Subgroup ($H$) information
| Description: | $C_{42}\times C_{420}$ |
| 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: |
$\left(\begin{array}{rr}
312 & 0 \\
0 & 112
\end{array}\right), \left(\begin{array}{rr}
376 & 0 \\
0 & 199
\end{array}\right), \left(\begin{array}{rr}
370 & 0 \\
0 & 33
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 370
\end{array}\right), \left(\begin{array}{rr}
307 & 0 \\
0 & 347
\end{array}\right), \left(\begin{array}{rr}
376 & 0 \\
0 & 318
\end{array}\right), \left(\begin{array}{rr}
311 & 0 \\
0 & 310
\end{array}\right), \left(\begin{array}{rr}
122 & 0 \\
0 & 176
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and metacyclic. 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: | $D_5$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 \(3870720\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $Q_8.(C_2\times C_6^2).C_2^4.\SO(3,7)$ |
| $\card{W}$ | 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 |