Subgroup ($H$) information
| Description: | not computed |
| Order: | \(104976\)\(\medspace = 2^{4} \cdot 3^{8} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$a^{3}e^{7}f^{16}gh, bcd^{2}f^{2}g^{5}h^{8}, h^{3}, d^{2}e^{6}g^{3}h^{7}, e^{3}f^{12}gh^{5}, e^{3}f^{8}g^{5}h^{7}, f^{6}g^{6}h^{3}, cd^{2}f^{4}g^{4}h^{5}, g^{3}h^{6}, e, e^{3}, e^{3}h^{7}$
|
| 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: | $D_9\wr C_2^2.C_3$ |
| Order: | \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| Exponent: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $A_4$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_9\wr A_4.C_3$, of order \(3779136\)\(\medspace = 2^{6} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | 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 |