Subgroup ($H$) information
| Description: | $D_{35}$ |
| Order: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Index: | \(2401\)\(\medspace = 7^{4} \) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Generators: |
$ad^{5}e^{6}f^{3}, b^{5}, b^{21}c^{4}d^{3}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_7^5:D_5$ |
| Order: | \(168070\)\(\medspace = 2 \cdot 5 \cdot 7^{5} \) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and an A-group. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^5.C_{120}.C_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_5\times F_7$, of order \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| $W$ | $D_{35}$, of order \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_1$ |
| Normalizer: | $D_{35}$ |
| Normal closure: | $C_7^5:D_5$ |
| Core: | $C_7$ |
Other information
| Number of subgroups in this autjugacy class | $2401$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_7^5:D_5$ |