Subgroup ($H$) information
| Description: | $\He_3:C_4$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rrr}
0 & z_{2} & 0 \\
1 & 0 & 0 \\
0 & 0 & z_{2} + 1
\end{array}\right), \left(\begin{array}{rrr}
z_{2} & z_{2} & z_{2} \\
z_{2} + 1 & 0 & z_{2} \\
1 & 0 & z_{2}
\end{array}\right), \left(\begin{array}{rrr}
0 & z_{2} + 1 & 1 \\
z_{2} & 0 & z_{2} \\
1 & z_{2} + 1 & 1
\end{array}\right), \left(\begin{array}{rrr}
0 & z_{2} + 1 & 0 \\
z_{2} & 1 & 0 \\
0 & z_{2} & 1
\end{array}\right), \left(\begin{array}{rrr}
z_{2} + 1 & 0 & 0 \\
0 & z_{2} + 1 & 0 \\
0 & 0 & z_{2} + 1
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and solvable.
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 \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_3^2:C_4$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3$ | |
| Normalizer: | $\He_3:C_4$ | |
| Normal closure: | $C_3.A_6$ | |
| Core: | $C_3$ | |
| Minimal over-subgroups: | $C_3.A_6$ | |
| Maximal under-subgroups: | $C_3^2:S_3$ | $C_{12}$ |
Other information
| Number of subgroups in this conjugacy class | $10$ |
| Möbius function | $-1$ |
| Projective image | $A_6$ |