Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$af^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
Ambient group ($G$) information
| Description: | $(C_6\times \He_3).D_6$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_2^3.C_2.\SU(4,2).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(207360\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2\times C_{12}$ | ||
| Normalizer: | $C_2\times C_{12}$ | ||
| Normal closure: | $(C_6\times \He_3).S_3$ | ||
| Core: | $C_2$ | ||
| Minimal over-subgroups: | $C_{12}$ | $C_3:C_4$ | $C_2\times C_4$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $162$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $(C_6\times \He_3):S_3$ |