Subgroup ($H$) information
| Description: | $C_{99}:D_4$ |
| Order: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Index: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
160 & 0 \\
0 & 276
\end{array}\right), \left(\begin{array}{rr}
321 & 0 \\
0 & 372
\end{array}\right), \left(\begin{array}{rr}
140 & 0 \\
0 & 108
\end{array}\right), \left(\begin{array}{rr}
124 & 0 \\
0 & 381
\end{array}\right), \left(\begin{array}{rr}
290 & 0 \\
0 & 256
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{396}.D_{198}$ |
| Order: | \(156816\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 11^{2} \) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(5702400\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{11}:(C_2^2\times C_{30})$ |
| $W$ | $D_{22}$, of order \(44\)\(\medspace = 2^{2} \cdot 11 \) |
Related subgroups
| Centralizer: | $C_{396}$ |
| Normalizer: | $D_{44}.C_{198}$ |
| Normal closure: | $D_{198}:C_{18}$ |
| Core: | $C_2\times C_{198}$ |
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |