Subgroup ($H$) information
| Description: | $C_3^6.C_3:S_3^3$ |
| Order: | \(472392\)\(\medspace = 2^{3} \cdot 3^{10} \) |
| Index: | \(3\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(4,18,29)(5,17,28)(6,16,30)(7,8,9)(10,23,35)(11,22,36)(12,24,34)(19,20,21) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and supersolvable (hence solvable and monomial). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^7.C_3:S_3^3$ |
| Order: | \(1417176\)\(\medspace = 2^{3} \cdot 3^{11} \) |
| 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
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_3^4.C_3^3.C_2^3.C_6.C_2^2$, of order \(306110016\)\(\medspace = 2^{6} \cdot 3^{14} \) |
| $\operatorname{Aut}(H)$ | $C_3^6.C_3^4.C_3^3.C_2^3.C_6.C_2^2$, of order \(306110016\)\(\medspace = 2^{6} \cdot 3^{14} \) |
| $W$ | $C_3^7.C_3:S_3^3$, of order \(1417176\)\(\medspace = 2^{3} \cdot 3^{11} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^7.C_3:S_3^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7.C_3:S_3^3$ |