Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(3\) |
| Generators: |
$\left(\begin{array}{rr}
10 & 3 \\
3 & 1
\end{array}\right)$
|
| 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), a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $D_4\times \GL(2,\mathbb{Z}/4)$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4.C_2^6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\card{W}$ | \(2\) |
Related subgroups
| Centralizer: | $C_{12}:C_2^3$ | ||||||||
| Normalizer: | $C_2^4:D_6$ | ||||||||
| Normal closure: | $A_4$ | ||||||||
| Core: | $C_1$ | ||||||||
| Minimal over-subgroups: | $A_4$ | $C_6$ | $C_6$ | $C_6$ | $C_6$ | $C_6$ | $C_6$ | $S_3$ | $S_3$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |