Subgroup ($H$) information
| Description: | not computed |
| Order: | \(13122\)\(\medspace = 2 \cdot 3^{8} \) |
| Index: | \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| Exponent: | not computed |
| Generators: |
$\langle(13,14,15)(22,24,23)(34,36,35), (1,3,2)(10,11,12)(13,15,14)(16,17,18)(19,21,20) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence 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: | $C_3^6.(C_6^4.(D_4\times S_4))$ |
| Order: | \(181398528\)\(\medspace = 2^{10} \cdot 3^{11} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $D_6^2:(C_2^2\times S_4)$ |
| Order: | \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2\times C_6^2.(C_2\times A_4).C_2^6$ |
| Outer Automorphisms: | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
| Derived length: | $3$ |
The quotient is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^{10}.C_2^4.C_6.C_2.C_2^5$ |
| $\operatorname{Aut}(H)$ | 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 |