Subgroup ($H$) information
| Description: | $\GL(2,3):D_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | $1$ |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
16 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
25 & 36 \\
6 & 25
\end{array}\right), \left(\begin{array}{rr}
22 & 45 \\
27 & 46
\end{array}\right), \left(\begin{array}{rr}
4 & 34 \\
0 & 38
\end{array}\right), \left(\begin{array}{rr}
35 & 0 \\
0 & 35
\end{array}\right), \left(\begin{array}{rr}
16 & 0 \\
0 & 16
\end{array}\right), \left(\begin{array}{rr}
34 & 42 \\
36 & 34
\end{array}\right), \left(\begin{array}{rr}
38 & 17 \\
17 & 4
\end{array}\right)$
|
| Derived length: | $4$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, and a Hall subgroup.
Ambient group ($G$) information
| Description: | $\GL(2,3):D_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6\times S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2^6\times S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $W$ | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $C_2^2\times S_4$ |