Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$d^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_2^4.D_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| 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: | $D_4:C_2^2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $S_4\wr C_2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Outer Automorphisms: | $\SOPlus(4,2)$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.D_4$, of order \(2048\)\(\medspace = 2^{11} \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1024\)\(\medspace = 2^{10} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2^3:C_8$ | |||||||||
| Normalizer: | $C_2^4.D_4$ | |||||||||
| Minimal over-subgroups: | $C_2\times C_4$ | $C_2\times C_4$ | $C_2\times C_4$ | $C_2\times C_4$ | $D_4$ | $D_4$ | $Q_8$ | $Q_8$ | $C_8$ | $C_8$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_2^3:D_4$ |