Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(235200\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2} \) |
| Exponent: | \(3\) |
| Generators: |
$\left(\begin{array}{rrrr}
4 & 0 & 0 & 0 \\
0 & 4 & 0 & 0 \\
0 & 0 & 4 & 0 \\
0 & 0 & 0 & 4
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, and simple. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $\SL(2,49).C_6$ |
| Order: | \(705600\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
| Exponent: | \(8400\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $\SL(2,49):C_2$ |
| Order: | \(235200\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2} \) |
| Exponent: | \(8400\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Automorphism Group: | $\PSL(2,49):C_2^2$, of order \(235200\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $1$ |
The quotient 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times \PSL(2,49).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |