Subgroup ($H$) information
| Description: | $C_7:C_{28}$ |
| Order: | \(196\)\(\medspace = 2^{2} \cdot 7^{2} \) |
| Index: | \(531441\)\(\medspace = 3^{12} \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$\langle(1,35,21,11,37,30,13,6,31,22,9,40,26,16,2,36,20,10,38,28,15,4,32,24,8,42,27,18) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_3^{12}.C_7.C_{28}$ |
| Order: | \(104162436\)\(\medspace = 2^{2} \cdot 3^{12} \cdot 7^{2} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and an A-group. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(64997360064\)\(\medspace = 2^{6} \cdot 3^{13} \cdot 7^{2} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_{14}:C_6^2$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_7:C_{28}$ |
| Normal closure: | $C_3^{12}.C_7.C_{28}$ |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $531441$ |
| Möbius function | not computed |
| Projective image | not computed |