Subgroup ($H$) information
| Description: | $C_{10}.S_4$ |
| Order: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Index: | \(7455\)\(\medspace = 3 \cdot 5 \cdot 7 \cdot 71 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
25 & 0 \\
0 & 25
\end{array}\right), \left(\begin{array}{rr}
70 & 0 \\
0 & 70
\end{array}\right), \left(\begin{array}{rr}
60 & 69 \\
20 & 10
\end{array}\right), \left(\begin{array}{rr}
40 & 22 \\
40 & 31
\end{array}\right), \left(\begin{array}{rr}
28 & 32 \\
53 & 43
\end{array}\right), \left(\begin{array}{rr}
29 & 69 \\
66 & 42
\end{array}\right)$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, and solvable.
Ambient group ($G$) information
| Description: | $C_5\times \SL(2,71)$ |
| Order: | \(1789200\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 71 \) |
| Exponent: | \(178920\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times \PSL(2,71).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^4.D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $7455$ |
| Möbius function | $-1$ |
| Projective image | $\PSL(2,71)$ |