Subgroup ($H$) information
Description: | $(C_2\times D_4^2):D_4$ |
Order: | \(1024\)\(\medspace = 2^{10} \) |
Index: | \(6561\)\(\medspace = 3^{8} \) |
Exponent: | \(8\)\(\medspace = 2^{3} \) |
Generators: |
$\langle(14,15)(16,17), (1,9)(2,7)(3,8)(4,11)(5,12)(6,10), (1,15,2,14)(3,13)(4,17,5,16) \!\cdots\! \rangle$
|
Nilpotency class: | $4$ |
Derived length: | $3$ |
The subgroup is maximal, 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: | $C_3^8:D_4^2.(C_2\times D_4)$ |
Order: | \(6718464\)\(\medspace = 2^{10} \cdot 3^{8} \) |
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 $W = N_G(H) / Z_G(H)$.
$\operatorname{Aut}(G)$ | $C_3^8.C_2^4.C_2^4.D_4^2$, of order \(107495424\)\(\medspace = 2^{14} \cdot 3^{8} \) |
$\operatorname{Aut}(H)$ | $C_2^9.C_2\wr D_4$, of order \(65536\)\(\medspace = 2^{16} \) |
$\card{W}$ | not computed |
Related subgroups
Centralizer: | not computed |
Normalizer: | $(C_2\times D_4^2):D_4$ |
Normal closure: | $C_3^8:D_4^2.(C_2\times D_4)$ |
Core: | $C_1$ |
Other information
Number of subgroups in this autjugacy class | $6561$ |
Number of conjugacy classes in this autjugacy class | $1$ |
Möbius function | not computed |
Projective image | not computed |