Subgroup ($H$) information
| Description: | not computed |
| Order: | \(4782969\)\(\medspace = 3^{14} \) |
| Index: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,25,14)(2,26,15)(3,27,13)(7,31,19)(8,32,20)(9,33,21), (4,6,5)(16,18,17) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.C_3^8:C_2\wr D_4$ |
| Order: | \(5509980288\)\(\medspace = 2^{7} \cdot 3^{16} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_{12}^2:D_4$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_3:(C_2^4.C_2^5.C_2^2)$ |
| Outer Automorphisms: | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(99179645184\)\(\medspace = 2^{8} \cdot 3^{18} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |