Compact Groups of Operators on Subproportional Quotients of $l^m_1$ It is proved that a typical'' $n$-dimensional quotient $X_n$ of $l^m_1$ with $n = m^{\sigma}$, $0 < \sigma < 1$, has the property $$\Average \int_G \|Tx\|_{X_n} \,dh_G(T) \geq \frac{c}{\sqrt{n\log^3 n}} \biggl( n - \int_G |\tr T| \,dh_G (T) \biggr),$$ for every compact group $G$ of operators acting on $X_n$, where $d_G(T)$ stands for the normalized Haar measure on $G$ and the average is taken over all extreme points of the unit ball of $X_n$. Several consequences of this estimate are presented. Categories:46B20, 46B09