Subgroup ($H$) information
| Description: | $C_3:Q_{16}$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
4 & 15 \\
15 & 28
\end{array}\right), \left(\begin{array}{rr}
3 & 21 \\
7 & 28
\end{array}\right), \left(\begin{array}{rr}
19 & 8 \\
24 & 27
\end{array}\right), \left(\begin{array}{rr}
25 & 16 \\
16 & 9
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_2\times C_4.D_{24}$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 \(8\)\(\medspace = 2^{3} \) |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^2.C_2^6.C_2^3)$ |
| $\operatorname{Aut}(H)$ | $D_8:D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{W}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | not computed |