Subgroup ($H$) information
| Description: | $C_{105}:D_4$ |
| Order: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
199 & 0 \\
0 & 104
\end{array}\right), \left(\begin{array}{rr}
33 & 0 \\
0 & 370
\end{array}\right), \left(\begin{array}{rr}
176 & 0 \\
0 & 122
\end{array}\right), \left(\begin{array}{rr}
27 & 0 \\
0 & 291
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right), \left(\begin{array}{rr}
112 & 0 \\
0 & 312
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{210}.C_6.C_2^5.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4\times C_2\times S_3\times F_7$ |
| $W$ | $D_{42}$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | $1$ |
| Projective image | $C_2\times D_{210}$ |