Subgroup ($H$) information
| Description: | $C_{39}$ |
| Order: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Index: | \(79\) |
| Exponent: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Generators: |
$a, b^{79}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is maximal, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,13$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_{1027}:C_3$ |
| Order: | \(3081\)\(\medspace = 3 \cdot 13 \cdot 79 \) |
| Exponent: | \(3081\)\(\medspace = 3 \cdot 13 \cdot 79 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{79}:(C_6\times C_{156})$, of order \(73944\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 13 \cdot 79 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_{12}$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{39}$ | |
| Normalizer: | $C_{39}$ | |
| Normal closure: | $C_{1027}:C_3$ | |
| Core: | $C_{13}$ | |
| Minimal over-subgroups: | $C_{1027}:C_3$ | |
| Maximal under-subgroups: | $C_{13}$ | $C_3$ |
Other information
| Number of subgroups in this conjugacy class | $79$ |
| Möbius function | $-1$ |
| Projective image | $C_{79}:C_3$ |