Subgroup ($H$) information
| Description: | $C_{15}\times \SL(2,11)$ |
| Order: | \(19800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
5 & 4 & 8 & 1 \\
10 & 1 & 10 & 0 \\
9 & 6 & 8 & 5 \\
0 & 3 & 2 & 5
\end{array}\right), \left(\begin{array}{rrrr}
8 & 7 & 2 & 8 \\
10 & 7 & 3 & 5 \\
9 & 8 & 4 & 2 \\
9 & 3 & 2 & 4
\end{array}\right), \left(\begin{array}{rrrr}
4 & 1 & 2 & 3 \\
4 & 2 & 2 & 9 \\
4 & 4 & 6 & 9 \\
1 & 1 & 2 & 10
\end{array}\right), \left(\begin{array}{rrrr}
9 & 0 & 0 & 0 \\
0 & 9 & 0 & 0 \\
0 & 0 & 9 & 0 \\
0 & 0 & 0 & 9
\end{array}\right), \left(\begin{array}{rrrr}
8 & 10 & 2 & 9 \\
4 & 10 & 2 & 10 \\
1 & 9 & 8 & 3 \\
6 & 7 & 6 & 9
\end{array}\right)$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and nonsolvable.
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.
Quotient group ($Q$) structure
| Description: | $Q_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^5\times S_3).C_2^2.\PSL(2,11).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4\times \PSL(2,11).C_2$ |
| $W$ | $C_2\times \PSL(2,11)$, of order \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | not computed |