Subgroup ($H$) information
| Description: | $D_4:D_4$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,2)(4,5,7,6)(8,13)(10,12)(14,15), (4,7)(5,6)(8,9,13,11)(10,12,15,14), (4,5) \!\cdots\! \rangle$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3:D_4\times \PGL(2,7)$ |
| Order: | \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times \SO(3,7)\times D_6$, of order \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^6:D_4$, of order \(512\)\(\medspace = 2^{9} \) |
| $W$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $63$ |
| Möbius function | $-1$ |
| Projective image | $D_6\times \PGL(2,7)$ |