Subgroup ($H$) information
| Description: | $C_3^5$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | \(2\) |
| Exponent: | \(3\) |
| Generators: |
$\left(\begin{array}{rr}
13 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
10 & 9 \\
9 & 1
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
13 & 12 \\
6 & 13
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
0 & 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, maximal, a semidirect factor, abelian (hence metabelian and an A-group), a $3$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $S_3\times C_3^4$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2.\PSL(4,3).C_2\times S_3$, of order \(145566720\)\(\medspace = 2^{10} \cdot 3^{7} \cdot 5 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $\GL(5,3)$, of order \(475566474240\)\(\medspace = 2^{10} \cdot 3^{10} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(48522240\)\(\medspace = 2^{10} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_3^5$ | ||
| Normalizer: | $S_3\times C_3^4$ | ||
| Complements: | $C_2$ | ||
| Minimal over-subgroups: | $S_3\times C_3^4$ | ||
| Maximal under-subgroups: | $C_3^4$ | $C_3^4$ | $C_3^4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $S_3$ |