Subgroup ($H$) information
| Description: | $C_{11}\times D_{210}$ |
| Order: | \(4620\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Index: | \(19\) |
| Exponent: | \(2310\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Generators: |
$a, b^{37620}, b^{27930}, b^{14630}, b^{21945}, b^{17556}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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: | $C_{19}$ |
| Order: | \(19\) |
| Exponent: | \(19\) |
| Automorphism Group: | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Outer Automorphisms: | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_{90}\times F_5\times S_3\times F_7$ |
| $\operatorname{Aut}(H)$ | $D_5:F_5^2$, of order \(100800\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| $W$ | $D_{105}$, of order \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_{19}\times D_{105}$ |