Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(2834352\)\(\medspace = 2^{4} \cdot 3^{11} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(1,3,2)(4,5,6)(7,8,9)(10,12,11)(13,14,15)(16,17,18)(19,20,21)(22,24,23)(25,27,26)(28,29,30)(31,33,32)(34,36,35)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), stem, a $p$-group, and simple. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.C_3^3:\GL(2,3)$ |
| Order: | \(8503056\)\(\medspace = 2^{4} \cdot 3^{12} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $6$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Order: | \(2834352\)\(\medspace = 2^{4} \cdot 3^{11} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times C_3^5.C_3^4.Q_8.S_3^2$, of order \(17006112\)\(\medspace = 2^{5} \cdot 3^{12} \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\card{W}$ | $1$ |
Related subgroups
| Centralizer: | $C_3^8.C_3^3:\GL(2,3)$ |
| Normalizer: | $C_3^8.C_3^3:\GL(2,3)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |