Subgroup ($H$) information
| Description: | $\OD_{16}:C_{34}$ |
| Order: | \(544\)\(\medspace = 2^{5} \cdot 17 \) |
| Index: | \(3\) |
| Exponent: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 0 \\
0 & 408
\end{array}\right), \left(\begin{array}{rr}
408 & 0 \\
0 & 408
\end{array}\right), \left(\begin{array}{rr}
48 & 0 \\
0 & 48
\end{array}\right), \left(\begin{array}{rr}
354 & 0 \\
0 & 354
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right), \left(\begin{array}{rr}
259 & 0 \\
0 & 259
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a Hall subgroup, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $\OD_{16}:C_{102}$ |
| Order: | \(1632\)\(\medspace = 2^{5} \cdot 3 \cdot 17 \) |
| Exponent: | \(408\)\(\medspace = 2^{3} \cdot 3 \cdot 17 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_8\times S_4).C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{16}\times S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $(C_2^2\times C_{16}) \times S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_2\times C_6$ |