Subgroup ($H$) information
| Description: | $D_4:(C_2\times C_{16})$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$bg, ch, d^{2}g, f^{7}hi$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_4^3.(C_2\times D_4^2)$ |
| Order: | \(8192\)\(\medspace = 2^{13} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $D_4:C_2^2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2\wr D_6$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Outer Automorphisms: | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(16777216\)\(\medspace = 2^{24} \) |
| $\operatorname{Aut}(H)$ | $C_2^6.C_2^5.C_2^3$ |
| $W$ | $(C_2^3\times C_4) . C_2^4$, of order \(512\)\(\medspace = 2^{9} \) |
Related subgroups
| Centralizer: | $C_2\times C_8$ |
| Normalizer: | $C_4^3.(C_2\times D_4^2)$ |
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | not computed |
| Projective image | $C_4^2.(C_2\times D_4^2)$ |