Subgroup ($H$) information
| Description: | $C_2\times C_{52}$ |
| Order: | \(104\)\(\medspace = 2^{3} \cdot 13 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(52\)\(\medspace = 2^{2} \cdot 13 \) |
| Generators: |
$b^{4}, c^{2}, b^{8}, c^{13}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $D_{52}.C_8$ |
| Order: | \(832\)\(\medspace = 2^{6} \cdot 13 \) |
| Exponent: | \(208\)\(\medspace = 2^{4} \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{13}.(C_4^2\times C_{12}).C_2$ |
| $\operatorname{Aut}(H)$ | $D_4\times C_{12}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(208\)\(\medspace = 2^{4} \cdot 13 \) |
| $W$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_2\times C_{52}$ | |||
| Normalizer: | $D_{52}.C_8$ | |||
| Minimal over-subgroups: | $D_{52}:C_2$ | $C_{13}\times \OD_{16}$ | $C_{26}:C_8$ | |
| Maximal under-subgroups: | $C_2\times C_{26}$ | $C_{52}$ | $C_{52}$ | $C_2\times C_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $D_{26}:C_4$ |