Subgroup ($H$) information
| Description: | $C_3^2\times C_3^4:C_3$ |
| Order: | \(2187\)\(\medspace = 3^{7} \) |
| Index: | \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(16,17,18)(22,23,24)(28,30,29)(34,36,35), (1,2,3)(7,9,8)(13,14,15)(19,21,20) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is a direct factor, a semidirect factor, metacyclic, monomial, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.(C_3\times S_3^2)\wr C_2$ |
| Order: | \(153055008\)\(\medspace = 2^{5} \cdot 3^{14} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3^5:D_6\wr C_2$ |
| Order: | \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_3^2.C_3^5.C_2^3.C_2^4$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and solvable. 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8.C_3^4.C_6^3.C_2^3$, of order \(918330048\)\(\medspace = 2^{6} \cdot 3^{15} \) |
| $\operatorname{Aut}(H)$ | Group of order \(1785233613312\)\(\medspace = 2^{9} \cdot 3^{20} \) |
| $\card{W}$ | 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 |