Subgroup ($H$) information
| Description: | $(C_2\times C_6):\PGL(2,7)$ |
| Order: | \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \) |
| Index: | \(2\) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(1,2,3)(8,15)(9,12)(10,13)(11,14), (4,7)(5,6), (2,3)(4,6,7,5)(8,11,9,14,12,10), (1,3,2), (4,5)(6,7)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times \SO(3,7)\times D_6$, of order \(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_6\times \PGL(2,7)$, of order \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \) |
| $W$ | $D_6\times \PGL(2,7)$, of order \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $D_6\times \PGL(2,7)$ |