> < ^ Date: Sun, 10 Mar 2002 16:51:03 -0000
> < ^ From: Roger Beresford <rogerberesford@supanet.com >
> < ^ Subject: SmallGroup(40,3)

On Friday Mar. 8th 2002, Igor Schein asked about GAP SmallGroup(40,3).
Using g4003 and similar abbreviations, g4003 has generators (a^5=b^8=1,
ba=aab), with the second part being a non-abelian re-write rule. It has
g2001 (Q20, with a^5=b^4=1, ba=aaaab) as a subgroup, and g2003 (with
a^5=b^4=1, ba=aab) as a factor (quotient) group. The CayleyTable determinant
has two linear factors, one quadratic factor, one unrepeated fourth order
and two fourth order factors that are each repeated four times.
Similar groups:- g2401 is (a^3=b^8=1, ba=aab), g4001 is (a^5=b^8=1,
ba=aaaab), g4007 is (a^5=b^4=c^2=1, ba=aaaab), g4012 is (a^5=b^4=c^2=1,
ba=aab), g4013 is (a^5=b^2=c^2=d^2=1, ba=aaaab), g5601 is (a^7=b^8=1,
ba=aaaaaab), g7201 is (a^9=b^8=1, ba=aaaaaaaab), and g10403 is (a^13=b^8=1,
ba=aaaaab). I have not looked at 88-element groups; other rewrite rules give
the same groups, or non-associative tables, with my procedure.
This information was obtained in a few minutes investigation using my
Mathematica "concast.ma" package, after transcribing g10403 from a GAP
session. it took 1040 secs (on a 700mHz PC) to find and identify the g10403
subgroups as {{1, "C2"}, {13, "C4"}, {13, "C8"}, {1, "C13"}, {1, "C26"}}.
This package is mainly concerned with conservative Cayley tables as
multiplication rules for renormalizing algebras, but includes a database of
groups (primarily with less than 73 elements) and procedures to generate,
investigate, and identify them.Contact me at
mail@beresford22.freeserve.com.uk if you would like a (600kb) Mathematica
demonstration notebook that includes the package.
Roger Beresford.

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