Subgroup ($H$) information
| Description: | $C_{51}$ |
| Order: | \(51\)\(\medspace = 3 \cdot 17 \) |
| Index: | \(544\)\(\medspace = 2^{5} \cdot 17 \) |
| Exponent: | \(51\)\(\medspace = 3 \cdot 17 \) |
| Generators: |
$\left(\begin{array}{rr}
309 & 0 \\
0 & 364
\end{array}\right), \left(\begin{array}{rr}
5 & 0 \\
0 & 82
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,17$), 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: | $C_{136}.D_{102}$ |
| Order: | \(27744\)\(\medspace = 2^{5} \cdot 3 \cdot 17^{2} \) |
| Exponent: | \(408\)\(\medspace = 2^{3} \cdot 3 \cdot 17 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $\OD_{16}:C_{34}$ |
| Order: | \(544\)\(\medspace = 2^{5} \cdot 17 \) |
| Exponent: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Automorphism Group: | $C_2^2\times C_{16}\times S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Outer Automorphisms: | $C_{48}:C_2^3$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{51}:((C_2\times C_{16}^2).C_2^5)$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{W}$ | 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 |