Subgroup ($H$) information
| Description: | $\OD_{16}:C_2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a, b^{2}, c^{5}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{40}.D_4$ |
| Order: | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| Exponent: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $D_5$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_4\times C_{10}:C_4^2$, of order \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_{40}$ | ||||
| Normalizer: | $C_{40}.D_4$ | ||||
| Minimal over-subgroups: | $\OD_{16}:C_{10}$ | $\OD_{32}:C_2$ | |||
| Maximal under-subgroups: | $D_4:C_2$ | $C_2\times C_8$ | $\OD_{16}$ | $C_2\times C_8$ | $\OD_{16}$ |
Other information
| Möbius function | $5$ |
| Projective image | $C_5:D_4$ |