Subgroup ($H$) information
| Description: | $C_{13}$ |
| Order: | \(13\) |
| Index: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Exponent: | \(13\) |
| Generators: |
$bc^{4}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, 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: | $C_{13}^2:C_3$ |
| Order: | \(507\)\(\medspace = 3 \cdot 13^{2} \) |
| Exponent: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{13}:C_3$ |
| Order: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Exponent: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Automorphism Group: | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{13}^2.\GL(2,13)$, of order \(4429152\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \cdot 13^{3} \) |
| $\operatorname{Aut}(H)$ | $C_{12}$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_{12}$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(26364\)\(\medspace = 2^{2} \cdot 3 \cdot 13^{3} \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
Other information
| Möbius function | $13$ |
| Projective image | $C_{13}^2:C_3$ |