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: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
0 & 0 & 10 & 1 \\
9 & 9 & 3 & 10 \\
1 & 1 & 4 & 0 \\
0 & 1 & 2 & 2
\end{array}\right), \left(\begin{array}{rrrr}
5 & 0 & 0 & 0 \\
3 & 8 & 9 & 0 \\
10 & 10 & 2 & 0 \\
10 & 10 & 8 & 5
\end{array}\right), \left(\begin{array}{rrrr}
5 & 9 & 0 & 5 \\
4 & 5 & 1 & 0 \\
1 & 3 & 8 & 2 \\
5 & 1 & 7 & 8
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 1 & 0 \\
2 & 10 & 6 & 1 \\
10 & 0 & 0 & 0 \\
8 & 10 & 9 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
5 & 6 & 4 & 0 \\
2 & 2 & 7 & 0 \\
2 & 2 & 6 & 1
\end{array}\right), \left(\begin{array}{rrrr}
8 & 3 & 1 & 0 \\
9 & 0 & 6 & 6 \\
1 & 1 & 1 & 8 \\
7 & 7 & 6 & 2
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
9 & 8 & 5 & 0 \\
8 & 8 & 1 & 0 \\
8 & 8 & 2 & 10
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{11}^3:(D_{12}\times C_5^2)$ |
| Order: | \(798600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11^{3} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(63888000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \cdot 11^{3} \) |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_{15}.C_{20}.C_2^4$ |
| $W$ | $C_{11}^2:(S_3\times C_{10})$, of order \(7260\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11^{2} \) |
Related subgroups
| Centralizer: | $C_{110}$ |
| Normalizer: | $C_{11}^3:(D_{12}\times C_5^2)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |