Subgroup ($H$) information
| Description: | $D_{132}:C_{18}$ |
| Order: | \(4752\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 11 \) |
| Index: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
314 & 0 \\
0 & 110
\end{array}\right), \left(\begin{array}{rr}
228 & 0 \\
0 & 131
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 34
\end{array}\right), \left(\begin{array}{rr}
142 & 0 \\
0 & 137
\end{array}\right), \left(\begin{array}{rr}
290 & 0 \\
0 & 256
\end{array}\right), \left(\begin{array}{rr}
374 & 0 \\
0 & 90
\end{array}\right), \left(\begin{array}{rr}
282 & 0 \\
0 & 252
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{396}.D_{396}$ |
| Order: | \(313632\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 11^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(11404800\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5^{2} \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_{66}.C_{30}.C_2^5$ |
| $\card{W}$ | 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 | $3$ |
| Möbius function | not computed |
| Projective image | not computed |