Subgroup ($H$) information
| Description: | $C_2^4:F_5$ |
| Order: | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
11 & 0 \\
10 & 11
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 10 \\
10 & 1
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
15 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 8 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
10 & 1
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_2\times F_5\times \GL(2,\mathbb{Z}/4)$ |
| Order: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times A_4).C_2^4.C_2^6$ |
| $\operatorname{Aut}(H)$ | $F_5\times C_2^6:(C_2\times S_4)$, of order \(61440\)\(\medspace = 2^{12} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $D_{10}.C_2^5$, of order \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32\)\(\medspace = 2^{5} \) |
| $W$ | $C_2^3\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2\times F_5\times S_4$ |