Subgroup ($H$) information
| Description: | $C_2^5:C_6$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
5 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 6 \\
6 & 1
\end{array}\right), \left(\begin{array}{rr}
5 & 0 \\
0 & 5
\end{array}\right), \left(\begin{array}{rr}
3 & 4 \\
8 & 3
\end{array}\right), \left(\begin{array}{rr}
7 & 0 \\
6 & 7
\end{array}\right), \left(\begin{array}{rr}
7 & 0 \\
0 & 7
\end{array}\right), \left(\begin{array}{rr}
7 & 9 \\
3 & 4
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $D_4.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4.C_2^6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\wr C_2^2\times S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\card{W}$ | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $2$ |
| Projective image | not computed |