Subgroup ($H$) information
| Description: | $C_2^4:F_7$ |
| Order: | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$ae^{42}, e^{28}, c^{3}e^{42}, d, b, e^{8}, c^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_8:C_2^3\times F_7$ |
| Order: | \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4:(D_4\times D_6)$, of order \(172032\)\(\medspace = 2^{13} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $F_7\times C_2^6:S_4$, of order \(64512\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7 \) |
| $\card{W}$ | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |