Subgroup ($H$) information
| Description: | $C_{12}$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rrr}
0 & z_{2} & 0 \\
1 & 0 & 0 \\
0 & 0 & z_{2} + 1
\end{array}\right), \left(\begin{array}{rrr}
0 & z_{2} + 1 & 1 \\
z_{2} & 0 & z_{2} \\
1 & z_{2} + 1 & 1
\end{array}\right), \left(\begin{array}{rrr}
z_{2} + 1 & 0 & 0 \\
0 & z_{2} + 1 & 0 \\
0 & 0 & z_{2} + 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_3.A_6$ |
| Order: | \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and quasisimple (hence nonsolvable and perfect).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\operatorname{res}(S)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{12}$ | |
| Normalizer: | $C_3\times D_4$ | |
| Normal closure: | $C_3.A_6$ | |
| Core: | $C_3$ | |
| Minimal over-subgroups: | $\He_3:C_4$ | $C_3\times D_4$ |
| Maximal under-subgroups: | $C_6$ | $C_4$ |
Other information
| Number of subgroups in this conjugacy class | $45$ |
| Möbius function | $2$ |
| Projective image | $A_6$ |