Subgroup ($H$) information
| Description: | $C_{10}.S_4$ |
| Order: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Index: | \(7\) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$a, d^{14}, b, c^{5}, c^{2}d^{14}, d^{21}$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, a Hall subgroup, and solvable.
Ambient group ($G$) information
| Description: | $C_{70}.S_4$ |
| Order: | \(1680\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_7\times A_4).C_{12}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2^4.D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^4.D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $7$ |
| Möbius function | $-1$ |
| Projective image | $C_7:S_4$ |