Subgroup ($H$) information
| Description: | $C_3\times C_{192}$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
109 & 0 \\
0 & 85
\end{array}\right), \left(\begin{array}{rr}
108 & 0 \\
0 & 84
\end{array}\right), \left(\begin{array}{rr}
130 & 0 \\
0 & 49
\end{array}\right), \left(\begin{array}{rr}
5 & 0 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
21 & 0 \\
0 & 46
\end{array}\right), \left(\begin{array}{rr}
139 & 0 \\
0 & 25
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 84
\end{array}\right), \left(\begin{array}{rr}
55 & 0 \\
0 & 186
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $D_{192}:C_{12}$ |
| Order: | \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| Exponent: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_{32}.C_8.C_2^6.C_2)$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{16}\times \GL(2,3)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^3\times C_{16}$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_3 \rtimes (C_4\times D_{32})$ |