Subgroup ($H$) information
| Description: | $C_3^3:S_3$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,14,5)(2,16,13)(3,4,15)(6,7,18)(8,10,17)(9,12,11), (1,3,11)(2,10,6)(4,9,14) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_6^4.C_6^2:D_4$ |
| Order: | \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | not computed |
| Automorphism Group: | not computed |
| Outer Automorphisms: | not computed |
| Derived length: | not computed |
Properties have not been computed
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^4.C_3^2.C_2^6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $\AGL(4,3)$, of order \(1965150720\)\(\medspace = 2^{9} \cdot 3^{10} \cdot 5 \cdot 13 \) |
| $W$ | $C_{3088}.C_{24}$, of order \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \) |
Related subgroups
| Centralizer: | $C_2^5$ |
| Normalizer: | $C_6^4.C_6^2:D_4$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_6^4.C_6^2:D_4$ |