Subgroup ($H$) information
| Description: | $C_6:\PGL(2,7)$ |
| Order: | \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(1,2,3)(8,15)(9,12)(10,13)(11,14), (4,7)(5,6), (2,3)(4,7)(8,11,9,14,12,10), (1,3,2)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $C_3:D_4\times \PGL(2,7)$ |
| Order: | \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times \SO(3,7)\times D_6$, of order \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $D_6\times \PGL(2,7)$, of order \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \) |
| $W$ | $D_6\times \PGL(2,7)$, of order \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $D_6\times \PGL(2,7)$ |