Subgroup ($H$) information
| Description: | $C_{13}:C_3$ |
| Order: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Generators: |
$a^{4}, b^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Ambient group ($G$) information
| Description: | $C_{26}:C_{12}$ |
| Order: | \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| Exponent: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_4\times F_{13}$, of order \(1248\)\(\medspace = 2^{5} \cdot 3 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_{13}:C_6$, of order \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
Related subgroups
| Centralizer: | $C_2^2$ | ||
| Normalizer: | $C_{26}:C_{12}$ | ||
| Complements: | $C_2\times C_4$ | ||
| Minimal over-subgroups: | $C_{13}:C_6$ | $C_{13}:C_6$ | $C_{13}:C_6$ |
| Maximal under-subgroups: | $C_{13}$ | $C_3$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{26}:C_{12}$ |