Subgroup ($H$) information
| Description: | $C_2^5\times C_6$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(7\) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 14 \\
14 & 15
\end{array}\right), \left(\begin{array}{rr}
15 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 25
\end{array}\right), \left(\begin{array}{rr}
1 & 14 \\
14 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 14 \\
0 & 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is maximal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a Hall subgroup, and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_2^5\times F_7$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5.\GL(5,2)\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_2\times \GL(6,2)$, of order \(40317419520\)\(\medspace = 2^{16} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \) |
| $\operatorname{res}(S)$ | $C_2^5.\GL(5,2)$, of order \(319979520\)\(\medspace = 2^{15} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $7$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $F_7$ |