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: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
1 & 2 & 3 & 0 \\
7 & 7 & 0 & 0 \\
7 & 7 & 10 & 1
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 9 & 2 \\
8 & 7 & 9 & 9 \\
9 & 9 & 6 & 0 \\
7 & 9 & 3 & 3
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
1 & 4 & 4 & 0 \\
10 & 2 & 5 & 0 \\
2 & 5 & 2 & 1
\end{array}\right), \left(\begin{array}{rrrr}
10 & 5 & 8 & 7 \\
10 & 8 & 2 & 8 \\
1 & 7 & 5 & 6 \\
10 & 1 & 1 & 3
\end{array}\right), \left(\begin{array}{rrrr}
0 & 7 & 2 & 0 \\
7 & 8 & 7 & 7 \\
9 & 0 & 0 & 6 \\
7 & 0 & 8 & 1
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
10 & 9 & 8 & 0 \\
4 & 4 & 0 & 0 \\
4 & 4 & 1 & 10
\end{array}\right), \left(\begin{array}{rrrr}
5 & 0 & 0 & 0 \\
6 & 0 & 7 & 0 \\
9 & 9 & 10 & 0 \\
9 & 9 & 5 & 5
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $C_{11}^3:C_{10}^2.S_4$ |
| Order: | \(3194400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 11^{3} \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.(C_5\times A_4).C_{10}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_{15}.C_{20}.C_2^4$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | not computed |