Subgroup ($H$) information
| Description: | $C_2^5:C_{10}$ |
| Order: | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| Index: | \(3\) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
43 & 30 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 22 \\
22 & 1
\end{array}\right), \left(\begin{array}{rr}
21 & 22 \\
22 & 21
\end{array}\right), \left(\begin{array}{rr}
8 & 17 \\
35 & 14
\end{array}\right), \left(\begin{array}{rr}
1 & 22 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
43 & 22 \\
22 & 43
\end{array}\right), \left(\begin{array}{rr}
25 & 0 \\
0 & 25
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_2^2:D_{12}\times C_{10}$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^6.C_2^5.C_2^4)$ |
| $\operatorname{Aut}(H)$ | $C_4\times C_2^8.S_4.C_2$ |
| $\card{\operatorname{res}(S)}$ | \(16384\)\(\medspace = 2^{14} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2\times D_6$ |