Subgroup ($H$) information
| Description: | $D_4:C_2^2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$ab, b^{2}c^{2}, c^{3}d^{2}, d^{2}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_4^2.D_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| 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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6:D_4$, of order \(512\)\(\medspace = 2^{9} \) |
| $\operatorname{Aut}(H)$ | $S_4\wr C_2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\wr D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
Related subgroups
| Centralizer: | $C_2$ | |||||
| Normalizer: | $C_4^2.D_4$ | |||||
| Minimal over-subgroups: | $D_4:D_4$ | $C_2^3.D_4$ | $C_2^3.D_4$ | |||
| Maximal under-subgroups: | $C_2\times D_4$ | $C_2\times D_4$ | $C_2\times D_4$ | $D_4:C_2$ | $D_4:C_2$ | $C_2\times D_4$ |
Other information
| Möbius function | $2$ |
| Projective image | $C_2\wr C_2^2$ |