Subgroup ($H$) information
| Description: | $(C_2\times \SL(2,11)):C_{10}$ |
| Order: | \(26400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
5 & 9 & 8 & 0 \\
7 & 4 & 4 & 8 \\
2 & 8 & 6 & 2 \\
4 & 2 & 4 & 5
\end{array}\right), \left(\begin{array}{rrrr}
2 & 2 & 4 & 0 \\
4 & 9 & 0 & 7 \\
1 & 0 & 9 & 2 \\
0 & 10 & 4 & 2
\end{array}\right), \left(\begin{array}{rrrr}
10 & 10 & 4 & 5 \\
7 & 4 & 0 & 4 \\
3 & 8 & 7 & 1 \\
3 & 3 & 4 & 1
\end{array}\right), \left(\begin{array}{rrrr}
6 & 4 & 10 & 10 \\
6 & 2 & 5 & 10 \\
9 & 1 & 9 & 7 \\
5 & 9 & 5 & 5
\end{array}\right), \left(\begin{array}{rrrr}
9 & 0 & 0 & 0 \\
0 & 9 & 0 & 0 \\
0 & 0 & 9 & 0 \\
0 & 0 & 0 & 9
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $\GL(2,11):D_4$ |
| Order: | \(105600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2.C_2^5.\PSL(2,11).C_2$ |
| $\operatorname{Aut}(H)$ | $(C_4\times D_4).\PSL(2,11).C_2$ |
| $\card{W}$ | \(10560\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |