Subgroup ($H$) information
| Description: | $C_6\times C_{36}$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(74\)\(\medspace = 2 \cdot 37 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$b^{111}, a^{12}, b^{222}, a^{18}, a^{4}, b^{148}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and metacyclic.
Ambient group ($G$) information
| Description: | $C_{12}\times F_{37}$ |
| Order: | \(15984\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 37 \) |
| Exponent: | \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{222}.C_{18}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_6^2:D_6$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $37$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $F_{37}$ |