Subgroup ($H$) information
| Description: | $C_3^2\times \He_3$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$\left(\begin{array}{rr}
25 & 24 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
25 & 0 \\
0 & 25
\end{array}\right), \left(\begin{array}{rr}
25 & 4 \\
24 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 15 \\
27 & 10
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a $3$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_2\times C_3^3:C_6^2$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_6^2.C_3^3.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_3^6.(C_3^2:\GL(2,3)\times \GL(2,3))$, of order \(15116544\)\(\medspace = 2^{8} \cdot 3^{10} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^3:D_6^2$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $W$ | $C_3\times S_3$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-8$ |
| Projective image | $C_2\times C_3^3:D_6$ |