Subgroup ($H$) information
| Description: | $C_{11}:C_5$ |
| Order: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Index: | \(1331\)\(\medspace = 11^{3} \) |
| Exponent: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
5 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 \\
8 & 0 & 6 & 1
\end{array}\right), \left(\begin{array}{rrrr}
5 & 0 & 0 & 0 \\
0 & 9 & 9 & 0 \\
6 & 0 & 4 & 0 \\
4 & 2 & 2 & 5
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 5$.
Ambient group ($G$) information
| Description: | $C_{11}^3:(C_{11}:C_5)$ |
| Order: | \(73205\)\(\medspace = 5 \cdot 11^{4} \) |
| Exponent: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_{11}:C_{10}^2$, of order \(133100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11^{3} \) |
| $\operatorname{Aut}(H)$ | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| $\card{W}$ | \(55\)\(\medspace = 5 \cdot 11 \) |
Related subgroups
| Centralizer: | $C_{11}$ | |
| Normalizer: | $C_{11}:C_{55}$ | |
| Normal closure: | $C_{11}^3:(C_{11}:C_5)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_{11}:C_{55}$ | $C_{11}^2:C_5$ |
| Maximal under-subgroups: | $C_{11}$ | $C_5$ |
Other information
| Number of subgroups in this conjugacy class | $121$ |
| Möbius function | not computed |
| Projective image | not computed |