Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(74\)\(\medspace = 2 \cdot 37 \) |
| Exponent: | \(3\) |
| Generators: |
$b^{37}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $S_3\times C_{37}$ |
| Order: | \(222\)\(\medspace = 2 \cdot 3 \cdot 37 \) |
| Exponent: | \(222\)\(\medspace = 2 \cdot 3 \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_{74}$ |
| Order: | \(74\)\(\medspace = 2 \cdot 37 \) |
| Exponent: | \(74\)\(\medspace = 2 \cdot 37 \) |
| Automorphism Group: | $C_{36}$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_{36}$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,37$), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times C_{36}$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{111}$ | |
| Normalizer: | $S_3\times C_{37}$ | |
| Complements: | $C_{74}$ | |
| Minimal over-subgroups: | $C_{111}$ | $S_3$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $1$ |
| Projective image | $S_3\times C_{37}$ |