Subgroup ($H$) information
| Description: | $C_3^3$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Index: | \(702\)\(\medspace = 2 \cdot 3^{3} \cdot 13 \) |
| Exponent: | \(3\) |
| Generators: |
$be^{2}fg^{2}, cefg^{2}, de^{2}f^{2}$
|
| 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). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^3.F_{27}$ |
| Order: | \(18954\)\(\medspace = 2 \cdot 3^{6} \cdot 13 \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_{13}^2.C_6.C_2$ |
| $\operatorname{Aut}(H)$ | $\GL(3,3)$, of order \(11232\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 13 \) |
| $W$ | $C_{26}$, of order \(26\)\(\medspace = 2 \cdot 13 \) |
Related subgroups
| Centralizer: | $C_3^6$ | ||
| Normalizer: | $C_3^3.F_{27}$ | ||
| Minimal over-subgroups: | $C_3^3:C_{13}$ | $C_3^4$ | $C_3^2:S_3$ |
| Maximal under-subgroups: | $C_3^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-27$ |
| Projective image | $C_3^3.F_{27}$ |