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: | \(2\) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
120 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
111 & 22 \\
77 & 12
\end{array}\right), \left(\begin{array}{rr}
3 & 0 \\
0 & 3
\end{array}\right), \left(\begin{array}{rr}
34 & 0 \\
0 & 34
\end{array}\right), \left(\begin{array}{rr}
120 & 0 \\
0 & 120
\end{array}\right), \left(\begin{array}{rr}
1 & 55 \\
11 & 1
\end{array}\right), \left(\begin{array}{rr}
60 & 1 \\
90 & 60
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is normal, maximal, 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: | $C_5\times C_{11}^3:D_{12}$ |
| Order: | \(159720\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{3} \) |
| 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_6.C_{10}^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_{15}.C_{20}.C_2^4$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |