Subgroup ($H$) information
| Description: | $C_3^5$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(19,21,23)(22,25,26), (2,15,9)(4,18,16)(5,7,11)(10,12,17)(20,27,24)(22,26,25) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^7.S_3^3$ |
| Order: | \(472392\)\(\medspace = 2^{3} \cdot 3^{10} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_3^2.S_3^3$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Automorphism Group: | $(D_9\times S_3^2):C_6$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^7.C_3^3.C_6^2.C_2^3$, of order \(17006112\)\(\medspace = 2^{5} \cdot 3^{12} \) |
| $\operatorname{Aut}(H)$ | $\GL(5,3)$, of order \(475566474240\)\(\medspace = 2^{10} \cdot 3^{10} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $\card{W}$ | 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 |