Subgroup ($H$) information
| Description: | $C_5:D_4$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
19 & 3 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
11 & 10 \\
0 & 11
\end{array}\right), \left(\begin{array}{rr}
1 & 4 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
19 & 0 \\
10 & 19
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_2^3:D_{30}$ |
| Order: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times S_4^2$, of order \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_5$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_2^2\times F_5$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2^3$ | |||
| Normalizer: | $C_2^3:D_{10}$ | |||
| Normal closure: | $C_{10}:S_4$ | |||
| Core: | $C_5$ | |||
| Minimal over-subgroups: | $C_{10}:D_4$ | $C_{10}:D_4$ | ||
| Maximal under-subgroups: | $C_2\times C_{10}$ | $D_{10}$ | $C_5:C_4$ | $D_4$ |
Other information
| Number of subgroups in this autjugacy class | $36$ |
| Number of conjugacy classes in this autjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $C_2^3:D_{30}$ |