Subgroup ($H$) information
| Description: | $(C_4\times C_8).C_2^3$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(2\) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$a, b, c^{4}d^{5}$
|
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_8^2:(C_2\times C_4)$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^2\times D_4^2).C_2^5$, of order \(8192\)\(\medspace = 2^{13} \) |
| $\operatorname{Aut}(H)$ | $(C_2\times D_4^2).C_2^5$, of order \(4096\)\(\medspace = 2^{12} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $(C_2\times D_4^2).C_2^5$, of order \(4096\)\(\medspace = 2^{12} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_4^2.(C_2\times D_4)$, of order \(256\)\(\medspace = 2^{8} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_4^2.(C_2\times D_4)$ |