Subgroup ($H$) information
| Description: | $C_3:C_{72}$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
8 & 34 \\
19 & 30
\end{array}\right), \left(\begin{array}{rr}
31 & 35 \\
19 & 6
\end{array}\right), \left(\begin{array}{rr}
36 & 0 \\
0 & 36
\end{array}\right), \left(\begin{array}{rr}
4 & 0 \\
0 & 4
\end{array}\right), \left(\begin{array}{rr}
27 & 0 \\
0 & 27
\end{array}\right), \left(\begin{array}{rr}
2 & 0 \\
0 & 2
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $\SL(2,3):C_{36}$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_6\times S_4$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $C_6^2:C_2^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $C_6^2:C_2^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{36}$ | ||
| Normalizer: | $C_3:C_{72}$ | ||
| Normal closure: | $\SL(2,3):C_{36}$ | ||
| Core: | $C_{36}$ | ||
| Minimal over-subgroups: | $\SL(2,3):C_{36}$ | ||
| Maximal under-subgroups: | $C_3\times C_{36}$ | $C_3:C_{24}$ | $C_{72}$ |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $-1$ |
| Projective image | $S_4$ |