Subgroup ($H$) information
| Description: | $C_{19}\times D_{105}$ |
| Order: | \(3990\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 19 \) |
| Index: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(3990\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 19 \) |
| Generators: |
$b^{17556}, b^{37620}, b^{14630}, b^{2310}, a$
|
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{209}\times D_{210}$ |
| Order: | \(87780\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \) |
| Exponent: | \(43890\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{22}$ |
| Order: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| 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 ($p = 2,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_{90}\times F_5\times S_3\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_5:F_5\times S_4$, of order \(90720\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| $W$ | $D_{105}$, of order \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $C_{11}\times D_{210}$ |