Subgroup ($H$) information
Description: | $C_4\times C_{48}$ |
Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
Index: | \(16\)\(\medspace = 2^{4} \) |
Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Generators: |
$\left(\begin{array}{rr}
29 & 30 \\
10 & 19
\end{array}\right), \left(\begin{array}{rr}
25 & 0 \\
0 & 25
\end{array}\right), \left(\begin{array}{rr}
17 & 16 \\
16 & 1
\end{array}\right), \left(\begin{array}{rr}
7 & 8 \\
24 & 15
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
3 & 21 \\
7 & 28
\end{array}\right), \left(\begin{array}{rr}
21 & 0 \\
0 & 21
\end{array}\right)$
|
Nilpotency class: | $1$ |
Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
$\operatorname{Aut}(G)$ | $C_4^2.(C_2^3\times C_{12}).C_2^6$ |
$\operatorname{Aut}(H)$ | $C_2^6.D_4$, of order \(512\)\(\medspace = 2^{9} \) |
$\operatorname{res}(S)$ | $C_2^6.D_4$, of order \(512\)\(\medspace = 2^{9} \) |
$\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
$W$ | $C_1$, of order $1$ |
Related subgroups
Other information
Number of subgroups in this autjugacy class | $16$ |
Number of conjugacy classes in this autjugacy class | $1$ |
Möbius function | not computed |
Projective image | $C_4^2:C_6$ |