Subgroup ($H$) information
| Description: | not computed |
| Order: | \(139968\)\(\medspace = 2^{6} \cdot 3^{7} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | not computed |
| Generators: |
$\langle(2,12,4,16,15,13,14,3,11)(5,18,6)(7,9,17)(20,21,22)(23,24)(25,30)(27,29,28) \!\cdots\! \rangle$
|
| 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: | $C_6^4.(C_6^2:C_4\times S_3)$ |
| Order: | \(1119744\)\(\medspace = 2^{9} \cdot 3^{7} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 \(26873856\)\(\medspace = 2^{12} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $(C_3^2\times A_4^2):(C_4\times S_3)$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.(C_6^2:C_4\times S_3)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4.(A_4^2:C_4\times D_6)$ |