Subgroup ($H$) information
| Description: | $C_2^5.C_2\wr C_4$ |
| Order: | \(2048\)\(\medspace = 2^{11} \) |
| Index: | $1$ |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$\langle(1,4,6,8,2,3,5,7)(9,16,13,11,10,15,14,12), (3,4)(7,8)(11,12)(15,16), (7,8) \!\cdots\! \rangle$
|
| Nilpotency class: | $8$ |
| Derived length: | $3$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the radical, a direct factor, nonabelian, a $2$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_2^5.C_2\wr C_4$ |
| Order: | \(2048\)\(\medspace = 2^{11} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $8$ |
| Derived length: | $3$ |
The ambient group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^5.C_2^5.C_2^3$, of order \(16384\)\(\medspace = 2^{14} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_2^5.C_2^5.C_2^3$, of order \(16384\)\(\medspace = 2^{14} \) |
| $\card{W}$ | \(1024\)\(\medspace = 2^{10} \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $C_2^5.C_2\wr C_4$ | ||
| Complements: | $C_1$ | ||
| Maximal under-subgroups: | $C_2^5.C_2^4.C_2$ | $C_2^6.\OD_{16}$ | $C_2^6.\OD_{16}$ |
Other information
| Möbius function | $1$ |
| Projective image | not computed |