Subgroup ($H$) information
| Description: | $C_6^2$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$c^{3}, d^{3}, d^{2}, c^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and metacyclic.
Ambient group ($G$) information
| Description: | $C_9\times C_3^2:S_4$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| 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: | $S_3\times C_9$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Automorphism Group: | $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2:C_3.C_3^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $S_3\times \GL(2,3)$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_9\times C_3:S_4$ |