Subgroup ($H$) information
| Description: | $C_7$ |
| Order: | \(7\) |
| Index: | \(63360\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(7\) |
| Generators: |
$\langle(1,18,6,2,12,14,17)(3,16,13,11,20,10,19)(4,5,8,15,21,22,9)\rangle$
|
| 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 $7$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $M_{22}$ |
| Order: | \(443520\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \) |
| Exponent: | \(9240\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $M_{22}:C_2$, of order \(887040\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_7$ | |
| Normalizer: | $C_7:C_3$ | |
| Normal closure: | $M_{22}$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $F_8$ | $C_7:C_3$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this conjugacy class | $21120$ |
| Möbius function | $0$ |
| Projective image | $M_{22}$ |