Subgroup ($H$) information
| Description: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Index: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(2,6,5)(3,7,4), (10,11,12)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $A_4\times D_6\times A_5$ |
| Order: | \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6\times S_4\times S_5$, of order \(34560\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $120$ |
| Projective image | $A_4\times D_6\times A_5$ |