Subgroup ($H$) information
| Description: | $A_5$ |
| Order: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rrr}
1 & 0 & 0 \\
0 & z_{2} & z_{2} \\
0 & 1 & z_{2}
\end{array}\right), \left(\begin{array}{rrr}
0 & 1 & 1 \\
1 & z_{2} & z_{2} + 1 \\
0 & z_{2} + 1 & z_{2}
\end{array}\right)$
|
| Derived length: | $0$ |
The subgroup is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
Ambient group ($G$) information
| Description: | $C_3.A_6$ |
| Order: | \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and quasisimple (hence nonsolvable and perfect).
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 18T262.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $S_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $A_5$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_3$ | ||
| Normalizer: | $\GL(2,4)$ | ||
| Normal closure: | $C_3.A_6$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $\GL(2,4)$ | ||
| Maximal under-subgroups: | $A_4$ | $D_5$ | $S_3$ |
| Autjugate subgroups: | 1080.260.18.a1.b1 |
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | $C_3.A_6$ |