Subgroup ($H$) information
| Description: | $C_{337}:C_{56}$ |
| Order: | \(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \) |
| Index: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Exponent: | \(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \) |
| Generators: |
$b^{7}, a^{24}b, a^{21}, a^{84}, a^{42}$
|
| 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: | $C_{2359}:C_{168}$ |
| Order: | \(396312\)\(\medspace = 2^{3} \cdot 3 \cdot 7^{2} \cdot 337 \) |
| 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: | $C_{21}$ |
| Order: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| 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 ($p = 3,7$), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{2359}.C_{21}.C_{24}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_{337}$, of order \(226464\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 337 \) |
| $W$ | $C_{337}:C_{84}$, of order \(28308\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 337 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $7$ |
| Number of conjugacy classes in this autjugacy class | $7$ |
| Möbius function | $1$ |
| Projective image | $C_{2359}:C_{84}$ |