Dear GAP Forum,
Ella Shalec asked:
I have a central extension of a given permutation group in which the
kernel is an elementary abelian group which can also be represented
as a permutation group. I would like to represent the extension as
a permutation group. I can do it by using the wreath product. Can
someone tell me is there a permutation representation of such
extension which is of lesser degree then the representation obtained
from the wreath product?
There are two standard permutation representations of a wreath product
of two permutation groups of which the imprimitive one (except for
trivial examples) has the smaller degree, which is the product of the
degrees of the factorgroup and the normal subgroup. Taking the
quaternion group of oder 8 and its center as normal subgroup, the
factor group is a Klein Four Group, hence of degree 4, while the
center is cyclic of order 2 hence of degree 2, so the degree of the
imprimitive representation of the wreath product is 8. However the
Quaternion group has no faithful permutation representation of degree
smaller than 8. Hence it provides a case, in which you cannot find a
permutation representation of smaller degree.
Hope this answers your question.