Subgroup ($H$) information
| Description: | $C_3^{12}.C_2^8.C_3^4.C_2^6.C_2^2$ |
| Order: | \(2821109907456\)\(\medspace = 2^{16} \cdot 3^{16} \) |
| Index: | \(2\) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(28,29,30), (25,26,27), (14,15)(20,21)(23,24)(26,27)(32,33)(35,36), (16,18,17) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, monomial, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^{12}.C_2^8.C_3^4.C_2^6.C_2^3$ |
| Order: | \(5642219814912\)\(\medspace = 2^{17} \cdot 3^{16} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial or rational has not been computed.
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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(22568879259648\)\(\medspace = 2^{19} \cdot 3^{16} \) |
| $\operatorname{Aut}(H)$ | Group of order \(22568879259648\)\(\medspace = 2^{19} \cdot 3^{16} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |