Subgroup ($H$) information
| Description: | $\SL(2,3)$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rrrr}
4 & 0 & 0 & 0 \\
0 & 4 & 0 & 0 \\
0 & 0 & 4 & 0 \\
0 & 0 & 0 & 4
\end{array}\right), \left(\begin{array}{rrrr}
3 & 1 & 1 & 4 \\
2 & 1 & 1 & 1 \\
0 & 0 & 3 & 4 \\
0 & 0 & 3 & 1
\end{array}\right), \left(\begin{array}{rrrr}
0 & 3 & 3 & 2 \\
3 & 4 & 1 & 2 \\
1 & 2 & 0 & 4 \\
1 & 1 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
0 & 1 & 1 & 4 \\
1 & 1 & 3 & 2 \\
3 & 2 & 1 & 1 \\
0 & 3 & 4 & 3
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable.
Ambient group ($G$) information
| Description: | $\SL(2,5):D_5$ |
| Order: | \(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times F_5\times S_5$, of order \(4800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(S)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_5:C_4$ | ||
| Normalizer: | $\SL(2,3):D_5$ | ||
| Normal closure: | $\SL(2,5)$ | ||
| Core: | $C_2$ | ||
| Minimal over-subgroups: | $\SL(2,5)$ | $C_5\times \SL(2,3)$ | $\SL(2,3):C_2$ |
| Maximal under-subgroups: | $Q_8$ | $C_6$ |
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | $-5$ |
| Projective image | $D_5\times A_5$ |