Subgroup ($H$) information
| Description: | $D_7$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Index: | \(36288\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\langle(2,4)(3,9)(5,7)(6,8)(11,17)(12,16)(13,15)(14,18), (1,4,5,8,6,7,2)(10,11,15,12,16,13,17)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $\SOPlus(4,8)$ |
| Order: | \(508032\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 7^{2} \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SL(2,8)^2:C_6$, of order \(1524096\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $W$ | $D_7$, of order \(14\)\(\medspace = 2 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $D_{14}$ | ||
| Normal closure: | $\SL(2,8)^2$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $\SL(2,8)$ | $C_7:D_7$ | $D_{14}$ |
| Maximal under-subgroups: | $C_7$ | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $18144$ |
| Möbius function | $-1$ |
| Projective image | $\SOPlus(4,8)$ |