Subgroup ($H$) information
| Description: | not computed |
| Order: | \(19683\)\(\medspace = 3^{9} \) |
| Index: | \(702\)\(\medspace = 2 \cdot 3^{3} \cdot 13 \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,3,2)(10,12,11)(13,14,15)(19,20,21)(25,27,26)(28,30,29)(31,33,32)(34,35,36) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is normal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). 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: | $C_3^{12}.C_{26}$ |
| Order: | \(13817466\)\(\medspace = 2 \cdot 3^{12} \cdot 13 \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group. Whether it is rational has not been computed.
Quotient group ($Q$) structure
| Description: | $F_{27}$ |
| Order: | \(702\)\(\medspace = 2 \cdot 3^{3} \cdot 13 \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Automorphism Group: | $F_{27}:C_3$, of order \(2106\)\(\medspace = 2 \cdot 3^{4} \cdot 13 \) |
| Outer Automorphisms: | $C_3$, of order \(3\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(2914269388992\)\(\medspace = 2^{6} \cdot 3^{13} \cdot 13^{4} \) |
| $\operatorname{Aut}(H)$ | 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 |