Subgroup ($H$) information
| Description: | $C_{42}$ |
| Order: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Index: | \(5676\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \cdot 43 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
7 & 0 \\
0 & 7
\end{array}\right), \left(\begin{array}{rr}
40 & 40 \\
30 & 18
\end{array}\right), \left(\begin{array}{rr}
6 & 0 \\
0 & 6
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_3\times \SL(2,43)$ |
| Order: | \(238392\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 43 \) |
| Exponent: | \(39732\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 43 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \PSL(2,43).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $946$ |
| Möbius function | $0$ |
| Projective image | $\PSL(2,43)$ |