Subgroup ($H$) information
| Description: | $\OD_{16}:C_2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(3822\)\(\medspace = 2 \cdot 3 \cdot 7^{2} \cdot 13 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{63} & 0 \\ 0 & \alpha^{63} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{88} & \alpha^{84} \\ \alpha^{59} & \alpha^{4} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{84} & 0 \\ 0 & \alpha^{84} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{93} & \alpha^{165} \\ \alpha^{94} & \alpha^{9} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{126} & 0 \\ 0 & \alpha^{126} \\ \end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{56}.\PGL(2,13)$ |
| Order: | \(122304\)\(\medspace = 2^{6} \cdot 3 \cdot 7^{2} \cdot 13 \) |
| Exponent: | \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3\times C_6\times \PSL(2,13).C_2$, of order \(104832\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $273$ |
| Möbius function | $-2$ |
| Projective image | not computed |