Subgroup ($H$) information
| Description: | $C_3^3:S_3$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(10,13,11)(12,14,15), (10,11,13)(12,14,15)(16,17,18), (1,8,4)(2,3,5)(6,7,9), (2,9)(3,7)(4,8)(5,6)(11,13)(14,15)(17,18), (1,3,7)(2,6,4)(5,9,8)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $\He_3^2:C_2^3$ |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and rational.
Quotient group ($Q$) structure
| Description: | $C_6:S_3$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2\times C_3^2:\GL(2,3)$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4:D_6\wr C_2$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $\AGL(4,3)$, of order \(1965150720\)\(\medspace = 2^{9} \cdot 3^{10} \cdot 5 \cdot 13 \) |
| $W$ | $\He_3^2:C_2^3$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $54$ |
| Projective image | $\He_3^2:C_2^3$ |