Subgroup ($H$) information
| Description: | $D_4^2:D_4$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Index: | \(729\)\(\medspace = 3^{6} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,17)(2,4)(3,9)(5,10)(6,11)(7,18)(8,12)(13,16)(14,15)(19,20,22,23)(21,26,25,24) \!\cdots\! \rangle$
|
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The subgroup is nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and rational.
Ambient group ($G$) information
| Description: | $C_6^4.D_6\wr C_2$ |
| Order: | \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^4.D_6\wr C_2$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_2^{10}.D_4^2.C_2$, of order \(131072\)\(\medspace = 2^{17} \) |
| $W$ | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $D_4^2:D_4$ |
| Normal closure: | $C_6^4.D_6\wr C_2$ |
| Core: | $C_2^4$ |
Other information
| Number of subgroups in this autjugacy class | $729$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_6^4.D_6\wr C_2$ |