Subgroups of the Monster in mmgroup
The Monster can be defined as the automorphism group of a
(conjecturally to be unique) extremal self-dual (or holomorphic) vertex operator algebra of central charge c=24. It is the largest of the 26 sporadic simple groups and was found by Bernd Fischer and Robert Griess and first constructed by Griess as automorphisms of the Griess algebra.
Martin Seysen's newly developed mmgroup package
allows fast computations inside the Monster.
We list generators of subgroups of the Monster and some simple python scripts for accessing this information.
Gerald Höhn, Fall 2023.
The Monster itself
Standart generators of the Monster as in the
ATLAS of Finite Group Representations - Version 3 and computed for mmgroup by Martin Seysen.
- a = M <d_200h>
- b = M <y_756h*x_1b0eh*d_29ah*p_142818544*l_2*p_1415040*l_1*p_12992112*l_1*t_1*l_2*p_2787840*l_2*p_31997178*l_1*t_1*l_2*p_960*l_2*p_10666896*l_1*p_4292160*t_1*l_1*p_1499520*l_1*p_21378434*t_1*l_2*p_1858560*l_2*p_464880*l_1*p_1927680*t_2*l_2*p_1393920*l_1*p_85409856*t_1*l_2*p_2956800*l_1*p_85837058>
Also available as file on github.
Algebraic conjugacy classes
If possible, representatives are in 21+24.Co1. The two classes of order 27 cannot
be distinguished yet. Computed by Gerald Höhn.
An entry of the list li in the file classes.py contains:
- conjugacy class representative g
- order of the class [g]
- trace of g on Griess algebra
- trace of g^2 on Griess algebra (if required for identification)
- trace of g^3 on Griess algebra (if required for identification)
- ATLAS name (not yet complete)
The file find_class.py provides the following
functions:
- class_name(x): The ATLAS name of the i-th class or an monster element g
- class_rep(i): A class representative for i-th class
- con_class(i): The number i of an element g in the Monster
Maximal subgroups
Generators of maximal subgroups. If possible, we try to find standart generators as in the
ATLAS of Finite Group Representations - Version 3
- 2.B
-- a=M<y_515h*x_55h*d_0c7ch*p_182792688*l_2*p_47942400*l_1*p_182772480*l_2*t_2*l_2*p_44394240*l_1*p_51469440*l_1*t_2*l_2*p_70118400*l_2*p_159709440*l_2*t_2*l_2*p_139751040>, b=M<y_4d0h*x_69h*d_658h*p_213819354*l_1*p_48829440*l_2*p_63909312*t_2*l_2*p_50603520*l_1*p_172128000*l_1*t_1*l_2*p_79875840*l_1*p_129550080*l_1*t_1*l_1*p_4012800*l_2*t_2*l_1*p_67900800>
- 21+24.Co1 -- gx0.py by Dietrich, Lee, Popiel (arXiv:2304.14646)
- 22+11+22.(M24×S3) -- a=M < y_383h*x_0a2dh*d_0ef1h*p_36582201*t_1 >, b=M < y_157h*x_1f92h*d_8f5h*p_107480988 > of order 8 and 12. (M. Seysen and G.H.)
- PSL2(13):2 -- pgl13.py by Dietrich, Lee, Popiel (arXiv:2304.14646)
-
Sylow subgroups
Conjugacy classes of commuting pairs