Subgroup ($H$) information
| Description: | $C_{11}^2$ |
| Order: | \(121\)\(\medspace = 11^{2} \) |
| Index: | \(14520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(11\) |
| Generators: |
$\left(\begin{array}{rr}
89 & 0 \\
0 & 89
\end{array}\right), \left(\begin{array}{rr}
12 & 99 \\
22 & 111
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_5\times C_{11}^4:D_{12}$ |
| Order: | \(1756920\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{4} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_5\times C_{11}^2:D_{12}$ |
| Order: | \(14520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $C_{11}^2.C_6.C_{10}.C_4.C_2^3$ |
| Outer Automorphisms: | $D_4\times C_{20}$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(255552000\)\(\medspace = 2^{9} \cdot 3 \cdot 5^{3} \cdot 11^{3} \) |
| $\operatorname{Aut}(H)$ | $\GL(2,11)$, of order \(13200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{11}^4:C_{60}$ |
| Normalizer: | $C_5\times C_{11}^4:D_{12}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_5\times C_{11}^2:D_{132}$ |