Subgroup ($H$) information
| Description: | $D_4.\PGL(2,11)$ |
| Order: | \(10560\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 11 \) |
| Index: | \(5\) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
4 & 6 & 9 & 0 \\
8 & 7 & 0 & 2 \\
4 & 0 & 7 & 6 \\
0 & 7 & 8 & 4
\end{array}\right), \left(\begin{array}{rrrr}
0 & 4 & 5 & 1 \\
3 & 10 & 5 & 8 \\
2 & 6 & 8 & 3 \\
5 & 9 & 4 & 4
\end{array}\right), \left(\begin{array}{rrrr}
6 & 5 & 1 & 0 \\
5 & 5 & 0 & 10 \\
6 & 0 & 5 & 5 \\
0 & 5 & 5 & 6
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 4 & 9 \\
4 & 3 & 9 & 4 \\
4 & 2 & 8 & 0 \\
9 & 4 & 7 & 10
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $\GL(2,11):C_2^2$ |
| Order: | \(52800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_5$ |
| Order: | \(5\) |
| Exponent: | \(5\) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| 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, and simple.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_4\times \PSL(2,11).C_2\times D_4$ |
| $\operatorname{Aut}(H)$ | $C_2\times \PSL(2,11).C_2\times D_4$ |
| $\card{W}$ | \(5280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | not computed |