Subgroup ($H$) information
| Description: | $C_2^4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(43\) |
| Exponent: | \(2\) |
| Generators: |
$a, b, c, d^{43}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $2$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and rational.
Ambient group ($G$) information
| Description: | $C_2^3\times C_{86}$ |
| Order: | \(688\)\(\medspace = 2^{4} \cdot 43 \) |
| Exponent: | \(86\)\(\medspace = 2 \cdot 43 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
Quotient group ($Q$) structure
| Description: | $C_{43}$ |
| Order: | \(43\) |
| Exponent: | \(43\) |
| Automorphism Group: | $C_{42}$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_{42}$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| 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, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{42}\times A_8$ |
| $\operatorname{Aut}(H)$ | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^3\times C_{86}$ |
| Normalizer: | $C_2^3\times C_{86}$ |
| Complements: | $C_{43}$ |
| Minimal over-subgroups: | $C_2^3\times C_{86}$ |
| Maximal under-subgroups: | $C_2^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_{43}$ |