Subgroup ($H$) information
| Description: | $C_4^3.C_8$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$\left(\begin{array}{rr}
29 & 6 \\
18 & 19
\end{array}\right), \left(\begin{array}{rr}
7 & 8 \\
24 & 15
\end{array}\right), \left(\begin{array}{rr}
25 & 0 \\
24 & 9
\end{array}\right), \left(\begin{array}{rr}
1 & 16 \\
16 & 1
\end{array}\right)$
|
| 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: | $C_4^3:C_{48}$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2.(C_2^3\times C_{12}).C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2^9.C_2^6.C_2^5$ |
| $\card{\operatorname{res}(S)}$ | \(8192\)\(\medspace = 2^{13} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
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 | $C_4^2:C_6$ |