Subgroup ($H$) information
| Description: | $C_2^4:C_{10}$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(5\) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$a, d^{50}, d^{25}, d^{20}, b, c$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_2^4:C_{50}$ |
| Order: | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| Exponent: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Nilpotency class: | $2$ |
| 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: | $C_5$ |
| Order: | \(5\) |
| Exponent: | \(5\) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:C_3.(C_{20}\times D_4)$ |
| $\operatorname{Aut}(H)$ | $C_2^7.D_6$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^7.D_6$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(5\) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_2\times C_{50}$ | |||
| Normalizer: | $C_2^4:C_{50}$ | |||
| Minimal over-subgroups: | $C_2^4:C_{50}$ | |||
| Maximal under-subgroups: | $D_4\times C_{10}$ | $C_2^2:C_{20}$ | $C_2^3\times C_{10}$ | $C_2^2\wr C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2^2\times C_{10}$ |