Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(3\) |
| Generators: |
$d^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $(C_3^2\times C_9).S_4$ |
| 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: | $C_3^3:S_4$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $S_3\times C_6^2:D_6$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Outer Automorphisms: | $S_3^2$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6^2.C_3^3.C_6$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_6^2.C_3^3$ | |||||||||||||
| Normalizer: | $(C_3^2\times C_9).S_4$ | |||||||||||||
| Minimal over-subgroups: | $C_3^2$ | $C_3^2$ | $C_3^2$ | $C_3^2$ | $C_9$ | $C_9$ | $C_9$ | $C_9$ | $C_9$ | $C_9$ | $C_9$ | $C_9$ | $C_6$ | $S_3$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $(C_3^2\times C_9).S_4$ |