Subgroup ($H$) information
| Description: | $C_{10}.S_4$ |
| Order: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Index: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rrrr}
3 & 1 & 9 & 6 \\
10 & 2 & 7 & 9 \\
6 & 4 & 9 & 10 \\
2 & 6 & 1 & 8
\end{array}\right), \left(\begin{array}{rrrr}
3 & 2 & 7 & 1 \\
9 & 10 & 7 & 7 \\
1 & 4 & 1 & 9 \\
9 & 1 & 2 & 8
\end{array}\right), \left(\begin{array}{rrrr}
5 & 4 & 9 & 9 \\
1 & 4 & 0 & 9 \\
9 & 9 & 7 & 7 \\
10 & 9 & 10 & 6
\end{array}\right), \left(\begin{array}{rrrr}
9 & 0 & 0 & 0 \\
0 & 9 & 0 & 0 \\
0 & 0 & 9 & 0 \\
0 & 0 & 0 & 9
\end{array}\right), \left(\begin{array}{rrrr}
8 & 0 & 0 & 0 \\
0 & 8 & 0 & 0 \\
0 & 0 & 8 & 0 \\
0 & 0 & 0 & 8
\end{array}\right), \left(\begin{array}{rrrr}
3 & 4 & 3 & 2 \\
9 & 10 & 7 & 3 \\
6 & 5 & 7 & 7 \\
9 & 6 & 2 & 3
\end{array}\right)$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, and solvable.
Ambient group ($G$) information
| Description: | $C_5\times \SL(2,9)$ |
| Order: | \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times S_6:C_2$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^4.D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^4.D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $15$ |
| Möbius function | $-1$ |
| Projective image | $A_6$ |