Subgroup ($H$) information
| Description: | $D_{30}$ |
| Order: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Index: | \(77\)\(\medspace = 7 \cdot 11 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$a, b^{770}, b^{1386}, b^{1155}$
|
| 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: | $C_{11}\times D_{210}$ |
| Order: | \(4620\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Exponent: | \(2310\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5:F_5^2$, of order \(100800\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $D_6\times F_5$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $D_6\times F_5$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| $W$ | $D_{15}$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $7$ |
| Möbius function | $1$ |
| Projective image | $C_{11}\times D_{105}$ |