Subgroup ($H$) information
| Description: | not computed |
| Order: | \(118098\)\(\medspace = 2 \cdot 3^{10} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,27,2,26,3,25)(4,30,5,28,6,29)(8,9)(10,24,11,22,12,23)(13,14)(16,17)(19,32,21,31,20,33) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, and supersolvable (hence solvable and monomial). Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.C_3:S_3^3$ |
| Order: | \(472392\)\(\medspace = 2^{3} \cdot 3^{10} \) |
| 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
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 \(306110016\)\(\medspace = 2^{6} \cdot 3^{14} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_3^6.C_3:S_3^3$, of order \(472392\)\(\medspace = 2^{3} \cdot 3^{10} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^6.C_3:S_3^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.C_3:S_3^3$ |