Subgroup ($H$) information
| Description: | $C_{20}.C_2^3$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
36 & 0 \\
0 & 59
\end{array}\right), \left(\begin{array}{rr}
1 & 68 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 16
\end{array}\right), \left(\begin{array}{rr}
21 & 0 \\
0 & 81
\end{array}\right), \left(\begin{array}{rr}
69 & 0 \\
0 & 69
\end{array}\right), \left(\begin{array}{rr}
16 & 0 \\
0 & 16
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_2\times C_{20}.S_4$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times A_4).C_2^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^4:S_4\times F_5$, of order \(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \) |
| $\card{W}$ | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | not computed |
| Projective image | not computed |