Subgroup ($H$) information
| Description: | $C_{165}:Q_8$ |
| Order: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Index: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 6 & 0 & 2 \\
10 & 7 & 9 & 10 \\
6 & 2 & 9 & 2 \\
5 & 8 & 8 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 4 & 3 & 8 \\
6 & 4 & 3 & 4 \\
7 & 8 & 1 & 10 \\
9 & 5 & 9 & 8
\end{array}\right), \left(\begin{array}{rrrr}
8 & 7 & 2 & 8 \\
10 & 7 & 3 & 5 \\
9 & 8 & 4 & 2 \\
9 & 3 & 2 & 4
\end{array}\right), \left(\begin{array}{rrrr}
9 & 0 & 0 & 0 \\
0 & 9 & 0 & 0 \\
0 & 0 & 9 & 0 \\
0 & 0 & 0 & 9
\end{array}\right), \left(\begin{array}{rrrr}
9 & 2 & 5 & 2 \\
3 & 9 & 9 & 0 \\
2 & 1 & 4 & 5 \\
8 & 1 & 10 & 6
\end{array}\right), \left(\begin{array}{rrrr}
2 & 10 & 5 & 8 \\
0 & 0 & 8 & 4 \\
5 & 6 & 3 & 9 \\
8 & 7 & 7 & 8
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{15}:Q_8\times \SL(2,11)$ |
| Order: | \(158400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^5\times S_3).C_2^2.\PSL(2,11).C_2$ |
| $\operatorname{Aut}(H)$ | $C_{15}:(C_4^2.C_2^3)$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| $W$ | $S_3\times C_{10}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $12$ |
| Möbius function | not computed |
| Projective image | not computed |