Subgroup ($H$) information
| Description: | not computed |
| Order: | \(59049\)\(\medspace = 3^{10} \) |
| Index: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | not computed |
| Generators: |
$\langle(19,21,20)(22,24,23)(31,33,32)(34,35,36), (7,9,8)(10,11,12)(19,21,20)(22,23,24) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
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, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.C_3^3:\GL(2,3)$ |
| Order: | \(8503056\)\(\medspace = 2^{4} \cdot 3^{12} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $6$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3\times \GL(2,3)$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times C_3^5.C_3^4.Q_8.S_3^2$, of order \(17006112\)\(\medspace = 2^{5} \cdot 3^{12} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(104976\)\(\medspace = 2^{4} \cdot 3^{8} \) |
Related subgroups
| Centralizer: | $C_3^4$ |
| Normalizer: | $C_3^8.C_3^3:\GL(2,3)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |