Subgroup ($H$) information
| Description: | $C_{37}:C_9$ |
| Order: | \(333\)\(\medspace = 3^{2} \cdot 37 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(333\)\(\medspace = 3^{2} \cdot 37 \) |
| Generators: |
$a, a^{3}, c^{2}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Ambient group ($G$) information
| Description: | $(C_2\times C_{74}):C_9$ |
| Order: | \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
| Exponent: | \(666\)\(\medspace = 2 \cdot 3^{2} \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{37}.(C_{36}\times A_4)$ |
| $\operatorname{Aut}(H)$ | $F_{37}$, of order \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
| $\operatorname{res}(S)$ | $F_{37}$, of order \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_{37}:C_9$, of order \(333\)\(\medspace = 3^{2} \cdot 37 \) |
Related subgroups
| Centralizer: | $C_1$ | |
| Normalizer: | $C_{37}:C_9$ | |
| Normal closure: | $(C_2\times C_{74}):C_9$ | |
| Core: | $C_{37}:C_3$ | |
| Minimal over-subgroups: | $(C_2\times C_{74}):C_9$ | |
| Maximal under-subgroups: | $C_{37}:C_3$ | $C_9$ |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $-1$ |
| Projective image | $(C_2\times C_{74}):C_9$ |