Subgroup ($H$) information
| Description: | $C_{54}$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Generators: |
$\left(\begin{array}{rr}
80 & 0 \\
0 & 80
\end{array}\right), \left(\begin{array}{rr}
19 & 7 \\
51 & 52
\end{array}\right), \left(\begin{array}{rr}
28 & 9 \\
54 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 27 \\
0 & 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{18}.C_3^4$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Exponent: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3^3$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(3\) |
| Automorphism Group: | $\GL(3,3)$, of order \(11232\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 13 \) |
| Outer Automorphisms: | $\GL(3,3)$, of order \(11232\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 13 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.(C_3\times Q_8).C_3^4.C_2^2$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_3^2\times C_{54}$ | ||
| Normalizer: | $C_{18}.C_3^4$ | ||
| Minimal over-subgroups: | $C_3\times C_{54}$ | $C_3\times C_{54}$ | $C_{27}:C_6$ |
| Maximal under-subgroups: | $C_{27}$ | $C_{18}$ |
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $24$ |
| Möbius function | $-27$ |
| Projective image | $C_3^4$ |