Subgroup ($H$) information
| Description: | $C_3.A_6$ |
| Order: | \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \) |
| Index: | $1$ |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rrr}
z_{2} & z_{2} & z_{2} \\
0 & z_{2} + 1 & z_{2} + 1 \\
z_{2} & z_{2} + 1 & z_{2}
\end{array}\right), \left(\begin{array}{rrr}
1 & 1 & z_{2} \\
z_{2} & z_{2} & z_{2} \\
z_{2} + 1 & z_{2} & 0
\end{array}\right), \left(\begin{array}{rrr}
z_{2} + 1 & 0 & 0 \\
0 & z_{2} + 1 & 0 \\
0 & 0 & z_{2} + 1
\end{array}\right)$
|
| Derived length: | $0$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a direct factor, nonabelian, a $1080$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), and quasisimple (hence nonsolvable and perfect).
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 group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), and perfect.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| $W$ | $A_6$, of order \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_3$ | ||||
| Normalizer: | $C_3.A_6$ | ||||
| Complements: | $C_1$ | ||||
| Maximal under-subgroups: | $\GL(2,4)$ | $\GL(2,4)$ | $\He_3:C_4$ | $C_3\times S_4$ | $C_3\times S_4$ |
Other information
| Möbius function | $1$ |
| Projective image | $A_6$ |