Properties

Label 256.5425.8.j1.a1
Order $ 2^{5} $
Index $ 2^{3} $
Normal No

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Subgroup ($H$) information

Description:$C_2\times Q_{16}$
Order: \(32\)\(\medspace = 2^{5} \)
Index: \(8\)\(\medspace = 2^{3} \)
Exponent: \(8\)\(\medspace = 2^{3} \)
Generators: $abc, b^{6}, c^{4}$ Copy content Toggle raw display
Nilpotency class: $3$
Derived length: $2$

The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

Ambient group ($G$) information

Description: $C_{16}.D_8$
Order: \(256\)\(\medspace = 2^{8} \)
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.

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.C_4^3.C_2^5$
$\operatorname{Aut}(H)$ $C_4.D_4^2$, of order \(256\)\(\medspace = 2^{8} \)
$\operatorname{res}(S)$$D_8:C_2^3$, of order \(128\)\(\medspace = 2^{7} \)
$\card{\operatorname{ker}(\operatorname{res})}$\(8\)\(\medspace = 2^{3} \)
$W$$D_8:C_2$, of order \(32\)\(\medspace = 2^{5} \)

Related subgroups

Centralizer:$C_2^2$
Normalizer:$C_8.D_8$
Normal closure:$C_4:Q_{16}$
Core:$C_2\times C_8$
Minimal over-subgroups:$C_4:Q_{16}$$Q_{32}:C_2$$C_4.D_8$
Maximal under-subgroups:$C_2\times C_8$$C_2\times Q_8$$Q_{16}$$Q_{16}$
Autjugate subgroups:256.5425.8.j1.b1

Other information

Number of subgroups in this conjugacy class$2$
Möbius function$0$
Projective image$C_8:D_8$