Subgroup ($H$) information
| Description: | $C_3^3$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Index: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(1,2,3)(7,9,8)(10,11,12)(13,15,14)(19,21,20)(22,23,24)(25,27,26)(31,33,32) \!\cdots\! \rangle$
|
| 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: | $C_3^6.(S_3\times \GL(2,3))$ |
| Order: | \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.(S_3\times \GL(2,3))$, of order \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $\GL(3,3)$, of order \(11232\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 13 \) |
| $W$ | $C_2\times C_3^2:\GL(2,3)$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_3^5$ |
| Normalizer: | $C_3^6.(S_3\times \GL(2,3))$ |
| Maximal under-subgroups: | $C_3^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.(S_3\times \GL(2,3))$ |