Subgroup ($H$) information
| Description: | $A_4\times D_6:D_4$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(4,7)(5,6), (11,13)(12,14), (4,6,5)(8,15)(9,10)(11,13)(12,14), (1,3,2)(5,6,7) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), and metabelian. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $(C_6\times \GL(2,\mathbb{Z}/4)).D_4$ |
| Order: | \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_6\times A_4).C_2^6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $(C_6\times A_4).C_2^6.C_2^2$ |
| $\card{W}$ | 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 |