Subgroup ($H$) information
| Description: | $C_2\times C_4\times C_{48}$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
19 & 6 \\
2 & 17
\end{array}\right), \left(\begin{array}{rr}
21 & 24 \\
8 & 13
\end{array}\right), \left(\begin{array}{rr}
17 & 16 \\
16 & 1
\end{array}\right), \left(\begin{array}{rr}
9 & 8 \\
24 & 17
\end{array}\right), \left(\begin{array}{rr}
3 & 21 \\
7 & 28
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
25 & 16 \\
16 & 9
\end{array}\right), \left(\begin{array}{rr}
31 & 0 \\
0 & 31
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_8\times C_2^3:C_{48}$ |
| 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), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4:C_4^2:C_2^2$, of order \(786432\)\(\medspace = 2^{18} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2.C_2^6.C_2^6$ |
| $\card{W}$ | $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |