# Two questions on the Schur multiplier of groups of order $p^4$

1. I tried to find a reference for the computation of the Schur multiplier of groups of order $p^4$. The case in which $p=2$ is well known, see e.g. Table 1 at http://pages.bangor.ac.uk/~mas010/pdffiles/non-abel-tensor.pdf. However, I didn't find any reference for the odd case. Since there is a well known (not too big) classification of such groups, it seems reasonable that this work was already done. Is there a paper or a book in which I can find such computation?
2. I am particularly interested in special elements of the Schur multiplier. These elements of $M(G)$ correspond to projective representation of $G$ of dimension $\sqrt{|G|}$. Groups admitting such elements are called of central type (non-classical). Is there a known classification of groups of central type of order $p^4$?

• Chapter III of [Karpilovsky, Gregory. The Schur multiplier. London Mathematical Society Monographs. New Series, 2. The Clarendon Press, Oxford University Press, New York, 1987. x+302 pp. ISBN: 0-19-853554-6 MR1200015 (93j:20002)] is devoted to the Schur multiplier of $p$-groups and contains several useful calculations. – Leandro Vendramin Dec 29 '14 at 10:56
• Any finite $p$-group $P$ which has $Z(P)$ cyclic, $[P:Z(P)] = p^{4}$ and $P^{\prime} \leq Z(P)$ has a faithful irreducible complex character of degree $p^{2} = \sqrt{[P:Z(P)]}.$ One such group $P$ is an extraspecial group of order $p^{5}$. – Geoff Robinson Dec 29 '14 at 23:24
• Geoff Robinson- Thank you for your answer. This says that such $P$ (if exists) is a group of central type in the classical sense. Do you know which non-abelian groups $G$ of order $p^4$ admits a projective representation of dimension $p^2$. In other words, which non-abelian groups $G$ of order $p^4$ are central type in the non-classical sense? – Ofir Schnabel Dec 30 '14 at 8:43
If I understand the question, the problem is to find all groups $X$ of order $p^4$ such that $X \cong P/Z(P)$ for some group $P$, where $P$ has an irreducible character of degree $p^2$. To do this, it seems that we need consider only groups $P$ of order $p^5$ or $p^6$. I have done this by "brute force" for $p = 3$, using the Magma software. Of the 15 groups of order $3^4$, exactly five have the desired property. They are numbers 2,4,9,12 and 15 of the small groups data base.
A group $X$ is often called a "central-type factor group" (CTFG) if $X \cong P/Z(P)$, where $P$ has an irreducible character of degree $\sqrt |X|$. There are various things known about CTFGs, and in particular, an abelian group is a CTFG if and only if it has the form $A \times B$, where $A \cong B$. It is also known that a CTFG cannot have a self-centralizing cyclic subgroup. (This condition eliminates five of the 15 groups of order $3^4$.)
• Thanks a lot for your answer. I almost done with the classification of CFTG for general prime $p$. To rule out groups I used arrguments which are very similar to those you mentiond above. But in order to show that a group $G$ is CTFG (in the non-abelian case) I construct complex simple twisted group algebra over $G$. I don't know if anyone but me will be intersted in this work however. – Ofir Schnabel Feb 13 '15 at 8:43