Subgroup ($H$) information
| Description: | not computed |
| Order: | \(4374\)\(\medspace = 2 \cdot 3^{7} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,7,9)(2,11,13)(3,16,15)(4,17,5)(6,8,18)(10,12,14)(19,25,24)(20,23,26) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), and metabelian. 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: | $C_3^8.D_6$ |
| Order: | \(78732\)\(\medspace = 2^{2} \cdot 3^{9} \) |
| 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: | $C_3\times S_3$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5\times C_{11}^2:C_{40}$, of order \(8503056\)\(\medspace = 2^{4} \cdot 3^{12} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_3^6:D_6$, of order \(8748\)\(\medspace = 2^{2} \cdot 3^{7} \) |
Related subgroups
| Centralizer: | $C_3^2$ |
| Normalizer: | $C_3^8.D_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |