Subgroup ($H$) information
| Description: | $C_2^4\times F_5$ |
| Order: | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
37 & 20 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 20 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
21 & 0 \\
0 & 21
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
1 & 8 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
11 & 20 \\
20 & 31
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $F_5\times C_2^6$ |
| Order: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.\GL(6,2)\times F_5$ |
| $\operatorname{Aut}(H)$ | $C_2^4.A_8\times F_5$, of order \(6451200\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| $\card{\operatorname{res}(S)}$ | \(6451200\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $W$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2^6$ | ||
| Normalizer: | $F_5\times C_2^6$ | ||
| Complements: | $C_2^2$ $C_2^2$ | ||
| Minimal over-subgroups: | $F_5\times C_2^5$ | ||
| Maximal under-subgroups: | $C_2^3\times F_5$ | $C_2^3\times D_{10}$ | $C_2^4\times C_4$ |
Other information
| Number of subgroups in this autjugacy class | $2604$ |
| Number of conjugacy classes in this autjugacy class | $2604$ |
| Möbius function | $2$ |
| Projective image | $C_2^2\times F_5$ |