Subgroup ($H$) information
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(234\)\(\medspace = 2 \cdot 3^{2} \cdot 13 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$a, c$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $\He_3:D_{26}$ |
| Order: | \(1404\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 13 \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{13}\times C_3^2:\GL(2,3)$, of order \(67392\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3744\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 13 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $9$ |
| Möbius function | $39$ |
| Projective image | $C_{39}:D_6$ |