Subgroup ($H$) information
| Description: | $C_{13}:C_3$ |
| Order: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Index: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Exponent: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Generators: |
$a, bc^{21}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Ambient group ($G$) information
| Description: | $C_{13}^2:C_3^2$ |
| Order: | \(1521\)\(\medspace = 3^{2} \cdot 13^{2} \) |
| Exponent: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times C_{13}^2.C_{12}.\PSL(2,13).C_2$ |
| $\operatorname{Aut}(H)$ | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| $\operatorname{res}(S)$ | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| $W$ | $C_{13}:C_3$, of order \(39\)\(\medspace = 3 \cdot 13 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $13$ |
| Möbius function | $1$ |
| Projective image | $C_{13}^2:C_3^2$ |