Subgroup ($H$) information
| Description: | not computed |
| Order: | \(148224\)\(\medspace = 2^{8} \cdot 3 \cdot 193 \) |
| Index: | \(2\) |
| Exponent: | not computed |
| Generators: |
$b^{9264}, b^{96}, a^{6}b^{3558}, b^{1158}, a^{3}b^{3}, b^{6176}, b^{4632}, b^{2316}, a^{12}b^{12684}, a^{8}b^{7408}$
|
| Derived length: | not computed |
The subgroup is normal, maximal, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is elementary, metacyclic, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_{18528}.C_{16}$ |
| Order: | \(296448\)\(\medspace = 2^{9} \cdot 3 \cdot 193 \) |
| 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1544}.C_{24}.C_4^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_{193}:C_{16}$, of order \(3088\)\(\medspace = 2^{4} \cdot 193 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_{386}:C_{16}$ |