Subgroup ($H$) information
| Description: | $C_3^{12}.C_2^8.C_3^4.D_8.C_2$ |
| Order: | \(352638738432\)\(\medspace = 2^{13} \cdot 3^{16} \) |
| Index: | \(2\) |
| Exponent: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Generators: |
$\langle(5,6)(7,20,32)(8,19,33)(9,21,31)(10,22,35,11,24,34,12,23,36)(13,14)(16,18) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is normal, maximal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^{12}.C_2^8.C_3^4.D_4:D_4$ |
| Order: | \(705277476864\)\(\medspace = 2^{14} \cdot 3^{16} \) |
| Exponent: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial 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
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 \(2821109907456\)\(\medspace = 2^{16} \cdot 3^{16} \) |
| $\operatorname{Aut}(H)$ | Group of order \(1410554953728\)\(\medspace = 2^{15} \cdot 3^{16} \) |
| $\card{W}$ | 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 |