Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(23760\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(2,3)(4,11)(5,8)(7,12), (2,4,3,11)(5,12,8,7)\rangle$
|
| 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) and a $p$-group.
Ambient group ($G$) information
| Description: | $M_{12}$ |
| Order: | \(95040\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $M_{12}:C_2$, of order \(190080\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $1485$ |
| Möbius function | $16$ |
| Projective image | $M_{12}$ |