Subgroup ($H$) information
| Description: | $C_5$ |
| Order: | \(5\) |
| Index: | \(9504\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 11 \) |
| Exponent: | \(5\) |
| Generators: |
$\langle(1,2,6,9,7)(4,11,8,5,10)\rangle$
|
| 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 $5$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $S_3\times M_{11}$ |
| Order: | \(47520\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times M_{11}$, of order \(47520\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_5\times S_3$ | ||||
| Normalizer: | $S_3\times F_5$ | ||||
| Normal closure: | $M_{11}$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $C_{11}:C_5$ | $C_{15}$ | $C_{10}$ | $D_5$ | $D_5$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this conjugacy class | $396$ |
| Möbius function | $0$ |
| Projective image | $S_3\times M_{11}$ |