Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(378\)\(\medspace = 2 \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(2\) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{23} & \alpha^{50} \\ \alpha^{35} & \alpha^{23} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{44} & \alpha^{5} \\ \alpha^{53} & \alpha^{44} \\ \end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_3\times \SL(2,8)$ |
| Order: | \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, an A-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SL(2,8):C_6$, of order \(3024\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_3$, of order \(3\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^2\times C_6$ | |
| Normalizer: | $C_2^2\times C_6$ | |
| Normal closure: | $\SL(2,8)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_2\times C_6$ | $C_2^3$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $63$ |
| Möbius function | $0$ |
| Projective image | $C_3\times \SL(2,8)$ |