Subgroup ($H$) information
| Description: | $C_5\times C_{210}$ |
| Order: | \(1050\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 0 \\
0 & 279
\end{array}\right), \left(\begin{array}{rr}
316 & 0 \\
0 & 4
\end{array}\right), \left(\begin{array}{rr}
33 & 0 \\
0 & 370
\end{array}\right), \left(\begin{array}{rr}
48 & 0 \\
0 & 307
\end{array}\right), \left(\begin{array}{rr}
79 & 0 \\
0 & 16
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the socle (hence characteristic and normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 5$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $D_{420}:C_{10}$ |
| Order: | \(8400\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \) |
| 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: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{210}.C_6.C_2^5.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_6\times \GL(2,5)$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $-8$ |
| Projective image | $C_2\times D_{210}$ |