Subgroup ($H$) information
| Description: | $C_{1544}.C_{16}$ |
| Order: | \(24704\)\(\medspace = 2^{7} \cdot 193 \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6176\)\(\medspace = 2^{5} \cdot 193 \) |
| Generators: |
$a^{36}b^{1692}, b^{48}, b^{4632}, a^{24}b^{3600}, b^{2316}, a^{9}b^{4659}, b^{1158}, a^{18}b^{6246}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{9264}.C_{48}$ |
| Order: | \(444672\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 193 \) |
| Exponent: | \(18528\)\(\medspace = 2^{5} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_3\times C_6$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{4632}.C_{96}.C_2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{772}.C_{96}.C_2^4$ |
| $\card{W}$ | \(9264\)\(\medspace = 2^{4} \cdot 3 \cdot 193 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-3$ |
| Projective image | not computed |