Subgroup ($H$) information
| Description: | $C_3^3:C_3^2$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | $1$ |
| Exponent: | \(3\) |
| Generators: |
$\left(\begin{array}{rrrr}
2 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 \\
2 & 0 & 0 & 0
\end{array}\right), \left(\begin{array}{rrrr}
2 & 2 & 0 & 1 \\
0 & 1 & 0 & 0 \\
1 & 2 & 1 & 1 \\
2 & 1 & 0 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 2 & 1 & 1 \\
0 & 1 & 0 & 0 \\
0 & 2 & 2 & 1 \\
0 & 1 & 2 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 \\
0 & 2 & 1 & 2
\end{array}\right), \left(\begin{array}{rrrr}
0 & 1 & 0 & 2 \\
0 & 2 & 2 & 2 \\
2 & 0 & 2 & 0 \\
1 & 1 & 1 & 0
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the radical, a direct factor, nonabelian, a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^3:C_3^2$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Exponent: | \(3\) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $0$ |
| 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 $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:S_3.\SO(5,3)$, of order \(8398080\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_3^3:S_3.\SO(5,3)$, of order \(8398080\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 5 \) |
| $W$ | $C_3^4$, of order \(81\)\(\medspace = 3^{4} \) |
Related subgroups
| Centralizer: | $C_3$ |
| Normalizer: | $C_3^3:C_3^2$ |
| Complements: | $C_1$ |
| Maximal under-subgroups: | $C_3\times \He_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3^4$ |