Subgroup ($H$) information
| Description: | $C_7:C_{12}$ |
| Order: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$ac, d^{3}, b^{3}, d^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{21}:S_4$ |
| Order: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times S_4\times F_7$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_{14}$, of order \(28\)\(\medspace = 2^{2} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_6$ | ||
| Normalizer: | $C_{21}:D_4$ | ||
| Normal closure: | $C_{21}:S_4$ | ||
| Core: | $C_{21}$ | ||
| Minimal over-subgroups: | $C_{21}:D_4$ | ||
| Maximal under-subgroups: | $C_{42}$ | $C_7:C_4$ | $C_{12}$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $C_7:S_4$ |