Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(507\)\(\medspace = 3 \cdot 13^{2} \) |
| Exponent: | \(3\) |
| Generators: |
$a^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), the Frattini subgroup, cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{13}^2:C_9$ |
| Order: | \(1521\)\(\medspace = 3^{2} \cdot 13^{2} \) |
| Exponent: | \(117\)\(\medspace = 3^{2} \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{13}^2:C_3$ |
| Order: | \(507\)\(\medspace = 3 \cdot 13^{2} \) |
| Exponent: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Automorphism Group: | $C_{13}^2.\GL(2,13)$, of order \(4429152\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \cdot 13^{3} \) |
| Outer Automorphisms: | $\SL(2,13):C_4$, of order \(8736\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 13 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times C_{13}^2.C_{12}.\PSL(2,13).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(13287456\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 7 \cdot 13^{3} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $-2197$ |
| Projective image | $C_{13}^2:C_3$ |