Q:

Question on the unsolvability a group

Let $G$ be a finite group. Let $\pi(G)=\{2,3,5\}$ be the set of prime
divisors of its order. If 6 divide the number of Sylow 5-subgroups of G
and 10 divide the number of Sylow $3$-subgroups of $G$, then whether the
group $G$ group with those properties is unsolvable?
In particular if the number of Sylow $5$-subgroups of $G$ is 6 or
the number of Sylow $3$-subgroups of $G$ is 10, then by the Hall's
theorem $G$ is unsolvable group. For example if the number of Sylow $5$-subgroups of $G$ is 6 and $G$ is solvable, then $2\equiv 1$ (mod $5$), a contradiction.
Hall's theorem: Let $G$ be a finite soluble group and $|G|=m.n$, where $m=p_{1}^{\alpha
_{1}}...p_{r}^{\alpha _{r}}$, $(m,n)=1$. Let $\pi =\{p_{1},...,p_{r}\}$ and $
h_{m}$ be the number of $\pi -$Hall subgroups of $G$. Then $
h_{m}=q_{1}^{\beta _{1}}...q_{s}^{\beta _{s}}$, satisfies the following
conditions for all $i\in \{1,2,...,s\}$:
1) $q_{i}^{\beta_{i}} \equiv 1$ (mod $p_{j}$), for some $p_{j}$.
2) The order of some chief factor of $G$ is divisible by $
q_{i}^{\beta_{i}}$.
Thank you so much.

A:

No. There is a solvable group of order $2^{22} 3^5 5^3$ with $n_2=1$, $n_3=2^{18} 5^2$, and $n_5 = 2^{20} 3^4$.$\newcommand{\GF}{\operatorname{GF}}\newcommand{\AGL}{\operatorname{AGL}}$
$$G = 
\left(\left(3\ltimes\GF(5^2)\right) \ltimes \left(\GF(2^4)^3\right)\right) ~ \times ~ \left(\left(5\ltimes\GF(3^4)\right) \ltimes \left(\GF(2^2)^5\right)\right)$$
I'll describe the pieces:
$H = 3\ltimes\GF(5^2) \leq \AGL(1,5^2)$ is the collection of affine maps $x\mapsto ax+b$ where $a,x,b$ are elements of the finite field $\GF(5^2)$ of order 25 and $a^3=1$.
$H$ has an irreducible $\GF(2^4)$ module $V$ of dimension 3 formed by inducing a non-principal one-dimensional module from $\GF(5^2) \leq H$.
$H \ltimes V =\left(3\ltimes\GF(5^2)\right) \ltimes \left(\GF(2^4)^3\right) \leq \AGL(3,2^4)$ consists of all maps $x\mapsto ax+b$ where $a \in H$ and $x,b \in V$.
$K = 5 \ltimes \GF(3^4) \leq \AGL(1,3^4)$ is the collection of affine maps $x\mapsto ax+b$ where $a,x,b$ are elements of the finite field $\GF(3^4)$ of order 81 and $a^5=1$.
$K$ has an irreducible $\GF(2^2)$ module $W$ of dimension 5 formed by inducing a non-principal one-dimensional module from $\GF(3^4) \leq K$.
$K \ltimes W = \left(5\ltimes\GF(3^4)\right) \ltimes \left(\GF(2^2)^5\right) \leq \AGL(5,2^2)$ consists of all maps $x\mapsto ax+b$ where $a \in K$ and $x,b \in W$.
$G = \left( H \ltimes V \right) \times \left( K \ltimes W \right)$ is the direct product of these two groups.
$\begin{array}{c|cccc}
            &         o      &  n_2   &     n_3    &      n_5     \\ \hline
H           & 2^{~0} 3^1 5^2 & 3^0 5^0 & 2^{~0} 5^2 &  2^{~0} 3^0  \\
H \ltimes V & 2^{12} 3^1 5^2 & 3^0 5^0 & 2^{~8} 5^2 &  2^{12} 3^0  \\
K           & 2^{~0} 3^4 5^1 & 3^0 5^0 & 2^{~0} 5^0 &  2^{~0} 3^4  \\
K \ltimes W & 2^{10} 3^4 5^1 & 3^0 5^0 & 2^{10} 5^0 &  2^{~8} 3^4  \\
G           & 2^{22} 3^5 5^3 & 3^0 5^0 & 2^{18} 5^2 &  2^{20} 3^4  \\
\end{array}$
I think this is approximately minimal order. I think I misunderstand Hall's theorem if there is any example of order less than $2^{22} 3^4 5^2$.

