Subgroup ($H$) information
Description: | $D_4$ |
Order: | \(8\)\(\medspace = 2^{3} \) |
Index: | \(128\)\(\medspace = 2^{7} \) |
Exponent: | \(4\)\(\medspace = 2^{2} \) |
Generators: |
$\langle(1,10,15,5)(2,9,16,6)(3,12,14,8)(4,11,13,7), (1,15)(2,16)(3,14)(4,13)(5,10)(6,9)(7,11)(8,12), (1,5)(2,6)(3,7)(4,8)(9,16)(10,15)(11,14)(12,13)\rangle$
|
Nilpotency class: | $2$ |
Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
Ambient group ($G$) information
Description: | $(C_2\times D_4^2):D_4$ |
Order: | \(1024\)\(\medspace = 2^{10} \) |
Exponent: | \(8\)\(\medspace = 2^{3} \) |
Nilpotency class: | $4$ |
Derived length: | $3$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and rational.
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 $W = N_G(H) / Z_G(H)$.
$\operatorname{Aut}(G)$ | $C_2^9.C_2\wr D_4$, of order \(65536\)\(\medspace = 2^{16} \) |
$\operatorname{Aut}(H)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
$\card{W}$ | \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Centralizer: | $C_2^3$ | ||
Normalizer: | $C_2^2\times D_4$ | ||
Normal closure: | $C_2^4:D_4$ | ||
Core: | $C_1$ | ||
Minimal over-subgroups: | $C_2\times D_4$ | $C_2\times D_4$ | |
Maximal under-subgroups: | $C_2^2$ | $C_2^2$ | $C_4$ |
Other information
Number of subgroups in this autjugacy class | $256$ |
Number of conjugacy classes in this autjugacy class | $8$ |
Möbius function | not computed |
Projective image | not computed |