Subgroup ($H$) information
| Description: | not computed |
| Order: | \(23887872\)\(\medspace = 2^{15} \cdot 3^{6} \) |
| Index: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(15,16)(17,18), (1,2)(5,6)(21,22)(23,24), (15,16)(17,18)(31,32)(35,36), (7,8) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, and solvable. 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: | $C_2^{12}.(C_3^6.(C_2^8.\GL(2,\mathbb{Z}/4)))$ |
| Order: | \(73383542784\)\(\medspace = 2^{25} \cdot 3^{7} \) |
| Exponent: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Derived length: | $6$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^7:S_4$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_2\times C_4^2:A_4.C_2^5.C_2$ |
| Outer Automorphisms: | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
| Derived length: | $4$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(1174136684544\)\(\medspace = 2^{29} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | 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 |