Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(78732\)\(\medspace = 2^{2} \cdot 3^{9} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,15,14)(16,18,17)(19,21,20)(22,23,24)(25,27,26)(28,30,29)(31,33,32)(34,35,36)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), stem, a $p$-group, and simple. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.C_3\wr C_2^2$ |
| Order: | \(236196\)\(\medspace = 2^{2} \cdot 3^{10} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Order: | \(78732\)\(\medspace = 2^{2} \cdot 3^{9} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.C_3^5.C_6^3.C_2$, of order \(25509168\)\(\medspace = 2^{4} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3^6.C_3\wr C_2^2$ |
| Normalizer: | $C_3^6.C_3\wr C_2^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |