Properties

Label 16384.ji.2.D
Order $ 2^{13} $
Index $ 2 $
Normal Yes

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Subgroup ($H$) information

Description:not computed
Order: \(8192\)\(\medspace = 2^{13} \)
Index: \(2\)
Exponent: not computed
Generators: $\langle(1,6)(2,5)(3,10)(4,21)(7,22)(8,24)(9,17)(11,20)(12,19)(13,18)(14,16)(15,23) \!\cdots\! \rangle$ Copy content Toggle raw display
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