Subgroup ($H$) information
| Description: | $C_2\times C_{84}$ |
| Order: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$b^{2}, d, c^{8}, c^{42}, c^{28}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{12}.(C_2\times D_{28})$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{42}.(C_2^5\times C_6).C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_{12}:C_2^3$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\card{W}$ | \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | not computed |