Subgroup ($H$) information
| Description: | $C_{337}:C_{56}$ |
| Order: | \(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \) |
| Generators: |
$b^{12}, a^{8}b^{674}, a^{42}b^{1455}, b^{2022}, a^{28}b^{2800}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Ambient group ($G$) information
| Description: | $D_{337}:C_{336}$ |
| Order: | \(226464\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 337 \) |
| Exponent: | \(113232\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 337 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{12}$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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 ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{674}.C_{168}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_{337}$, 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
| Centralizer: | $C_{12}$ | ||
| Normalizer: | $D_{337}:C_{336}$ | ||
| Minimal over-subgroups: | $C_{1011}:C_{56}$ | $D_{337}:C_{56}$ | |
| Maximal under-subgroups: | $C_{337}:C_{28}$ | $C_{337}:C_8$ | $C_{56}$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{2022}:C_{56}$ |