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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.

Joachim Neubueser

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