Subgroup ($H$) information
| Description: | $C_3\times S_6$ |
| Order: | \(2160\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,6)(7,8,9)(10,11,12)(13,14,15), (1,2,3,4,5)(7,8,9)(10,11,12)(13,14,15), (7,9,8)(10,12,11)(13,15,14)\rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $C_3\wr S_3\times S_6$ |
| Order: | \(116640\)\(\medspace = 2^{5} \cdot 3^{6} \cdot 5 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_3^2:C_6$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times C_3.S_3^2.A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $S_6:C_2^2$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| $W$ | $S_6$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^2:C_6\times S_6$ |