Subgroup ($H$) information
| Description: | $C_{84}\times D_{105}$ |
| Order: | \(17640\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 0 \\
0 & 370
\end{array}\right), \left(\begin{array}{rr}
307 & 0 \\
0 & 48
\end{array}\right), \left(\begin{array}{rr}
2 & 0 \\
0 & 156
\end{array}\right), \left(\begin{array}{rr}
4 & 0 \\
0 & 316
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 122
\end{array}\right), \left(\begin{array}{rr}
16 & 0 \\
0 & 79
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right), \left(\begin{array}{rr}
370 & 0 \\
0 & 33
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{420}.D_{210}$ |
| Order: | \(176400\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(3870720\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^3\times C_6\times F_5\times S_3\times F_7$ |
| $\card{W}$ | 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 |