Subgroup ($H$) information
| Description: | $C_{20}\times D_{35}$ |
| Order: | \(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Index: | \(3\) |
| Exponent: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Generators: |
$a^{5}, b^{60}, a^{2}b^{210}, b^{210}, b^{105}, b^{252}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_{20}\times D_{105}$ |
| Order: | \(4200\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_4\times F_5\times S_3\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_4\times F_5\times F_7$ |
| $\card{\operatorname{res}(S)}$ | \(13440\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{35}$, 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 | $D_{105}$ |