Subgroup ($H$) information
| Description: | $C_2\times \PGL(2,7)$ |
| Order: | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(8,15)(9,12)(10,13)(11,14), (4,7)(5,6), (4,5)(6,7)(8,11,9,14,12,10)\rangle$
|
| Derived length: | $1$ |
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: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, 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 \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2\times \PGL(2,7)$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $W$ | $C_2\times \PGL(2,7)$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $-6$ |
| Projective image | $D_6\times \PGL(2,7)$ |