Subgroup ($H$) information
| Description: | $C_3^6$ |
| Order: | \(729\)\(\medspace = 3^{6} \) |
| Index: | \(15840\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(28,30,29), (13,14,15) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), abelian (hence 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: | $C_3^5.(S_3\times M_{11})$ |
| Order: | \(11547360\)\(\medspace = 2^{5} \cdot 3^{8} \cdot 5 \cdot 11 \) |
| Exponent: | \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2\times M_{11}$ |
| Order: | \(15840\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $M_{11}$, of order \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $1$ |
The quotient is nonabelian and nonsolvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.(S_3\times M_{11})$, of order \(11547360\)\(\medspace = 2^{5} \cdot 3^{8} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $\GL(6,3)$, of order \(84129611558952960\)\(\medspace = 2^{13} \cdot 3^{15} \cdot 5 \cdot 7 \cdot 11^{2} \cdot 13^{2} \) |
| $W$ | $C_2\times M_{11}$, of order \(15840\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \) |
Related subgroups
| Centralizer: | $C_3^6$ |
| Normalizer: | $C_3^5.(S_3\times M_{11})$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^5.(S_3\times M_{11})$ |