Subgroup ($H$) information
| Description: | $C_3^{12}.C_2^6.D_6.D_4$ |
| Order: | \(3265173504\)\(\medspace = 2^{11} \cdot 3^{13} \) |
| Index: | \(2\) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(17,18)(20,21)(23,24)(26,27)(29,30)(32,33), (1,2,3)(4,6,5)(7,9,8)(10,11,12) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
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: | $C_3^{12}.C_2^6.D_6.(C_2\times D_4)$ |
| Order: | \(6530347008\)\(\medspace = 2^{12} \cdot 3^{13} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $4$ |
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 \(13060694016\)\(\medspace = 2^{13} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | Group of order \(6530347008\)\(\medspace = 2^{12} \cdot 3^{13} \) |
| $\card{W}$ | 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 |