Subgroup ($H$) information
Description: | not computed |
Order: | \(8192\)\(\medspace = 2^{13} \) |
Index: | \(2\) |
Exponent: | not computed |
Generators: |
$\langle(1,6)(2,5)(3,10)(4,21)(7,22)(8,24)(9,17)(11,20)(12,19)(13,18)(14,16)(15,23) \!\cdots\! \rangle$
|
Nilpotency class: | not computed |
Derived length: | not computed |
The subgroup is characteristic (hence normal), maximal, 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: | $C_2^8.D_4^2$ |
Order: | \(16384\)\(\medspace = 2^{14} \) |
Exponent: | \(8\)\(\medspace = 2^{3} \) |
Nilpotency class: | $5$ |
Derived length: | $3$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and rational.
Quotient group ($Q$) structure
Description: | $C_2$ |
Order: | \(2\) |
Exponent: | \(2\) |
Automorphism Group: | $C_1$, of order $1$ |
Outer Automorphisms: | $C_1$, of order $1$ |
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, 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 $W = G / Z_G(H)$.
$\operatorname{Aut}(G)$ | Group of order \(16777216\)\(\medspace = 2^{24} \) |
$\operatorname{Aut}(H)$ | not computed |
$\card{W}$ | not computed |
Related subgroups
Centralizer: | not computed |
Normalizer: | $C_2^8.D_4^2$ |
Other information
Number of conjugacy classes in this autjugacy class | $1$ |
Möbius function | not computed |
Projective image | not computed |