Subgroup ($H$) information
| Description: | $C_{273}$ |
| Order: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
| Index: | \(416\)\(\medspace = 2^{5} \cdot 13 \) |
| Exponent: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
| Generators: |
$a^{28}, bc^{24}, a^{12}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,7,13$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{13}^2:(C_4\times C_{168})$ |
| Order: | \(113568\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 13^{2} \) |
| Exponent: | \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{104}:C_4$ |
| Order: | \(416\)\(\medspace = 2^{5} \cdot 13 \) |
| Exponent: | \(104\)\(\medspace = 2^{3} \cdot 13 \) |
| Automorphism Group: | $C_2\times D_4\times F_{13}$, of order \(2496\)\(\medspace = 2^{6} \cdot 3 \cdot 13 \) |
| Outer Automorphisms: | $C_6\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(850397184\)\(\medspace = 2^{11} \cdot 3^{3} \cdot 7 \cdot 13^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_6\times C_{12}$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |