Subgroup ($H$) information
| Description: | $C_3^2\wr C_2$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(2,5,3)(7,8,9)(10,11,14), (10,11)(13,15), (1,6,4)(2,3,5)(7,8,9)(10,14,11), (10,14,11)(12,15,13), (10,14,11)\rangle$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^4:(D_4\times D_6)$ |
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| 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: | $C_2^2\times D_6$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3\times C_2^3:\GL(3,2)$, of order \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \) |
| Outer Automorphisms: | $C_2^3:\GL(3,2)$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, an A-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3:S_3\times \He_3).D_4^2$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)\times \GL(2,3)$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $W$ | $S_3^3:C_2^2$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-192$ |
| Projective image | $C_3^4:(D_4\times D_6)$ |