Subgroup ($H$) information
| Description: | $C_2^2\times C_{86}$ |
| Order: | \(344\)\(\medspace = 2^{3} \cdot 43 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(86\)\(\medspace = 2 \cdot 43 \) |
| Generators: |
$a, bc, d^{2}, d^{43}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_2^3:D_{86}$ |
| Order: | \(1376\)\(\medspace = 2^{5} \cdot 43 \) |
| Exponent: | \(172\)\(\medspace = 2^{2} \cdot 43 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_2^4.C_{129}.C_{42}.C_2$ |
| $\operatorname{Aut}(H)$ | $C_{42}\times \PSL(2,7)$ |
| $\operatorname{res}(S)$ | $S_4\times C_{42}$, of order \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(344\)\(\medspace = 2^{3} \cdot 43 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_{43}:D_4$ |