Subgroup ($H$) information
| Description: | not computed |
| Order: | \(177147\)\(\medspace = 3^{11} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | not computed |
| Generators: |
$\langle(10,11,12)(22,24,23)(34,35,36), (13,15,14)(19,20,21)(25,27,26)(31,32,33) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), a semidirect factor, nonabelian, a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and metabelian. Whether it is a direct factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^7.(C_3^2\wr C_2.D_4)$ |
| Order: | \(2834352\)\(\medspace = 2^{4} \cdot 3^{11} \) |
| 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: | $C_2^2:C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 \(102036672\)\(\medspace = 2^{6} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_3^5.S_3^2:C_4$, of order \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
Related subgroups
| Centralizer: | $C_3^4$ |
| Normalizer: | $C_3^7.(C_3^2\wr C_2.D_4)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7.(C_3^2\wr C_2.D_4)$ |