2

2

0

3

2

0

0

2

1

2

3

1

2

2

0

2

1

4

8

11

18

21

0

1

3

1

2

0

3

1

1

3

0

0

3

2

3

1

1

1

0

1

3

2

6

8

14

18

20

TI

0

2

0

3

0

4

0

2

0

4

0

0

0

0

0

0

2

3

4

2

4

2

6

10

12

16

22

TI

0

2

0

3

0

4

0

2

0

4

0

0

0

0

0

0

2

3

4

2

4

2

4

9

13

18

22

0

2

0

3

2

1

1

1

4

0

1

0

0

2

1

4

2

3

1

1

0

1

3

8

12

16

20

0

2

0

4

0

2

0

4

0

3

0

0

0

0

0

0

2

4

2

4

3

1

5

10

12

15

21

0

2

1

1

2

2

1

1

2

1

2

0

1

2

1

2

2

1

2

1

1

1

5

7

13

18

20

0

2

1

1

2

2

2

1

2

1

1

0

1

2

2

2

2

1

2

1

1

3

6

11

13

16

18

0

2

2

0

3

0

3

1

1

3

0

0

2

3

3

1

2

0

0

1

3

3

6

8

10

19

21

0

3

1

2

1

1

2

0

3

0

2

0

1

1

2

3

3

2

1

0

0

1

4

9

14

15

20

1

0

2

0

3

3

0

2

1

1

2

1

2

3

0

1

0

0

3

2

1

1

6

9

12

18

19

TI

1

0

2

1

2

2

1

1

2

2

1

1

2

2

1

2

0

1

2

1

2

1

6

10

11

17

20

TI

1

0

2

1

2

2

1

1

2

2

1

1

2

2

1

2

0

1

2

1

2

1

4

7

12

17

22

1

0

2

1

3

2

1

1

1

2

1

1

2

3

1

1

0

1

2

1

2

2

3

7

12

18

21

1

0

2

2

2

2

1

1

2

1

1

1

2

2

1

2

0

2

2

1

1

2

6

7

13

17

19

1

1

0

2

2

2

2

0

1

2

2

1

0

2

2

1

1

2

2

0

2

1

5

9

11

16

22

1

1

0

2

2

2

2

1

1

1

2

1

0

2

2

1

1

2

2

1

1

2

3

9

14

16

19

1

1

1

0

3

2

2

1

1

2

1

1

1

3

2

1

1

0

2

1

2

3

5

11

14

20

21

1

1

1

1

1

3

2

1

2

1

1

1

1

1

2

2

1

1

3

1

1

2

3

10

13

15

20

1

1

1

1

3

0

2

1

1

3

1

1

1

3

2

1

1

1

0

1

3

1

3

9

13

17

21

1

1

1

3

1

0

2

2

2

1

1

1

1

1

2

2

1

3

0

2

1

3

5

12

13

19

22

1

1

2

0

1

2

2

2

2

2

0

1

2

1

2

2

1

0

2

2

2

2

3

8

11

17

22

1

1

2

0

2

2

1

3

2

0

1

1

2

2

1

2

1

0

2

3

0

3

5

8

9

15

18

1

1

2

1

1

2

2

0

2

3

0

1

2

1

2

2

1

1

2

0

3

4

6

7

10

15

18

1

1

3

1

1

1

0

3

1

1

2

1

3

1

0

1

1

1

1

3

1

1

6

7

14

16

21

TI

1

2

0

0

2

1

4

2

2

1

0

1

0

2

4

2

2

0

1

2

1

1

6

8

13

15

22

TI

1

2

0

0

2

1

4

2

2

1

0

1

0

2

4

2

2

0

1

2

1

2

5

7

14

15

22

1

2

1

0

2

0

3

2

2

2

0

1

1

2

3

2

2

0

0

2

2

3

5

7

10

16

17

1

2

1

1

1

1

2

1

2

2

1

1

1

1

2

2

2

1

1

1

2

7

10

12

14

19

20

TI

1

2

1

1

2

1

2

1

2

2

0

1

1

2

2

2

2

1

1

1

2

9

10

15

17

19

22

TI

1

2

1

1

2

1

2

1

2

2

0

1

1

2

2

2

2

1

1

1

2

4

5

8

10

20

22

1

2

1

2

2

2

1

1

0

3

0

1

1

2

1

0

2

2

2

1

3

7

9

11

14

17

18

TI

1

2

2

2

1

1

2

1

1

1

1

1

2

1

2

1

2

2

1

1

1

11

12

15

18

20

22

TI

1

2

2

2

1

1

2

1

1

1

1

1

2

1

2

1

2

2

1

1

1

3

6

12

14

15

17

1

2

3

0

1

1

0

2

2

1

2

1

3

1

0

2

2

0

1

2

1

