Subgroup ($H$) information
| Description: | $C_{11}:C_{10}$ |
| Order: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Index: | \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
6 & 3 & 6 & 1 \\
8 & 8 & 3 & 6 \\
6 & 4 & 5 & 8 \\
3 & 6 & 3 & 7
\end{array}\right), \left(\begin{array}{rrrr}
7 & 0 & 7 & 3 \\
2 & 3 & 9 & 7 \\
9 & 4 & 4 & 0 \\
8 & 9 & 9 & 0
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| 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: | $C_5\times \SL(2,11):D_6$ |
| Order: | \(79200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_4\times S_3\times D_4).\PSL(2,11).C_2$ |
| $\operatorname{Aut}(H)$ | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| $W$ | $C_{11}:C_5$, of order \(55\)\(\medspace = 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $12$ |
| Möbius function | $-6$ |
| Projective image | not computed |