Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(8,12)(9,15)(10,14)(11,13)\rangle$
|
| 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: | $C_3\times C_2^5:S_4$ |
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_6.C_2^5$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\operatorname{res}(S)$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^2:S_4\times C_6$ | ||||||
| Normalizer: | $C_2^2:S_4\times C_6$ | ||||||
| Normal closure: | $C_2^3$ | ||||||
| Core: | $C_1$ | ||||||
| Minimal over-subgroups: | $C_6$ | $C_6$ | $C_6$ | $C_2^2$ | $C_2^2$ | $C_2^2$ | $C_2^2$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3\times C_2^5:S_4$ |