Subgroup ($H$) information
| Description: | $\SL(2,5):D_5$ |
| Order: | \(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \) |
| Index: | \(2\) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rrrr}
3 & 0 & 3 & 3 \\
2 & 1 & 3 & 3 \\
0 & 0 & 1 & 0 \\
2 & 0 & 3 & 4
\end{array}\right), \left(\begin{array}{rrrr}
2 & 4 & 1 & 0 \\
1 & 2 & 4 & 3 \\
2 & 0 & 1 & 3 \\
1 & 4 & 4 & 1
\end{array}\right), \left(\begin{array}{rrrr}
0 & 1 & 4 & 0 \\
0 & 0 & 0 & 1 \\
4 & 0 & 0 & 1 \\
0 & 1 & 0 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 3 & 0 & 2 \\
1 & 0 & 4 & 0 \\
1 & 2 & 0 & 2 \\
1 & 1 & 4 & 4
\end{array}\right), \left(\begin{array}{rrrr}
0 & 3 & 4 & 3 \\
0 & 1 & 3 & 0 \\
4 & 3 & 3 & 3 \\
2 & 0 & 0 & 4
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a direct factor, nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $\SL(2,5):D_{10}$ |
| Order: | \(2400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_4\times F_5\times S_5$, of order \(19200\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_5\times S_5$, of order \(4800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \) |
| $\operatorname{res}(S)$ | $C_2\times F_5\times S_5$, of order \(4800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_5\times A_5$, of order \(600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $D_{10}\times A_5$ |