Subgroup ($H$) information
| Description: | $C_5\times \SL(2,3)$ |
| Order: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Index: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rrrr}
6 & 8 & 8 & 0 \\
6 & 3 & 8 & 8 \\
8 & 3 & 9 & 3 \\
8 & 8 & 5 & 6
\end{array}\right), \left(\begin{array}{rrrr}
6 & 9 & 5 & 9 \\
0 & 6 & 0 & 5 \\
8 & 8 & 5 & 2 \\
0 & 8 & 0 & 5
\end{array}\right), \left(\begin{array}{rrrr}
3 & 6 & 6 & 1 \\
8 & 1 & 3 & 6 \\
9 & 3 & 2 & 5 \\
7 & 9 & 3 & 0
\end{array}\right), \left(\begin{array}{rrrr}
7 & 4 & 7 & 0 \\
6 & 0 & 7 & 7 \\
2 & 7 & 0 & 7 \\
3 & 2 & 5 & 4
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
0 & 10 & 0 & 0 \\
0 & 0 & 10 & 0 \\
0 & 0 & 0 & 10
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is the commutator subgroup (hence characteristic and normal), nonabelian, and solvable.
Ambient group ($G$) information
| Description: | $C_5\times \GL(2,3):D_{10}$ |
| Order: | \(4800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
Quotient group ($Q$) structure
| Description: | $C_2^2\times C_{10}$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4\times \GL(3,2)$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_4\times \GL(3,2)$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times A_4\times F_5).C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_4\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| $W$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $8$ |
| Projective image | not computed |