Subgroup ($H$) information
| Description: | $C_4^2.D_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(2\) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$ad, bd, c^{3}$
|
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).
Ambient group ($G$) information
| Description: | $(C_4\times C_8).D_4$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $5$ |
| 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_4^2.C_2^4$, of order \(1024\)\(\medspace = 2^{10} \) |
| $\operatorname{Aut}(H)$ | $C_2^6:D_4$, of order \(512\)\(\medspace = 2^{9} \) |
| $\operatorname{res}(S)$ | $D_4^2:C_2^2$, of order \(256\)\(\medspace = 2^{8} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2\wr D_4$, of order \(128\)\(\medspace = 2^{7} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_2\wr D_4$ |