Subgroup ($H$) information
| Description: | $D_4^2:D_4$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Index: | $1$ |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,5)(2,6)(3,8)(4,7)(9,15,10,16)(11,14,12,13), (1,3)(2,4)(5,8)(6,7)(9,11) \!\cdots\! \rangle$
|
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the radical, a direct factor, nonabelian, a $2$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $D_4^2:D_4$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The ambient group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.(C_2^2\times D_4^2)$, of order \(4096\)\(\medspace = 2^{12} \) |
| $\operatorname{Aut}(H)$ | $C_2^4.(C_2^2\times D_4^2)$, of order \(4096\)\(\medspace = 2^{12} \) |
| $\card{W}$ | \(256\)\(\medspace = 2^{8} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |