Subgroup ($H$) information
| Description: | $C_{10}\times C_{11}\wr S_3$ |
| Order: | \(79860\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11^{3} \) |
| Index: | \(11\) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
31 & 86 \\
62 & 90
\end{array}\right), \left(\begin{array}{rr}
45 & 33 \\
77 & 78
\end{array}\right), \left(\begin{array}{rr}
81 & 0 \\
0 & 81
\end{array}\right), \left(\begin{array}{rr}
89 & 0 \\
0 & 89
\end{array}\right), \left(\begin{array}{rr}
120 & 0 \\
0 & 120
\end{array}\right), \left(\begin{array}{rr}
111 & 0 \\
110 & 12
\end{array}\right), \left(\begin{array}{rr}
66 & 43 \\
111 & 54
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $C_5\times C_{11}^4:D_6$ |
| Order: | \(878460\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11^{4} \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and an A-group. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_3.C_5^3.C_4.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_{15}.C_{20}.C_2^4$ |
| $W$ | $C_{11}^2:S_3$, of order \(726\)\(\medspace = 2 \cdot 3 \cdot 11^{2} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_{10}\times C_{11}\wr S_3$ |
| Normal closure: | $C_5\times C_{11}^4:D_6$ |
| Core: | $C_{11}^2:C_{330}$ |
Other information
| Number of subgroups in this autjugacy class | $11$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_{11}^2:D_{33}$ |