Subgroup ($H$) information
| Description: | $S_4$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(650\)\(\medspace = 2 \cdot 5^{2} \cdot 13 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left[ \left(\begin{array}{rrrr}
1 & 0 & 4 & 2 \\
2 & 2 & 0 & 4 \\
1 & 1 & 3 & 0 \\
1 & 2 & 1 & 3
\end{array}\right) \right], \left[ \left(\begin{array}{rrrr}
2 & 4 & 4 & 2 \\
1 & 4 & 4 & 3 \\
4 & 4 & 2 & 0 \\
1 & 0 & 4 & 2
\end{array}\right) \right], \left[ \left(\begin{array}{rrrr}
3 & 0 & 1 & 4 \\
3 & 3 & 4 & 4 \\
0 & 4 & 1 & 1 \\
3 & 4 & 4 & 3
\end{array}\right) \right], \left[ \left(\begin{array}{rrrr}
1 & 1 & 1 & 4 \\
0 & 4 & 0 & 2 \\
0 & 0 & 4 & 0 \\
0 & 0 & 0 & 1
\end{array}\right) \right]$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $\POMinus(4,5)$ |
| Order: | \(15600\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| Exponent: | \(780\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 13 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, and nonsolvable.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(2,25).C_2^2$, of order \(31200\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $325$ |
| Möbius function | $0$ |
| Projective image | $\POMinus(4,5)$ |