Subgroup ($H$) information
| Description: | $C_2^7$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(2\) |
| Generators: |
$\left(\begin{array}{rr}
9 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
31 & 20 \\
30 & 1
\end{array}\right), \left(\begin{array}{rr}
39 & 0 \\
0 & 39
\end{array}\right), \left(\begin{array}{rr}
1 & 20 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
11 & 20 \\
20 & 31
\end{array}\right), \left(\begin{array}{rr}
21 & 0 \\
0 & 21
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.\GL(6,2)\times F_5$ |
| $\operatorname{Aut}(H)$ | $\GL(7,2)$, of order \(163849992929280\)\(\medspace = 2^{21} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \cdot 127 \) |
| $\card{\operatorname{res}(S)}$ | \(20158709760\)\(\medspace = 2^{15} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(256\)\(\medspace = 2^{8} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $5$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $F_5$ |