Subgroup ($H$) information
| Description: | $C_3^3:D_{18}$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Index: | \(2\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$b^{3}, e^{4}, d, c^{3}, e^{3}, b^{2}, c^{2}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_9:S_3^3$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:C_2^2.D_6\times D_9:C_3$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $S_3\wr C_2\times C_2\times D_9:C_3$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $\card{\operatorname{res}(S)}$ | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $D_9\times S_3^2$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_3$ | ||||
| Normalizer: | $C_9:S_3^3$ | ||||
| Complements: | $C_2$ $C_2$ $C_2$ | ||||
| Minimal over-subgroups: | $C_9:S_3^3$ | ||||
| Maximal under-subgroups: | $C_3^3:C_{18}$ | $C_3^3:D_9$ | $C_9:S_3^2$ | $C_3^2:D_{18}$ | $C_3\wr C_2^2$ |
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $C_9:S_3^3$ |