Subgroup ($H$) information
| Description: | $C_{337}:C_{56}$ |
| Order: | \(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \) |
| Generators: |
$b^{8}, a^{8}, a^{14}b^{1537}, b^{1348}, a^{28}b^{2210}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Ambient group ($G$) information
| Description: | $C_{2696}:C_{56}$ |
| Order: | \(150976\)\(\medspace = 2^{6} \cdot 7 \cdot 337 \) |
| Exponent: | \(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1348}.C_{336}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_{337}$, of order \(226464\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 337 \) |
| $W$ | $C_{337}:C_{56}$, of order \(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_{1348}:C_{56}$ |