Subgroup ($H$) information
| Description: | $C_3^2\times C_9$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$d^{7}f^{5}, e^{3}f^{6}, f^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_3^5.C_3\wr C_2^2$ |
| Order: | \(78732\)\(\medspace = 2^{2} \cdot 3^{9} \) |
| 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: | $(C_3\times C_9):S_3^2$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Automorphism Group: | $S_3\times C_3^3:\GL(2,3)$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Outer Automorphisms: | $C_3\times S_4$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and supersolvable (hence solvable and monomial).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{6176}.C_{32}$, of order \(2125764\)\(\medspace = 2^{2} \cdot 3^{12} \) |
| $\operatorname{Aut}(H)$ | $C_6.C_3^4:\GL(2,3)$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $\card{W}$ | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3\times C_9^3$ | |
| Normalizer: | $C_3^5.C_3\wr C_2^2$ | |
| Minimal over-subgroups: | $C_3^3\times C_9$ | $C_3.C_9^2$ |
| Maximal under-subgroups: | $C_3^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |