Subgroup ($H$) information
| Description: | $C_2^4:D_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(8,13)(10,11), (7,12)(8,10)(9,14)(11,13), (4,6)(9,10)(11,12), (7,11)(8,9) \!\cdots\! \rangle$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $D_4:C_6^2:D_{12}$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.A_4^2.C_4.C_2^5.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^5.(D_4\times S_4)$, of order \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| $W$ | $C_2^3:D_4$, of order \(64\)\(\medspace = 2^{6} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $54$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-1$ |
| Projective image | $(C_2^2\times C_6^2):D_{12}$ |