Subgroup ($H$) information
| Description: | $C_{11}$ |
| Order: | \(11\) |
| Index: | \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Exponent: | \(11\) |
| Generators: |
$b^{105}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a direct factor, cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), a $11$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{11}\times D_{105}$ |
| Order: | \(2310\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Exponent: | \(2310\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $D_{105}$ |
| Order: | \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Exponent: | \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Automorphism Group: | $S_3\times F_5\times F_7$, of order \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Outer Automorphisms: | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{10}\times F_5\times S_3\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{11}\times D_{105}$ | |||
| Normalizer: | $C_{11}\times D_{105}$ | |||
| Complements: | $D_{105}$ | |||
| Minimal over-subgroups: | $C_{77}$ | $C_{55}$ | $C_{33}$ | $C_{22}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $105$ |
| Projective image | $D_{105}$ |