Subgroup ($H$) information
| Description: | $C_{13}:C_4$ |
| Order: | \(52\)\(\medspace = 2^{2} \cdot 13 \) |
| Index: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Exponent: | \(52\)\(\medspace = 2^{2} \cdot 13 \) |
| Generators: |
$a, cd^{9}, b^{6}c^{4}d^{11}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 \(24336\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
| $\operatorname{Aut}(H)$ | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| $\operatorname{res}(S)$ | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_{13}:C_4$, of order \(52\)\(\medspace = 2^{2} \cdot 13 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $39$ |
| Möbius function | $0$ |
| Projective image | $C_{13}^2:(C_4\times S_3)$ |