Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(26400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(2\) |
| Generators: |
$\left(\begin{array}{rrrr}
9 & 9 & 10 & 8 \\
5 & 5 & 0 & 10 \\
3 & 7 & 6 & 2 \\
4 & 3 & 6 & 2
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup 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.
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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_4\times \PSL(2,11).C_2\times D_4$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $132$ |
| Möbius function | $0$ |
| Projective image | $\GL(2,11):C_2^2$ |