Subgroup ($H$) information
| Description: | $C_2^3\times C_{86}$ |
| Order: | \(688\)\(\medspace = 2^{4} \cdot 43 \) |
| Index: | \(2\) |
| Exponent: | \(86\)\(\medspace = 2 \cdot 43 \) |
| Generators: |
$d^{86}, d^{4}, c, b, a$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the socle (hence characteristic and normal), maximal, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_2^3\times C_{172}$ |
| Order: | \(1376\)\(\medspace = 2^{5} \cdot 43 \) |
| Exponent: | \(172\)\(\medspace = 2^{2} \cdot 43 \) |
| 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: | $C_2$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{42}.C_2^6.C_2.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $C_{42}\times A_8$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(56448\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^3\times C_{172}$ | ||
| Normalizer: | $C_2^3\times C_{172}$ | ||
| Minimal over-subgroups: | $C_2^3\times C_{172}$ | ||
| Maximal under-subgroups: | $C_2^2\times C_{86}$ | $C_2^2\times C_{86}$ | $C_2^4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2$ |