Subgroup ($H$) information
| Description: | $D_{420}$ |
| Order: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Index: | \(11\) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$a, b^{1320}, b^{2310}, b^{1540}, b^{1155}, b^{2772}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{11}\times D_{420}$ |
| Order: | \(9240\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Exponent: | \(4620\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| 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: | $C_{11}$ |
| Order: | \(11\) |
| Exponent: | \(11\) |
| Automorphism Group: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| 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, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4.C_4^2$, of order \(403200\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_{210}.C_6.C_2^5$ |
| $W$ | $D_{210}$, of order \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_{22}$ | |||||
| Normalizer: | $C_{11}\times D_{420}$ | |||||
| Complements: | $C_{11}$ | |||||
| Minimal over-subgroups: | $C_{11}\times D_{420}$ | |||||
| Maximal under-subgroups: | $D_{210}$ | $D_{210}$ | $C_{420}$ | $D_{140}$ | $D_{84}$ | $D_{60}$ |
Other information
| Möbius function | $-1$ |
| Projective image | not computed |