Subgroup ($H$) information
| Description: | $C_2^8.D_4^2$ |
| Order: | \(16384\)\(\medspace = 2^{14} \) |
| Index: | $1$ |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| 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: | $5$ |
| Derived length: | $3$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the radical, a semidirect factor, nonabelian, a $2$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and rational. Whether it is a direct factor 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: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 \(16777216\)\(\medspace = 2^{24} \) |
| $\operatorname{Aut}(H)$ | Group of order \(16777216\)\(\medspace = 2^{24} \) |
| $\card{W}$ | 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 |