3

6

7

9

20

22

1

2

3

1

1

2

1

1

2

0

1

1

3

1

1

2

2

1

2

1

0

4

5

12

14

16

18

1

3

0

2

0

1

1

2

2

2

1

1

0

0

1

2

3

2

1

2

2

4

5

11

13

15

17

1

3

0

2

0

2

1

1

2

2

1

1

0

0

1

2

3

2

2

1

2

4

5

7

9

19

21

1

3

1

1

2

1

1

2

0

3

0

1

1

2

1

0

3

1

1

2

3

2

5

10

11

18

19

2

0

1

0

2

2

1

4

3

0

0

2

1

2

1

3

0

0

2

4

0

4

6

12

13

20

21

2

1

0

0

1

2

3

4

2

0

0

2

0

1

3

2

1

0

2

4

0

7

8

15

17

20

21

TI

2

1

1

1

1

1

1

2

3

2

0

2

1

1

1

3

1

1

1

2

2

8

9

12

14

21

22

TI

2

1

1

1

1

1

1

2

3

2

0

2

1

1

1

3

1

1

1

2

2

7

8

16

18

19

22

TI

2

1

2

1

0

1

1

2

1

2

2

2

2

0

1

1

1

1

1

2

2

7

10

11

13

21

22

TI

2

1

2

1

0

1

1

2

1

2

2

2

2

0

1

1

1

1

1

2

2

4

6

8

9

16

17

2

2

1

1

1

0

1

2

2

2

1

2

1

1

1

2

2

1

0

2

2

8

9

11

13

19

20

TI

2

2

1

1

1

1

1

1

1

2

2

2

1

1

1

1

2

1

1

1

2

9

10

16

18

20

21

TI

2

2

1

1

1

1

1

1

1

2

2

2

1

1

1

1

2

1

1

1

2

8

10

12

13

17

18

TI

2

2

1

2

3

1

1

1

1

1

0

2

1

3

1

1

2

2

1

1

1

11

12

16

17

19

21

TI

2

2

1

2

3

1

1

1

1

1

0

2

1

3

1

1

2

2

1

1

1

7

9

12

13

15

16

2

2

3

2

1

2

1

1

1

0

0

2

3

1

1

1

2

2

2

1

0

5

6

15

16

19

20

3

0

1

2

1

0

1

2

2

2

1

3

1

1

1

2

0

2

0

2

2

3

4

7

8

13

14

I

3

0

1

2

2

2

1

0

1

2

1

3

1

2

1

1

0

2

2

0

2

3

4

15

16

21

22

I

3

0

1

2

2

2

1

0

1

2

1

3

1

2

1

1

0

2

2

0

2

5

6

17

18

21

22

3

0

1

2

2

2

1

0

1

2

1

3

1

2

1

1

0

2

2

0

2

5

6

9

10

13

14

3

0

2

4

2

0

1

2

1

0

0

3

2

2

1

1

0

4

0

2

0

13

14

15

18

19

22

3

1

2

3

2

1

1

1

1

0

0

3

2

2

1

1

1

3

1

1

0

13

14

16

17

20

21

3

1

3

3

1

1

2

1

0

0

0

3

3

1

2

0

1

3

1

1

0

8

10

11

14

15

16

3

2

2

2

2

2

1

1

0

0

0

3

2

2

1

0

2

2

2

1

0

1

2

11

12

13

14

4

2

1

0

0

0

0

0

2

4

2

4

1

0

0

2

2

0

0

0

4

1

2

7

8

9

10

I

4

2

1

0

1

2

2

2

1

0

0

4

1

1

2

1

2

0

2

2

0

1

2

15

16

17

18

I

4

2

1

0

1

2

2

2

1

0

0

4

1

1

2

1

2

0

2

2

0

3

4

9

10

11

12

T

4

2

1

0

1

2

2

2

1

0

0

4

1

1

2

1

2

0

2

2

0

3

4

17

18

19

20

TI

4

2

1

0

1

2

2

2

1

0

0

4

1

1

2

1

2

0

2

2

0

5

6

7

8

11

12

4

2

2

2

2

2

1

0

0

0

0

4

2

2

1

0

2

2

2

0

0

1

2

3

4

5

6

T

5

4

3

2

1

0

0

0

0

0

0

5

3

1

0

0

4

2

0

0

0

1

2

19

20

21

22

T

5

4

3

2

1

0

0

0

0

0

0

5

3

1

0

0

4

2

0

0

0

Oh! I should add them up, of course, then I need the duplicates. Here is
the final sum. Oops, a mistake somewhere, but it's close to FLID.

111

108

110

112

110

108

111

110

110

110

55