Subgroup ($H$) information
| Description: | $C_{11}^2:C_5$ |
| Order: | \(605\)\(\medspace = 5 \cdot 11^{2} \) |
| Index: | \(121\)\(\medspace = 11^{2} \) |
| Exponent: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
5 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 \\
8 & 0 & 6 & 1
\end{array}\right), \left(\begin{array}{rrrr}
5 & 0 & 0 & 0 \\
0 & 9 & 9 & 0 \\
6 & 0 & 4 & 0 \\
4 & 2 & 2 & 5
\end{array}\right), \left(\begin{array}{rrrr}
3 & 0 & 7 & 0 \\
1 & 3 & 9 & 7 \\
1 & 0 & 10 & 0 \\
6 & 1 & 10 & 10
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 \(133100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11^{3} \) |
| $\operatorname{Aut}(H)$ | $F_{11}^2$, of order \(12100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11^{2} \) |
| $\card{W}$ | \(605\)\(\medspace = 5 \cdot 11^{2} \) |
Related subgroups
| Centralizer: | $C_{11}$ | ||
| Normalizer: | $C_{11}^3:C_5$ | ||
| Normal closure: | $C_{11}^3:(C_{11}:C_5)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_{11}^3:C_5$ | ||
| Maximal under-subgroups: | $C_{11}^2$ | $C_{11}:C_5$ | $C_{11}:C_5$ |
Other information
| Number of subgroups in this conjugacy class | $11$ |
| Möbius function | not computed |
| Projective image | not computed |