Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(7,8)(10,12)(13,15)\rangle$
|
| 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, simple, and rational.
Ambient group ($G$) information
| Description: | $C_3:S_3^4$ |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3^5.D_6$, of order \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\operatorname{res}(S)$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $162$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | not computed |
| Projective image | $C_3:S_3^4$ |