Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(1,3,2)(7,8,11)(10,13,15)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_3\times S_3\times {}^2G(2,3)$ |
| Order: | \(27216\)\(\medspace = 2^{4} \cdot 3^{5} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Derived length: | $2$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3^2\times {}^2G(2,3)$, of order \(54432\)\(\medspace = 2^{5} \cdot 3^{5} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $84$ |
| Möbius function | $18$ |
| Projective image | $C_3\times S_3\times {}^2G(2,3)$ |