Subgroup ($H$) information
| Description: | $C_3\times C_{78}$ |
| Order: | \(234\)\(\medspace = 2 \cdot 3^{2} \cdot 13 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Generators: |
$d^{39}, c, d^{6}, b$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_3^2:D_{78}$ |
| Order: | \(1404\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 13 \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSU(3,2).C_{39}.C_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_{12}\times \GL(2,3)$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $C_{12}\times D_6$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(234\)\(\medspace = 2 \cdot 3^{2} \cdot 13 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $3$ |
| Projective image | $C_3:D_{39}$ |