Subgroup ($H$) information
| Description: | $C_2^3\times C_{20}$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 30 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 20 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
29 & 0 \\
20 & 9
\end{array}\right), \left(\begin{array}{rr}
19 & 0 \\
20 & 19
\end{array}\right), \left(\begin{array}{rr}
29 & 0 \\
0 & 29
\end{array}\right), \left(\begin{array}{rr}
1 & 32 \\
0 & 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), a semidirect factor, abelian (hence metabelian and an A-group), and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $D_{10}.C_2^5$ |
| Order: | \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| 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: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^5\times D_5).C_2^6.C_2^2.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $C_2^2.C_2^7:\GL(3,2)$, of order \(86016\)\(\medspace = 2^{12} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2.C_2^7:\GL(3,2)$, of order \(86016\)\(\medspace = 2^{12} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| $W$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2^3\times C_{20}$ | ||
| Normalizer: | $D_{10}.C_2^5$ | ||
| Complements: | $C_4$ | ||
| Minimal over-subgroups: | $C_{20}:C_2^4$ | ||
| Maximal under-subgroups: | $C_2^2\times C_{20}$ | $C_2^3\times C_{10}$ | $C_2^3\times C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2\times F_5$ |