Subgroup ($H$) information
| Description: | $C_3^3$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$\left(\begin{array}{rr}
61 & 0 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
61 & 30 \\
0 & 61
\end{array}\right), \left(\begin{array}{rr}
1 & 60 \\
0 & 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_3^5.C_6$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3^2:C_6$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2\times C_3^2:\GL(2,3)$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.(C_3^3\times Q_8).C_3^4.C_2^3$, of order \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $\GL(3,3)$, of order \(11232\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 13 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_3^4:C_6$ | |||
| Normalizer: | $C_3^5.C_6$ | |||
| Minimal over-subgroups: | $C_3^4$ | $C_9:C_3^2$ | $C_9:C_3^2$ | $C_3^2\times C_6$ |
| Maximal under-subgroups: | $C_3^2$ | $C_3^2$ | $C_3^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $27$ |
| Projective image | $C_3^2\wr C_2$ |