[GAP Forum] question on automorphism group presentation
Walter Becker
w_becker at hotmail.com
Sun Nov 8 22:04:51 GMT 2015
Gap Forum question on automorphism groups.
I am interested in getting a presentation for the automorphism group of the group
(288, number 873) whose presentation is given by the following relations:
a1^8=b1^2=c1^2=(a1,b1)*a1^2=(a1,c1)=(b1,c1)=
d1^p=e1^p=(d1,e1)=
d1^a1*e1=e1^a1*d1^-1=
(b1,d1)=e1^b1*e1=
d1^c1*d1,e1^c1*e1=1.
The4 structure of this automoprhism group takes the form:
[ (18, numbere 4) X (32, nuumber 340] @ D_4.
The prsentations of the groups (18,4) and (32, 34) are:
a^3=b^3=(a,b)=c^2=a^c*a=b^c*b=1
and
d^4=e^4=(d,e)=f^2=d^f*d=e^f*e=1.
The action of the D_4 quotient on the order 18 group gives the group
(144,182), and the action of the D_4 group on the order 32 group
gives rise to the group (256, number 16870). Note an alternate,
but less desirable for form for the D_4 action on the order
32 group yields the group (256,16885).
The problem here is how to identify the common quotient groups
so as to obtain a presentation in terms of the group generators
(a,b,c,d,e,f) plus the generators of the D_4 group. I have
explicit presentations for these quotient groups but my relations
do not yield the correct presentation for the automorphism group.
One presentation of the group [(C_3 X C_3) @ C_2] @ D_4
takes the form:
a^4=b^2=a^b*a=c^8=a^-1*c^2=b^c*a*b=(a,c)=d^3=e^3=(d,e)=
d^a*e^-1=e^a*d=(d,b)=e^b*e=d^c*d^-1*e=e^c*d^-1*e^-1=1
Note this involves a different form for the generators of
the order 18 group here. The extension is also not a
semi-direct product. Here (a,b,c) genrates the order 16
quotient group QD_8 rather than (18 number 4).
A presentation for the order 256 (number 16870) group is
a^4=b^4=c^2=(a,b)=a^c*a=b^c*b=d^4=e^2=f^2 =
a^2 * d^-2 =d * c * f =
(a, d)=b^-1 * d^-1 * b^-1 * d =b^e*b =
e * b * a^-1 * e * a=e * b * c * e * c=
(d^-1 * e)^4=1.
Again here the generators for this order 256 group
has a somewhat altered form (the order 34 group here
is generated by (a,b,c) rather than (d,e,f).
I also believe that this required automorphism group is a subgroup
but not a normal subgroup of the direct product of these
order 182 and 256 groups.
Any guesses or ideas on how to come up with the desired form
of the automorphism group of the group (288, number 873).
Walter Becker
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