Subgroup ($H$) information
| Description: | $C_{18}:C_4$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$b, c^{132}, c^{99}, b^{2}, c^{176}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $D_{22}.D_{18}$ |
| Order: | \(1584\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $D_{11}$ |
| Order: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Automorphism Group: | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_5$, of order \(5\) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{99}.C_{30}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $D_{36}:C_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_{18}:C_6$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| $W$ | $D_{18}$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
Related subgroups
Other information
| Möbius function | $11$ |
| Projective image | $D_9\times D_{22}$ |