Subgroup ($H$) information
| Description: | $C_4.D_4$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(41\) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a, b$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{164}.D_4$ |
| Order: | \(1312\)\(\medspace = 2^{5} \cdot 41 \) |
| Exponent: | \(328\)\(\medspace = 2^{3} \cdot 41 \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{41}$ |
| Order: | \(41\) |
| Exponent: | \(41\) |
| Automorphism Group: | $C_{40}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Outer Automorphisms: | $C_{40}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| 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, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{40}\times D_4^2.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\wr D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\wr D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $C_2^2:C_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_{82}$ | ||
| Normalizer: | $C_{164}.D_4$ | ||
| Complements: | $C_{41}$ | ||
| Minimal over-subgroups: | $C_{164}.D_4$ | ||
| Maximal under-subgroups: | $C_2\times Q_8$ | $\OD_{16}$ | $\OD_{16}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_2^2:C_{164}$ |