Subgroup ($H$) information
| Description: | $C_3^2:A_4$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,4)(2,3), (10,17,13)(12,14,16), (1,4,2)(10,16,11)(12,18,13)(14,15,17), (11,15,18)(12,14,16), (1,2)(3,4)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_6^3:S_3^2$ |
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| 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: | $S_3\times D_6$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6\wr C_2$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6^2.C_3^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_6^2:S_3^2$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| $W$ | $C_3^2:S_4$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-72$ |
| Projective image | $C_6^3:S_3^2$ |