Subgroup ($H$) information
| Description: | $D_{198}:C_{18}$ |
| Order: | \(7128\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 11 \) |
| Index: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
290 & 0 \\
0 & 256
\end{array}\right), \left(\begin{array}{rr}
228 & 0 \\
0 & 249
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 34
\end{array}\right), \left(\begin{array}{rr}
314 & 0 \\
0 & 110
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 93
\end{array}\right), \left(\begin{array}{rr}
0 & 5 \\
159 & 0
\end{array}\right), \left(\begin{array}{rr}
25 & 0 \\
0 & 270
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 312
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), and metabelian. Whether it is a direct factor or a semidirect factor has not been computed.
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.
Quotient group ($Q$) structure
| Description: | $C_{22}$ |
| Order: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Automorphism Group: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 \(5702400\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_{99}.C_{15}.C_6.C_2^4$ |
| $W$ | $D_{198}$, of order \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
Related subgroups
| Centralizer: | $C_{396}$ |
| Normalizer: | $C_{396}.D_{198}$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |