Subgroup ($H$) information
| Description: | $C_4:Q_{16}$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(2\) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$ad, bd, c$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $\OD_{16}:D_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_2^7:D_4$, of order \(16384\)\(\medspace = 2^{14} \) |
| $\operatorname{Aut}(H)$ | $(D_4\times C_2^3).D_4^2$, of order \(4096\)\(\medspace = 2^{12} \) |
| $\operatorname{res}(S)$ | $C_2^5.D_4^2$, of order \(2048\)\(\medspace = 2^{11} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
| Centralizer: | $C_2^2$ | |||
| Normalizer: | $\OD_{16}:D_4$ | |||
| Complements: | $C_2$ $C_2$ | |||
| Minimal over-subgroups: | $\OD_{16}:D_4$ | |||
| Maximal under-subgroups: | $C_4:Q_8$ | $C_4:Q_8$ | $C_4\times C_8$ | $C_2\times Q_{16}$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-1$ |
| Projective image | $C_2^2\times D_4$ |