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: | $1$ |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
120 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
78 & 33 \\
55 & 45
\end{array}\right), \left(\begin{array}{rr}
81 & 0 \\
0 & 81
\end{array}\right), \left(\begin{array}{rr}
89 & 0 \\
0 & 89
\end{array}\right), \left(\begin{array}{rr}
120 & 0 \\
0 & 120
\end{array}\right), \left(\begin{array}{rr}
67 & 0 \\
0 & 56
\end{array}\right), \left(\begin{array}{rr}
60 & 43 \\
64 & 60
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, monomial, and an A-group.
Ambient group ($G$) information
| Description: | $C_{10}\times C_{11}\wr S_3$ |
| Order: | \(79860\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11^{3} \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{15}.C_{20}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_{15}.C_{20}.C_2^4$ |
| $W$ | $C_{11}^2:S_3$, of order \(726\)\(\medspace = 2 \cdot 3 \cdot 11^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_{11}^2:S_3$ |