Subgroup ($H$) information
| Description: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$b^{3}de^{2}f^{2}, c^{3}e^{3}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
Ambient group ($G$) information
| Description: | $(S_3\times C_6^3):D_6$ |
| Order: | \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
| 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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.C_3^4.C_2^6$ |
| $\operatorname{Aut}(H)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $D_4\times S_4$ |
| Normal closure: | $C_3^3:D_{12}$ |
| Core: | $C_2$ |
| Minimal over-subgroups: | $C_2\times D_4$ |
Other information
| Number of subgroups in this autjugacy class | $81$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_6^2.S_3^3$ |