Subgroup ($H$) information
| Description: | $D_4\times C_2^3$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
31 & 20 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
29 & 10 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
19 & 0 \\
20 & 19
\end{array}\right), \left(\begin{array}{rr}
29 & 20 \\
20 & 29
\end{array}\right), \left(\begin{array}{rr}
21 & 0 \\
0 & 21
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $D_{10}.C_2^6$ |
| 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 $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $F_5$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Automorphism Group: | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(220200960\)\(\medspace = 2^{21} \cdot 3 \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^5.C_2^7:\GL(3,2)$, of order \(688128\)\(\medspace = 2^{15} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(688128\)\(\medspace = 2^{15} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2^2\times F_5$ |