Subgroup ($H$) information
| Description: | $C_{11}\times D_{70}$ |
| Order: | \(1540\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \cdot 11 \) |
| Index: | \(57\)\(\medspace = 3 \cdot 19 \) |
| Exponent: | \(770\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 11 \) |
| Generators: |
$a, b^{27930}, b^{17556}, b^{21945}, b^{37620}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{209}\times D_{210}$ |
| Order: | \(87780\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \) |
| Exponent: | \(43890\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_{90}\times F_5\times S_3\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{10}\times F_5\times F_7$ |
| $W$ | $D_{35}$, of order \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $1$ |
| Projective image | $C_{19}\times D_{105}$ |