Subgroup ($H$) information
| Description: | $C_3\times D_6^2$ |
| Order: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(7,8,9), (1,6,3)(2,5,4), (3,6)(4,5), (10,12)(11,13), (10,11)(12,13), (1,2)(3,5)(4,6), (2,4,5)\rangle$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $D_6^2:D_6$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2.C_2^4:C_3.C_4.C_2^3\times S_3$ |
| $\operatorname{Aut}(H)$ | $D_6^2:(C_2^2\times S_4)$, of order \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_2\times D_6^2:D_6$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $S_3^2:C_2^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $2$ |
| Projective image | $S_3^3:C_2$ |