Properties

Label 224.74.14.g1.a1
Order $ 2^{4} $
Index $ 2 \cdot 7 $
Normal No

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Subgroup ($H$) information

Description:$C_4:C_4$
Order: \(16\)\(\medspace = 2^{4} \)
Index: \(14\)\(\medspace = 2 \cdot 7 \)
Exponent: \(4\)\(\medspace = 2^{2} \)
Generators: $ab, c^{7}$ Copy content Toggle raw display
Nilpotency class: $2$
Derived length: $2$

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

Ambient group ($G$) information

Description: $C_2^3.D_{14}$
Order: \(224\)\(\medspace = 2^{5} \cdot 7 \)
Exponent: \(28\)\(\medspace = 2^{2} \cdot 7 \)
Derived length:$2$

The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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)$$F_7\times C_2^6$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \)
$\operatorname{Aut}(H)$ $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \)
$\operatorname{res}(S)$$C_2^4$, of order \(16\)\(\medspace = 2^{4} \)
$\card{\operatorname{ker}(\operatorname{res})}$\(24\)\(\medspace = 2^{3} \cdot 3 \)
$W$$C_2^3$, of order \(8\)\(\medspace = 2^{3} \)

Related subgroups

Centralizer:$C_2^2$
Normalizer:$C_4^2:C_2$
Normal closure:$C_{14}.D_4$
Core:$C_2\times C_4$
Minimal over-subgroups:$C_{14}.D_4$$C_4^2:C_2$
Maximal under-subgroups:$C_2\times C_4$$C_2\times C_4$$C_2\times C_4$

Other information

Number of subgroups in this conjugacy class$7$
Möbius function$1$
Projective image$C_2\times D_{14}$