Subgroup ($H$) information
| Description: | $S_3\times C_3^2.A_4$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$b^{9}, d^{3}, cd^{3}, a^{2}cd^{3}, d^{2}, b^{14}d^{3}, b^{6}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_6^3.D_6$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6^2.(C_6\times S_3).C_2$ |
| $\operatorname{Aut}(H)$ | $(A_4\times \He_3).D_6$ |
| $\operatorname{res}(S)$ | $C_6^2:D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_6^2:D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $C_6^3.D_6$ |