Subgroup ($H$) information
| Description: | $C_{11}$ |
| Order: | \(11\) |
| Index: | \(17569200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{4} \) |
| Exponent: | \(11\) |
| Generators: |
$\left(\begin{array}{rr}
56 & 0 \\
0 & 56
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, and simple. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{11}^4.\GL(2,11)$ |
| Order: | \(193261200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{5} \) |
| Exponent: | \(14520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_{11}^3.\GL(2,11)$ |
| Order: | \(17569200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{4} \) |
| Exponent: | \(14520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| Automorphism Group: | $C_2\times C_4\times C_{11}^3.\PSL(2,11).C_2$ |
| Outer Automorphisms: | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(140553600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \cdot 11^{4} \) |
| $\operatorname{Aut}(H)$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\card{W}$ | 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 |