Subgroup ($H$) information
| Description: | $C_{11}^3$ |
| Order: | \(1331\)\(\medspace = 11^{3} \) |
| Index: | \(11\) |
| Exponent: | \(11\) |
| Generators: |
$ad^{8}, cd^{3}, bd^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, maximal, central (hence abelian, 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: | $C_{11}^4$ |
| Order: | \(14641\)\(\medspace = 11^{4} \) |
| Exponent: | \(11\) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
Quotient group ($Q$) structure
| Description: | $C_{11}$ |
| Order: | \(11\) |
| Exponent: | \(11\) |
| Automorphism Group: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(41393302251840000\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5^{4} \cdot 7 \cdot 11^{6} \cdot 19 \cdot 61 \) |
| $\operatorname{Aut}(H)$ | $\GL(3,11)$, of order \(2124276000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \cdot 7 \cdot 11^{3} \cdot 19 \) |
| $\card{W}$ | 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 |