Subgroup ($H$) information
| Description: | $C_{21}:Q_8$ |
| Order: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$ab^{37}, c^{2}, d, c, b^{6}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $D_6.D_{42}$ |
| Order: | \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6\times C_2^2\times S_3\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_2\times D_6\times F_7$, of order \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2\times D_6\times F_7$, of order \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $S_3\times D_{14}$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $1$ |
| Projective image | $S_3\times D_{42}$ |