Subgroup ($H$) information
| Description: | $C_3^4$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(702\)\(\medspace = 2 \cdot 3^{3} \cdot 13 \) |
| Exponent: | \(3\) |
| Generators: |
$bfgh^{2}, cf^{2}h, dfh, ef^{2}g^{2}$
|
| 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: | $C_3^7:C_{26}$ |
| Order: | \(56862\)\(\medspace = 2 \cdot 3^{7} \cdot 13 \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metabelian (hence solvable), and an A-group. Whether it is monomial 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_{13}^2.C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2.\PSL(4,3).C_2$ |
| $W$ | $C_{26}$, of order \(26\)\(\medspace = 2 \cdot 13 \) |
Related subgroups
| Centralizer: | $C_3^7$ |
| Normalizer: | $C_3^7:C_{26}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7:C_{26}$ |