Subgroup ($H$) information
| Description: | $C_{11}^2$ |
| Order: | \(121\)\(\medspace = 11^{2} \) |
| Index: | \(1056\)\(\medspace = 2^{5} \cdot 3 \cdot 11 \) |
| Exponent: | \(11\) |
| Generators: |
$\left(\begin{array}{rrrr}
2 & 8 & 6 & 7 \\
7 & 3 & 0 & 6 \\
1 & 4 & 10 & 3 \\
2 & 1 & 4 & 0
\end{array}\right), \left(\begin{array}{rrrr}
2 & 6 & 2 & 5 \\
3 & 6 & 2 & 2 \\
6 & 0 & 7 & 5 \\
7 & 6 & 8 & 0
\end{array}\right)$
|
| 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: | $C_{11}^2.C_{24}:D_{22}$ |
| Order: | \(127776\)\(\medspace = 2^{5} \cdot 3 \cdot 11^{3} \) |
| Exponent: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_{24}:D_{22}$ |
| Order: | \(1056\)\(\medspace = 2^{5} \cdot 3 \cdot 11 \) |
| Exponent: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Automorphism Group: | $C_{66}.C_{10}.C_2^5$ |
| Outer Automorphisms: | $C_2^2\times C_{10}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_{30}.C_5.C_2^6$ |
| $\operatorname{Aut}(H)$ | $\GL(2,11)$, of order \(13200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| $W$ | $D_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{11}^2:C_{44}$ |
| Normalizer: | $C_{11}^2.C_{24}:D_{22}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_{11}^2.C_{24}:D_{22}$ |