Subgroup ($H$) information
| Description: | not computed |
| Order: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Index: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(13,16,18)(14,15,17), (1,3,6)(2,4,5)(7,9,11)(8,10,12)(13,18,16)(14,17,15) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is characteristic (hence normal) and abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group). 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: | $C_3^6.(C_2^9.S_4)$ |
| Order: | \(8957952\)\(\medspace = 2^{12} \cdot 3^{7} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | not computed |
| Automorphism Group: | not computed |
| Outer Automorphisms: | not computed |
| Nilpotency class: | not computed |
| Derived length: | not computed |
Properties have not been computed
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 \(573308928\)\(\medspace = 2^{18} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_6^6$ |
| Normalizer: | $C_3^6.(C_2^9.S_4)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |