Subgroup ($H$) information
| Description: | $D_6\times S_4$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Index: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(3,4,6), (8,11)(9,10), (1,6)(3,4), (1,4)(3,6)(8,11)(9,10), (3,4)(8,11)(9,10), (2,5,7)(8,11)(9,10), (2,5)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_2^2\times S_7$ |
| Order: | \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, nonsolvable, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times S_7$, of order \(120960\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^2:D_6^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $W$ | $S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $210$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | $1$ |
| Projective image | $C_2\times S_7$ |