Subgroup ($H$) information
| Description: | $C_{11}\times D_{84}$ |
| Order: | \(1848\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 11 \) |
| Index: | \(5\) |
| Exponent: | \(924\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \) |
| Generators: |
$a, b^{1320}, b^{420}, b^{2310}, b^{1155}, b^{1540}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{11}\times D_{420}$ |
| Order: | \(9240\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Exponent: | \(4620\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4.C_4^2$, of order \(403200\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_{210}.C_6.C_2^4$ |
| $W$ | $D_{42}$, 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 | $D_{210}$ |