Subgroup ($H$) information
| Description: | $D_9:C_3^3$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Index: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$d^{9}, e^{3}, e^{7}, a^{2}d^{14}, c^{3}d^{12}, d^{6}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $D_9^3:C_6$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $\SOPlus(4,2)$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9^3.C_4.C_6^2.C_2$, of order \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $D_9:C_3\times \AGL(2,3)$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $W$ | $(D_9\times S_3^2):C_6$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $D_9^3:C_6$ |