Subgroup ($H$) information
| Description: | $D_4:C_8$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | $1$ |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a, c$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
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), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $D_4:C_8$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $3$ |
| 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: | $C_1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_4\times C_2^4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{Aut}(H)$ | $D_4\times C_2^4$, of order \(128\)\(\medspace = 2^{7} \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_2\times C_4$ | ||
| Normalizer: | $D_4:C_8$ | ||
| Complements: | $C_1$ | ||
| Maximal under-subgroups: | $C_4\times D_4$ | $C_4\times C_8$ | $C_4:C_8$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_4$ |