Subgroup ($H$) information
| Description: | $S_3\times C_7$ |
| Order: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Index: | \(6048\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(1,9)(2,6)(4,7)(5,8)(10,15,12,18,14,13,16), (1,3,9)(2,8,4)(5,6,7)(10,16,13,14,18,12,15), (10,14,15,13,12,16,18)\rangle$
|
| 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: | $\SL(2,8)^2$ |
| Order: | \(254016\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 7^{2} \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Derived length: | $0$ |
The ambient group is nonabelian, an A-group, and perfect (hence 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | ${}^2G(2,3)\wr C_2$, of order \(4572288\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_7$ | ||
| Normalizer: | $S_3\times D_7$ | ||
| Normal closure: | $\SL(2,8)^2$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $S_3\times F_8$ | $C_7\times D_9$ | $S_3\times D_7$ |
| Maximal under-subgroups: | $C_{21}$ | $C_{14}$ | $S_3$ |
Other information
| Number of subgroups in this autjugacy class | $6048$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $\SL(2,8)^2$ |