Subgroup ($H$) information
| Description: | $C_2^3\times C_{100}$ |
| Order: | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| Index: | \(2\) |
| Exponent: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
51 & 0 \\
0 & 51
\end{array}\right), \left(\begin{array}{rr}
49 & 0 \\
0 & 49
\end{array}\right), \left(\begin{array}{rr}
1 & 50 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 50 \\
50 & 1
\end{array}\right), \left(\begin{array}{rr}
93 & 50 \\
50 & 43
\end{array}\right), \left(\begin{array}{rr}
1 & 60 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
0 & 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), maximal, a semidirect factor, abelian (hence metabelian and an A-group), and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_{100}:C_2^4$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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), a $p$-group, simple, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{50}.(C_2^4\times C_{20}).C_2^6.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $(C_2\times C_{20}).C_2^6.\PSL(2,7)$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(430080\)\(\medspace = 2^{12} \cdot 3 \cdot 5 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $D_{25}$ |