Subgroup ($H$) information
| Description: | $C_2^2\times D_4\times F_7$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Index: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(5,7), (5,8)(6,7), (1,4)(2,3), (1,3)(2,4), (1,3)(2,4)(5,6)(7,8)(9,12,11) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_2^2\times S_7\times D_4$ |
| Order: | \(161280\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, nonsolvable, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5.C_2^6.D_6.A_7.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^5.C_2^6.(C_2\times S_3\times F_7)$ |
| $W$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^2\times D_4\times F_7$ |
| Normal closure: | $C_2^2\times S_7\times D_4$ |
| Core: | $C_2^2\times D_4$ |
Other information
| Number of subgroups in this autjugacy class | $120$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^2\times S_7$ |