Subgroup ($H$) information
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(386\)\(\medspace = 2 \cdot 193 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$a^{2}, b^{193}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and cyclic (hence elementary ($p = 2,3$), hyperelementary, metacyclic, and a Z-group).
Ambient group ($G$) information
| Description: | $C_{193}:C_{12}$ |
| Order: | \(2316\)\(\medspace = 2^{2} \cdot 3 \cdot 193 \) |
| Exponent: | \(2316\)\(\medspace = 2^{2} \cdot 3 \cdot 193 \) |
| 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: | $D_{193}$ |
| Order: | \(386\)\(\medspace = 2 \cdot 193 \) |
| Exponent: | \(386\)\(\medspace = 2 \cdot 193 \) |
| Automorphism Group: | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Outer Automorphisms: | $C_{96}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{193}.C_{96}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{193}:C_{12}$ | |
| Normalizer: | $C_{193}:C_{12}$ | |
| Minimal over-subgroups: | $C_{1158}$ | $C_{12}$ |
| Maximal under-subgroups: | $C_3$ | $C_2$ |
Other information
| Möbius function | $193$ |
| Projective image | $D_{193}$ |