Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(3\) |
| Generators: |
$f$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), the Frattini subgroup, the socle, cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), stem, a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $\Unitary(3,2):C_3$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable.
Quotient group ($Q$) structure
| Description: | $C_3^3:\SL(2,3)$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $S_3\times C_3^2:\GL(2,3)$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Outer Automorphisms: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and solvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4:\SL(2,3)$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $\Unitary(3,2):C_3$ | |||||||||
| Normalizer: | $\Unitary(3,2):C_3$ | |||||||||
| Minimal over-subgroups: | $C_9$ | $C_3^2$ | $C_3^2$ | $C_3^2$ | $C_9$ | $C_9$ | $C_9$ | $C_9$ | $C_9$ | $C_6$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_3^3:\SL(2,3)$ |