Subgroup ($H$) information
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(82\)\(\medspace = 2 \cdot 41 \) |
| Exponent: | \(2\) |
| Generators: |
$b, c, d^{41}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a direct factor, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
Ambient group ($G$) information
| Description: | $C_2^3\times C_{82}$ |
| Order: | \(656\)\(\medspace = 2^{4} \cdot 41 \) |
| Exponent: | \(82\)\(\medspace = 2 \cdot 41 \) |
| 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: | $C_{82}$ |
| Order: | \(82\)\(\medspace = 2 \cdot 41 \) |
| Exponent: | \(82\)\(\medspace = 2 \cdot 41 \) |
| Automorphism Group: | $C_{40}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Outer Automorphisms: | $C_{40}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,41$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{40}\times A_8$ |
| $\operatorname{Aut}(H)$ | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^3\times C_{82}$ | |
| Normalizer: | $C_2^3\times C_{82}$ | |
| Complements: | $C_{82}$ | |
| Minimal over-subgroups: | $C_2^2\times C_{82}$ | $C_2^4$ |
| Maximal under-subgroups: | $C_2^2$ |
Other information
| Number of subgroups in this autjugacy class | $15$ |
| Number of conjugacy classes in this autjugacy class | $15$ |
| Möbius function | $1$ |
| Projective image | $C_{82}$ |