Subgroup ($H$) information
| Description: | $C_3^2:D_{18}$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$b^{3}e^{4}, e^{2}, d^{3}, e^{3}, ce^{4}, d^{4}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_2\times C_3^3.S_3^2$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_3^3.C_3^4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_3^4.S_3^2$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_3^3.S_3^2$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $W$ | $C_3^2.S_3^2$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
Related subgroups
Other information
| Möbius function | $3$ |
| Projective image | $C_3^3.S_3^2$ |