Subgroup ($H$) information
| Description: | not computed |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Index: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,4,2,3)(5,7,6,8)(17,22)(18,21)(19,23)(20,24)(25,30,26,29)(27,31,28,32) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_2^5.C_2^8:C_{10}$ |
| Order: | \(81920\)\(\medspace = 2^{14} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^4:C_5$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $F_{16}:C_4$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $C_3:C_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6^2:S_4$, of order \(10485760\)\(\medspace = 2^{21} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(2560\)\(\medspace = 2^{9} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2^5$ |
| Normalizer: | $C_2^5.C_2^8:C_{10}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |