Subgroup ($H$) information
| Description: | $C_2^2:D_4^2$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(3\) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
11 & 5 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
31 & 9 \\
0 & 19
\end{array}\right), \left(\begin{array}{rr}
53 & 42 \\
36 & 17
\end{array}\right), \left(\begin{array}{rr}
41 & 0 \\
30 & 11
\end{array}\right), \left(\begin{array}{rr}
7 & 30 \\
24 & 43
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $D_6:D_4^2$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(786432\)\(\medspace = 2^{18} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2^{12}.(C_2^2\times S_4)$, of order \(393216\)\(\medspace = 2^{17} \cdot 3 \) |
| $\card{W}$ | \(32\)\(\medspace = 2^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | not computed |