Subgroup ($H$) information
| Description: | $C_2^2\times D_{14}$ |
| Order: | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$a, d, e^{28}, c^{3}, e^{8}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
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 \(172032\)\(\medspace = 2^{13} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $F_7\times C_2^3:\GL(3,2)$, of order \(56448\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7^{2} \) |
| $\card{W}$ | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |