powers of Antisymmetric Signed binary Matrices (n=5 included)

[Meeussen Wouter (bkarnd)]

just in case, 
count of antisymmetric (Tij = -Tji ;i<j) signed binary n*n matrices, 
whose powers T^k (k<=25) are antisymmetric binary signed matrices too.
a(n)= 2,14,92,796,7672,.. offset=1;

T^k =Identity for k in {1, 2, 3, 4, 6 or 12} seems to be independent of n.
Where does the T^(k=12) as maximum cycle size come from?
Why does it behave like a picture under 12 symmetry operations come from?
C1, C2, C3, C4=I, S_vert, S_diag, group order=12? or abouts? probably.

but if so rudimentary, then why not yet in EIS?
-------------------------------

details & examples:

n=1; 
a times cycle length (b) :: 1*(1) + 1*(2)
a times (b) non-zero elements :: 2*(1)
example :: {{-1}} and {{1}}

n=2; 
a times cycle length (b) :: 1*(1)+3*(2)+4*(3)+2*(4)+4*(6)
a times (b) non-zero elements :: 6*(2)+8*(3)
example :: {{1,1},{-1,0}}

n=3; 
a times cycle length (b) :: 1*(1)+7*(2)+12*(3)+36*(4)+36*(6)
a times (b) non-zero elements :: 2*(3)+48*(4)+24*(7)
example :: {{1,-1,0},{1,0,0},{0,0,-1}}

n=4; 
a times cycle length (b) :: 1*(1)+15*(2)+72*(3)+300*(4)+312*(6)+96*(12)
a times (b) non-zero elements :: 76*(4)+288*(5)+192*(6)+192*(8)+48*(12)
example :: {{1,-1,0,0},{1,0,0,0},{0,0,1,0},{0,0,0,1}}

n=5; 
a times cycle length (b) :: 1*(1)+31*(2)+280*(3)+2200*(4)+2280*(6)+2880*(12)
a times (b) non-zero elements ::
312*(5)+1600*(6)+1920*(7)+1440*(9)+1920*(10)+480*(13)
example :: {{1,-1,0,0,0},{1,0,0,0,0},{0,0,1,-1,0},{0,0,1,0,0},{0,0,0,0,-1}}

Wouter.

PS  to Neil: I'll send it properly asap.


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