Subgroup ($H$) information
| Description: | $C_3^6$ |
| Order: | \(729\)\(\medspace = 3^{6} \) |
| Index: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Exponent: | \(3\) |
| Generators: |
$bd^{2}fgh, gh^{2}, fgh^{2}, ce^{2}gh, de^{2}f^{2}gh, efg^{2}h^{2}$
|
| 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: | $S_3\times C_{13}$ |
| Order: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Automorphism Group: | $S_3\times C_{12}$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_{12}$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_{13}^2.C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $\GL(6,3)$, of order \(84129611558952960\)\(\medspace = 2^{13} \cdot 3^{15} \cdot 5 \cdot 7 \cdot 11^{2} \cdot 13^{2} \) |
| $W$ | $C_{26}$, 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 | $C_3^7:C_{26}$ |