Subgroup ($H$) information
| Description: | $C_{337}:C_{56}$ |
| Order: | \(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \) |
| Index: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Exponent: | \(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \) |
| Generators: |
$b^{84}, a^{4}b^{14154}, a^{7}b^{14175}, b^{14154}, a^{14}b^{10206}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{28308}.C_{28}$ |
| Order: | \(792624\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{2} \cdot 337 \) |
| Exponent: | \(56616\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 337 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{42}$ |
| Order: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Automorphism Group: | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{4718}.C_{21}.C_{24}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_{337}$, of order \(226464\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 337 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |