Subgroup ($H$) information
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(10,12)(11,13), (10,11)(12,13), (1,2)(3,5)(4,6)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
Ambient group ($G$) information
| Description: | $D_6^2:D_6$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.C_2^4:C_3.C_4.C_2^3\times S_3$ |
| $\operatorname{Aut}(H)$ | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $S_3^3:C_2$ |