Subgroup ($H$) information
| Description: | $C_{33}:Q_8$ |
| Order: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Index: | \(903\)\(\medspace = 3 \cdot 7 \cdot 43 \) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
30 & 22 \\
20 & 13
\end{array}\right), \left(\begin{array}{rr}
0 & 33 \\
13 & 0
\end{array}\right), \left(\begin{array}{rr}
7 & 0 \\
0 & 7
\end{array}\right), \left(\begin{array}{rr}
30 & 17 \\
35 & 11
\end{array}\right), \left(\begin{array}{rr}
6 & 0 \\
0 & 6
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_3\times \SL(2,43)$ |
| Order: | \(238392\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 43 \) |
| Exponent: | \(39732\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 43 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \PSL(2,43).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times D_4\times F_{11}$, of order \(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \) |
| $W$ | $D_{22}$, of order \(44\)\(\medspace = 2^{2} \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $903$ |
| Möbius function | $-1$ |
| Projective image | $\PSL(2,43)$ |