Subgroup ($H$) information
| Description: | $D_4\times \He_3$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(7,8,9)(10,12,11), (2,4)(5,6)(10,12,11)(13,14,15), (7,12,14)(8,10,15)(9,11,13) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^2:C_6\times A_6$ |
| Order: | \(19440\)\(\medspace = 2^{4} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3.S_3^2.A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2^4.\SL(3,3)$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| $W$ | $C_6\times D_6$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $45$ |
| Möbius function | $-1$ |
| Projective image | $C_3^2:C_6\times A_6$ |