Subgroup ($H$) information
| Description: | not computed |
| Order: | \(34012224\)\(\medspace = 2^{6} \cdot 3^{12} \) |
| Index: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(8,9)(10,12)(17,18)(26,27)(32,33)(35,36), (1,2,3)(10,12,11)(13,15,14)(19,20,21) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^{12}.C_2^6.C_2^6.C_6.C_2^3$ |
| Order: | \(104485552128\)\(\medspace = 2^{16} \cdot 3^{13} \) |
| 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: | $C_2^7:S_4$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_2\times C_4^2:A_4.C_2^5.C_2$ |
| Outer Automorphisms: | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
| Derived length: | $4$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(417942208512\)\(\medspace = 2^{18} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | 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 |