Subgroup ($H$) information
| Description: | $C_3\times \SL(2,43)$ |
| Order: | \(238392\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 43 \) |
| Index: | $1$ |
| Exponent: | \(39732\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 43 \) |
| Generators: |
$\left(\begin{array}{rr}
42 & 15 \\
9 & 1
\end{array}\right), \left(\begin{array}{rr}
7 & 0 \\
0 & 7
\end{array}\right), \left(\begin{array}{rr}
1 & 4 \\
34 & 13
\end{array}\right), \left(\begin{array}{rr}
6 & 0 \\
0 & 6
\end{array}\right)$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a Hall subgroup, and nonsolvable.
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.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \PSL(2,43).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times \PSL(2,43).C_2$ |
| $W$ | $\PSL(2,43)$, of order \(39732\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 43 \) |
Related subgroups
| Centralizer: | $C_6$ | ||||
| Normalizer: | $C_3\times \SL(2,43)$ | ||||
| Complements: | $C_1$ | ||||
| Maximal under-subgroups: | $\SL(2,43)$ | $C_{129}:C_{42}$ | $C_{33}:Q_8$ | $C_{21}:C_{12}$ | $C_3\times \SL(2,3)$ |
Other information
| Möbius function | $1$ |
| Projective image | $\PSL(2,43)$ |