Subgroup ($H$) information
| Description: | $C_{11}$ |
| Order: | \(11\) |
| Index: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Exponent: | \(11\) |
| Generators: |
$a^{952}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $11$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{5236}$ |
| Order: | \(5236\)\(\medspace = 2^{2} \cdot 7 \cdot 11 \cdot 17 \) |
| Exponent: | \(5236\)\(\medspace = 2^{2} \cdot 7 \cdot 11 \cdot 17 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7,11,17$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Quotient group ($Q$) structure
| Description: | $C_{476}$ |
| Order: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Exponent: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Automorphism Group: | $C_2^2\times C_{48}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Outer Automorphisms: | $C_2^2\times C_{48}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7,17$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3\times C_{240}$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{5236}$ | ||
| Normalizer: | $C_{5236}$ | ||
| Complements: | $C_{476}$ | ||
| Minimal over-subgroups: | $C_{187}$ | $C_{77}$ | $C_{22}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{476}$ |