Subgroup ($H$) information
| Description: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a, c^{13}$
|
| 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), a direct factor, a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_2\times C_4\times F_{13}$ |
| Order: | \(1248\)\(\medspace = 2^{5} \cdot 3 \cdot 13 \) |
| Exponent: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $F_{13}$ |
| Order: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Exponent: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Automorphism Group: | $F_{13}$, of order \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\wr C_2^2\times F_{13}$, of order \(9984\)\(\medspace = 2^{8} \cdot 3 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1248\)\(\medspace = 2^{5} \cdot 3 \cdot 13 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $F_{13}$ |