Subgroup ($H$) information
| Description: | $C_2^4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(3,8)(5,16)(13,19)(14,18), (1,4)(2,9)(13,19)(14,18), (7,15)(11,20)(13,19)(14,18), (6,12)(7,15)(10,17)(11,20)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^6.C_2\wr D_5$ |
| Order: | \(20480\)\(\medspace = 2^{12} \cdot 5 \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^6:D_{10}$ |
| Order: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Automorphism Group: | $F_{16}.C_2^4.C_2^3$ |
| Outer Automorphisms: | $D_4:D_6$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5\times C_5^2:D_6$, of order \(327680\)\(\medspace = 2^{16} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $W$ | $D_5$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2^6.C_2^5$ |
| Normalizer: | $C_2^6.C_2\wr D_5$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^6.C_2\wr D_5$ |