Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(468\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 13 \) |
| Exponent: | \(3\) |
| Generators: |
$c$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), stem (hence central), a $p$-group, and simple.
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: | $C_3:D_{78}$ |
| Order: | \(468\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 13 \) |
| Exponent: | \(78\)\(\medspace = 2 \cdot 3 \cdot 13 \) |
| Automorphism Group: | $\PSU(3,2).C_{39}.C_6.C_2^3$ |
| Outer Automorphisms: | $C_2\times \GL(2,3):C_6$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSU(3,2).C_{39}.C_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(67392\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 13 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3^2:D_{78}$ | |||||||
| Normalizer: | $C_3^2:D_{78}$ | |||||||
| Minimal over-subgroups: | $C_{39}$ | $C_3^2$ | $C_3^2$ | $C_3^2$ | $C_3^2$ | $C_6$ | $C_6$ | $C_6$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $-702$ |
| Projective image | $C_3:D_{78}$ |