Subgroup ($H$) information
| Description: | $C_3\times A_4$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Index: | \(12320\)\(\medspace = 2^{5} \cdot 5 \cdot 7 \cdot 11 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,2)(3,4)(5,16)(6,12)(8,20)(9,15)(11,19)(13,21), (1,20)(2,8)(3,11)(4,19) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 \(887040\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3$ | ||||
| Normalizer: | $C_3:S_4$ | ||||
| Normal closure: | $M_{22}$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $A_4^2$ | $C_3:S_4$ | |||
| Maximal under-subgroups: | $C_2\times C_6$ | $A_4$ | $A_4$ | $A_4$ | $C_3^2$ |
Other information
| Number of subgroups in this conjugacy class | $6160$ |
| Möbius function | $0$ |
| Projective image | $M_{22}$ |