Subgroup ($H$) information
| Description: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a^{2}b^{3}, c^{41}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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: | $C_{164}$ |
| Order: | \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Exponent: | \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Automorphism Group: | $C_2\times C_{40}$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Outer Automorphisms: | $C_2\times C_{40}$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,41$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{40}\times D_4^2.C_2$ |
| $\operatorname{Aut}(H)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(S)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| $W$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2\times C_{164}$ | |
| Normalizer: | $C_{164}.D_4$ | |
| Minimal over-subgroups: | $C_2\times C_{164}$ | $C_2\times Q_8$ |
| Maximal under-subgroups: | $C_2^2$ | $C_4$ |
| Autjugate subgroups: | 1312.8.164.b1.a1 |
Other information
| Möbius function | $0$ |
| Projective image | $C_2^2:C_{164}$ |