Subgroup ($H$) information
| Description: | $C_{220}:C_5$ |
| Order: | \(1100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
0 & 6 & 9 & 4 \\
1 & 5 & 10 & 9 \\
5 & 7 & 1 & 5 \\
8 & 5 & 10 & 6
\end{array}\right), \left(\begin{array}{rrrr}
4 & 0 & 0 & 0 \\
0 & 4 & 0 & 0 \\
0 & 0 & 4 & 0 \\
0 & 0 & 0 & 4
\end{array}\right), \left(\begin{array}{rrrr}
5 & 3 & 8 & 4 \\
1 & 4 & 0 & 5 \\
6 & 2 & 2 & 9 \\
9 & 1 & 3 & 0
\end{array}\right), \left(\begin{array}{rrrr}
7 & 0 & 0 & 0 \\
0 & 7 & 0 & 0 \\
0 & 0 & 7 & 0 \\
0 & 0 & 0 & 7
\end{array}\right), \left(\begin{array}{rrrr}
9 & 9 & 8 & 10 \\
3 & 8 & 0 & 8 \\
6 & 5 & 3 & 2 \\
6 & 6 & 8 & 2
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 5$, and an A-group.
Ambient group ($G$) information
| Description: | $\SL(2,11):C_{10}$ |
| Order: | \(13200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_4\times \PSL(2,11).C_2$ |
| $\operatorname{Aut}(H)$ | $D_{110}:C_{20}$, of order \(4400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11 \) |
| $W$ | $C_{11}:C_5$, of order \(55\)\(\medspace = 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $12$ |
| Möbius function | $-1$ |
| Projective image | $\PSL(2,11)$ |