Subgroup ($H$) information
| Description: | $C_{12}.D_{14}$ |
| Order: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$a, d^{6}, d^{21}, d^{14}, bc^{7}, c^{4}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{84}.C_2^4$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{42}\times A_4).C_6.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times D_4\times F_7$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2^2\times D_4\times F_7$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $C_2^2\times D_{14}$, of order \(112\)\(\medspace = 2^{4} \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $C_{42}.C_2^4$ |