Subgroup ($H$) information
| Description: | $C_{193}:C_8$ |
| Order: | \(1544\)\(\medspace = 2^{3} \cdot 193 \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(1544\)\(\medspace = 2^{3} \cdot 193 \) |
| Generators: |
$a^{8}b^{2173}, a^{16}b^{626}, b^{12}, b^{1158}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $D_{193}:C_{192}$ |
| Order: | \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \) |
| Exponent: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{48}$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Automorphism Group: | $C_2^2\times C_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Outer Automorphisms: | $C_2^2\times C_4$, of order \(16\)\(\medspace = 2^{4} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{386}.C_{96}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_{193}$, of order \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \) |
| $W$ | $C_{193}:C_{32}$, of order \(6176\)\(\medspace = 2^{5} \cdot 193 \) |
Related subgroups
| Centralizer: | $C_{12}$ | |
| Normalizer: | $D_{193}:C_{192}$ | |
| Minimal over-subgroups: | $C_{193}:C_{24}$ | $D_{193}:C_8$ |
| Maximal under-subgroups: | $C_{193}:C_4$ | $C_8$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{1158}:C_{32}$ |