Subgroup ($H$) information
| Description: | $C_{792}$ |
| Order: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Index: | \(2\) |
| Exponent: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Generators: |
$b^{99}, b^{396}, b^{72}, b^{198}, b^{704}, b^{528}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), maximal, a semidirect factor, and cyclic (hence abelian, elementary ($p = 2,3,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $D_{792}$ |
| Order: | \(1584\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{99}.C_{60}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^3\times C_{30}$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^3\times C_{30}$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{792}$ | ||
| Normalizer: | $D_{792}$ | ||
| Complements: | $C_2$ $C_2$ | ||
| Minimal over-subgroups: | $D_{792}$ | ||
| Maximal under-subgroups: | $C_{396}$ | $C_{264}$ | $C_{72}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $D_{396}$ |