Subgroup ($H$) information
| Description: | $C_{41}:C_{10}$ |
| Order: | \(410\)\(\medspace = 2 \cdot 5 \cdot 41 \) |
| Index: | \(2\) |
| Exponent: | \(410\)\(\medspace = 2 \cdot 5 \cdot 41 \) |
| Generators: |
$a^{10}, a^{4}, b$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Ambient group ($G$) information
| Description: | $C_{41}:C_{20}$ |
| Order: | \(820\)\(\medspace = 2^{2} \cdot 5 \cdot 41 \) |
| Exponent: | \(820\)\(\medspace = 2^{2} \cdot 5 \cdot 41 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{41}$, of order \(1640\)\(\medspace = 2^{3} \cdot 5 \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $F_{41}$, of order \(1640\)\(\medspace = 2^{3} \cdot 5 \cdot 41 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_{41}$, of order \(1640\)\(\medspace = 2^{3} \cdot 5 \cdot 41 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_{41}:C_{20}$, of order \(820\)\(\medspace = 2^{2} \cdot 5 \cdot 41 \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $C_{41}:C_{20}$ | ||
| Minimal over-subgroups: | $C_{41}:C_{20}$ | ||
| Maximal under-subgroups: | $C_{41}:C_5$ | $D_{41}$ | $C_{10}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{41}:C_{20}$ |