Subgroup ($H$) information
| Description: | $C_{396}$ |
| Order: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
124 & 0 \\
0 & 183
\end{array}\right), \left(\begin{array}{rr}
228 & 0 \\
0 & 218
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 396
\end{array}\right), \left(\begin{array}{rr}
314 & 0 \\
0 & 120
\end{array}\right), \left(\begin{array}{rr}
290 & 0 \\
0 & 256
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $D_{44}:C_{36}$ |
| Order: | \(3168\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{22}.C_{30}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{30}$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_2^2\times C_{30}$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $D_{44}:C_4$ |