Subgroup ($H$) information
| Description: | $C_{13}$ |
| Order: | \(13\) |
| Index: | \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| Exponent: | \(13\) |
| Generators: |
$cd^{9}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{13}^2:(C_4\times S_3)$ |
| Order: | \(4056\)\(\medspace = 2^{3} \cdot 3 \cdot 13^{2} \) |
| Exponent: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{13}^2.(C_6\times S_3)$, of order \(24336\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
| $\operatorname{Aut}(H)$ | $C_{12}$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_{12}$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(338\)\(\medspace = 2 \cdot 13^{2} \) |
| $W$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $C_{13}^2:(C_4\times S_3)$ |