Subgroup ($H$) information
| Description: | $Q_{16}.D_{14}$ |
| Order: | \(448\)\(\medspace = 2^{6} \cdot 7 \) |
| Index: | \(3\) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Generators: |
$a, c^{12}, c^{3}, d^{7}, b, d^{2}, c^{6}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $D_{56}.D_6$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_{84}).C_6.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_{28}.(C_2^5\times C_6).C_2$ |
| $\card{W}$ | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |