Subgroup ($H$) information
| Description: | $C_{99}:D_4$ |
| Order: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
396 & 0 \\
0 & 396
\end{array}\right), \left(\begin{array}{rr}
34 & 0 \\
0 & 34
\end{array}\right), \left(\begin{array}{rr}
304 & 0 \\
0 & 93
\end{array}\right), \left(\begin{array}{rr}
312 & 0 \\
0 & 312
\end{array}\right), \left(\begin{array}{rr}
31 & 0 \\
0 & 333
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{36}.D_{396}$ |
| Order: | \(28512\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 11 \) |
| Exponent: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{99}.C_{30}.C_6.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{11}:(C_2^2\times C_{30})$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $18$ |
| Möbius function | not computed |
| Projective image | not computed |