Subgroup ($H$) information
| Description: | not computed |
| Order: | \(236196\)\(\medspace = 2^{2} \cdot 3^{10} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(4,5,6)(13,14,15)(19,20,21)(22,23,24)(25,27,26)(28,30,29)(31,33,32)(34,36,35) \!\cdots\! \rangle$
|
| 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: | $C_3^6.C_3^4:(C_4\times S_3)$ |
| Order: | \(1417176\)\(\medspace = 2^{3} \cdot 3^{11} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(459165024\)\(\medspace = 2^{5} \cdot 3^{15} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_3^6.C_3^4:(C_4\times S_3)$, of order \(1417176\)\(\medspace = 2^{3} \cdot 3^{11} \) |
Related subgroups
| Centralizer: | $C_1$ |
| Normalizer: | $C_3^6.C_3^4:(C_4\times S_3)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.C_3^4:(C_4\times S_3)$ |