Subgroup ($H$) information
| Description: | $C_3^2:D_6$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$ade^{2}, ef^{3}, b^{3}, b^{2}def^{5}, cde$
|
| Derived length: | $3$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and rational.
Ambient group ($G$) information
| Description: | $C_3^4:S_3^2$ |
| Order: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 27T466.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.S_3^3$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $\He_3:D_4$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_1$ | ||||
| Normalizer: | $C_3^2:D_6$ | ||||
| Normal closure: | $C_3^4:S_3^2$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $C_3^2:S_3^2$ | ||||
| Maximal under-subgroups: | $C_3^2:C_6$ | $C_3^2:C_6$ | $C_3^2:S_3$ | $S_3^2$ | $S_3^2$ |
Other information
| Number of subgroups in this conjugacy class | $27$ |
| Möbius function | not computed |
| Projective image | $C_3^4:S_3^2$ |