Subgroup ($H$) information
| Description: | $C_2^3:F_5$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 34 \\
30 & 23
\end{array}\right), \left(\begin{array}{rr}
1 & 8 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
21 & 0 \\
0 & 21
\end{array}\right), \left(\begin{array}{rr}
11 & 20 \\
0 & 11
\end{array}\right), \left(\begin{array}{rr}
31 & 1 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
21 & 16 \\
0 & 29
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $D_{10}.(C_4\times D_4)$ |
| Order: | \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5:(C_4\times C_2^8.C_2^3)$ |
| $\operatorname{Aut}(H)$ | $C_2\wr C_2^2\times F_5$, of order \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| $\card{W}$ | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | not computed |