Subgroup ($H$) information
| Description: | $C_{37}:C_9$ |
| Order: | \(333\)\(\medspace = 3^{2} \cdot 37 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(333\)\(\medspace = 3^{2} \cdot 37 \) |
| Generators: |
$a^{4}b, a^{12}b^{9}, b^{3}$
|
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Ambient group ($G$) information
| Description: | $C_{111}:C_{36}$ |
| Order: | \(3996\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 37 \) |
| 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), hyperelementary for $p = 3$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{12}$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{111}.C_{18}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_{37}$, of order \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
| $\operatorname{res}(S)$ | $F_{37}$, of order \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_{37}:C_9$, of order \(333\)\(\medspace = 3^{2} \cdot 37 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_{111}:C_{36}$ |