Subgroup ($H$) information
| Description: | $C_5\times D_{11}$ |
| Order: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Index: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
47 & 35 \\
51 & 74
\end{array}\right), \left(\begin{array}{rr}
12 & 99 \\
44 & 111
\end{array}\right), \left(\begin{array}{rr}
3 & 0 \\
0 & 3
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_5\times C_{11}^2:D_{12}$ |
| Order: | \(14520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_6.C_{10}.C_4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_4\times F_{11}$, of order \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| $W$ | $D_{11}$, of order \(22\)\(\medspace = 2 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | $C_{11}^2:D_{12}$ |