Subgroup ($H$) information
| Description: | $C_7:C_4$ |
| Order: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$abc^{3}, d^{12}, d^{42}$
|
| 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: | $C_2\times C_{12}.D_{28}$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_{42}).C_2^6.C_6.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $\card{W}$ | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2^3$ | |||
| Normalizer: | $C_{14}.C_2^4$ | |||
| Normal closure: | $C_{42}:C_4$ | |||
| Core: | $C_{14}$ | |||
| Minimal over-subgroups: | $C_{21}:C_4$ | $C_{14}:C_4$ | $C_{14}:C_4$ | $C_7:Q_8$ |
| Maximal under-subgroups: | $C_{14}$ | $C_4$ |
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | not computed |