Subgroup ($H$) information
| Description: | $F_8$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{23} & \alpha^{50} \\ \alpha^{35} & \alpha^{23} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{41} & \alpha^{20} \\ \alpha^{61} & \alpha^{13} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{44} & \alpha^{5} \\ \alpha^{53} & \alpha^{44} \\ \end{array}\right), \left(\begin{array}{ll}\alpha & \alpha^{32} \\ \alpha^{17} & \alpha \\ \end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Hall subgroup, monomial (hence solvable), metabelian, and an A-group.
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.
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 27T390.
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 \(3024\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $F_8:C_3$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $F_8:C_3$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $F_8$, of order \(56\)\(\medspace = 2^{3} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_3$ | |
| Normalizer: | $C_3\times F_8$ | |
| Normal closure: | $\SL(2,8)$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $\SL(2,8)$ | $C_3\times F_8$ |
| Maximal under-subgroups: | $C_2^3$ | $C_7$ |
Other information
| Number of subgroups in this conjugacy class | $9$ |
| Möbius function | $1$ |
| Projective image | $C_3\times \SL(2,8)$ |