Subgroup ($H$) information
| Description: | $C_{15}:C_{20}$ |
| Order: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Index: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 0 \\
0 & 279
\end{array}\right), \left(\begin{array}{rr}
38 & 0 \\
0 & 277
\end{array}\right), \left(\begin{array}{rr}
181 & 0 \\
0 & 107
\end{array}\right), \left(\begin{array}{rr}
377 & 0 \\
0 & 354
\end{array}\right), \left(\begin{array}{rr}
0 & 2 \\
244 & 0
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{210}.C_6.C_2^5.C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_{30}:C_4^2$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| $W$ | $D_{30}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $7$ |
| Möbius function | $-2$ |
| Projective image | $C_2\times D_{210}$ |