Subgroup ($H$) information
| Description: | $C_5\times C_{60}$ |
| Order: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Index: | \(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rrrr}
4 & 7 & 10 & 0 \\
3 & 8 & 8 & 6 \\
4 & 8 & 3 & 1 \\
0 & 2 & 3 & 1
\end{array}\right), \left(\begin{array}{rrrr}
8 & 3 & 10 & 7 \\
1 & 4 & 0 & 10 \\
2 & 9 & 6 & 8 \\
2 & 2 & 10 & 2
\end{array}\right), \left(\begin{array}{rrrr}
9 & 9 & 8 & 10 \\
3 & 8 & 0 & 8 \\
6 & 5 & 3 & 2 \\
6 & 6 & 8 & 2
\end{array}\right), \left(\begin{array}{rrrr}
9 & 0 & 0 & 0 \\
0 & 9 & 0 & 0 \\
0 & 0 & 9 & 0 \\
0 & 0 & 0 & 9
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 5$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $\GL(2,11):D_6$ |
| Order: | \(158400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_6).C_2^5.\PSL(2,11).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times \GL(2,5)$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $66$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $4$ |
| Projective image | not computed |