Subgroup ($H$) information
| Description: | $D_4\times C_3^2$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left[ \left(\begin{array}{rrrr}
6 & 1 & -1 & -1 \\
-1 & 6 & -1 & -1 \\
6 & 1 & 2 & 1 \\
-1 & 2 & -1 & 2
\end{array}\right) \right], \left[ \left(\begin{array}{rrrr}
4 & 7 & 4 & 3 \\
-1 & 4 & -1 & 0 \\
4 & 7 & 0 & 7 \\
-1 & 0 & -1 & 0
\end{array}\right) \right], \left[ \left(\begin{array}{rrrr}
0 & 3 & 4 & 3 \\
-1 & 0 & -1 & 0 \\
4 & 7 & 4 & 3 \\
-1 & 0 & -1 & 4
\end{array}\right) \right], \left[ \left(\begin{array}{rrrr}
0 & 0 & -1 & -1 \\
-1 & 0 & -1 & -1 \\
-1 & -1 & 0 & 4 \\
-1 & -1 & -1 & 0
\end{array}\right) \right], \left[ \left(\begin{array}{rrrr}
0 & 6 & -1 & -1 \\
-1 & 0 & -1 & -1 \\
-1 & -1 & 0 & 2 \\
-1 & -1 & -1 & 0
\end{array}\right) \right]$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $\PSOPlus(4,9)$ |
| Order: | \(259200\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_6^2.D_4$, of order \(4147200\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $D_4\times \GL(2,3)$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $W$ | $C_2^2:C_8$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $900$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $\PSOPlus(4,9)$ |