G = Syl2(M12) is Sylow 2-subgroup of Mathieu Group M_12
The Hall-Senior number of this group is 261.
G has 3 minimal generators, rank 3 and exponent 8. The centre has rank 1.
There are 5 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3, 3, 3, 3.
This cohomology ring calculation is complete.
Ring structure | Completion information | Koszul information | Restriction information | Poincaré series
The cohomology ring has 7 generators:
There are 9 minimal relations:
This minimal generating set constitutes a Gröbner basis for the relations ideal.
This cohomology ring was obtained from a calculation out to degree 12. The cohomology ring approximation is stable from degree 6 onwards, and Benson's tests detect stability from degree 6 onwards.
This cohomology ring has dimension 3 and depth 3. Here is a homogeneous system of parameters:
The first 3 terms h1, h2, h3 form a regular sequence of maximum length.
The first term h1 forms a complete Duflot regular sequence. That is, its restriction to the greatest central elementary abelian subgroup forms a regular sequence of maximal length.
Data for Benson's test:
A basis for R/(h1, h2, h3) is as follows.
(1 + 3t + 4t2 + 4t3 + 3t4 + t5) / (1 - t2)2 (1 - t4)