Subgroup ($H$) information
| Description: | $C_2^6.\OD_{16}$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Index: | \(2\) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$\langle(1,11,4,10,6,15,8,14,2,12,3,9,5,16,7,13), (3,4)(7,8)(11,12)(15,16), (5,6) \!\cdots\! \rangle$
|
| Nilpotency class: | $7$ |
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, 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: | $C_2$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^5.C_2^5.C_2^3$, of order \(16384\)\(\medspace = 2^{14} \) |
| $\operatorname{Aut}(H)$ | $C_2^7.C_2\wr D_4$, of order \(16384\)\(\medspace = 2^{14} \) |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(8192\)\(\medspace = 2^{13} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $\card{W}$ | \(1024\)\(\medspace = 2^{10} \) |
Related subgroups
| Centralizer: | $C_2$ | |
| Normalizer: | $C_2^5.C_2\wr C_4$ | |
| Complements: | $C_2$ $C_2$ | |
| Minimal over-subgroups: | $C_2^5.C_2\wr C_4$ | |
| Maximal under-subgroups: | $(C_2^3\times D_4).C_2^3$ | $C_2^6.C_8$ |
Other information
| Möbius function | $-1$ |
| Projective image | not computed |