Subgroup ($H$) information
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(296\)\(\medspace = 2^{3} \cdot 37 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$ab^{4}, c^{37}$
|
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, and rational.
Ambient group ($G$) information
| Description: | $S_3\times C_{37}:C_8$ |
| Order: | \(1776\)\(\medspace = 2^{4} \cdot 3 \cdot 37 \) |
| Exponent: | \(888\)\(\medspace = 2^{3} \cdot 3 \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{37}:C_8$ |
| Order: | \(296\)\(\medspace = 2^{3} \cdot 37 \) |
| Exponent: | \(296\)\(\medspace = 2^{3} \cdot 37 \) |
| Automorphism Group: | $C_2\times F_{37}$, of order \(2664\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 37 \) |
| Outer Automorphisms: | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{111}.C_{18}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(S)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2664\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 37 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $S_3\times C_{37}:C_8$ |