Subgroup ($H$) information
| Description: | $\SL(2,5)$ |
| Order: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
4 & 2 \\
1 & 2
\end{array}\right), \left(\begin{array}{rr}
4 & 0 \\
0 & 4
\end{array}\right), \left(\begin{array}{rr}
1 & 2 \\
4 & 4
\end{array}\right)$
|
| Derived length: | $0$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, nonabelian, and quasisimple (hence nonsolvable and perfect).
Ambient group ($G$) information
| Description: | $\GL(2,5)$ |
| Order: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 \(2\) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times S_5$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $S_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_4$ | ||
| Normalizer: | $\GL(2,5)$ | ||
| Complements: | $C_4$ $C_4$ | ||
| Minimal over-subgroups: | $\SL(2,5):C_2$ | ||
| Maximal under-subgroups: | $\SL(2,3)$ | $C_5:C_4$ | $C_3:C_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $A_5:C_4$ |