Subgroup ($H$) information
| Description: | $C_2\times Q_8\times C_{16}$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | $1$ |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$\left(\begin{array}{rr}
15 & 31 \\
16 & 17
\end{array}\right), \left(\begin{array}{rr}
31 & 0 \\
0 & 31
\end{array}\right), \left(\begin{array}{rr}
19 & 30 \\
0 & 23
\end{array}\right), \left(\begin{array}{rr}
23 & 7 \\
0 & 9
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
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), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_2\times Q_8\times C_{16}$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $2$ |
| 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: | $C_1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_6.C_2.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2^6.C_6.C_2.C_2^6$ |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2^2\times C_{16}$ | |||
| Normalizer: | $C_2\times Q_8\times C_{16}$ | |||
| Complements: | $C_1$ | |||
| Maximal under-subgroups: | $C_2\times C_4\times C_{16}$ | $C_4^2.C_8$ | $C_4^2.C_2^3$ | $Q_8\times C_{16}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_2^2$ |