Subgroup ($H$) information
| Description: | $C_{111}:C_9$ |
| Order: | \(999\)\(\medspace = 3^{3} \cdot 37 \) |
| Index: | \(3\) |
| Exponent: | \(333\)\(\medspace = 3^{2} \cdot 37 \) |
| Generators: |
$ab^{2}, b^{111}, b^{9}, a^{3}b^{330}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 3$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{333}:C_9$ |
| Order: | \(2997\)\(\medspace = 3^{4} \cdot 37 \) |
| Exponent: | \(333\)\(\medspace = 3^{2} \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 3$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{333}.(C_6\times C_{36})$ |
| $\operatorname{Aut}(H)$ | $S_3\times F_{37}$, of order \(7992\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 37 \) |
| $\operatorname{res}(S)$ | $S_3\times F_{37}$, of order \(7992\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 37 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_{37}:C_9$, of order \(333\)\(\medspace = 3^{2} \cdot 37 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_{111}:C_9$ |