Subgroup ($H$) information
| Description: | $C_3^4:C_6$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(7,8,9)(10,11,12)(13,14,15), (10,12,11)(13,14,15), (1,3,2), (1,2)(4,5), (7,14,11)(8,15,12)(9,13,10), (1,3,2)(4,6,5)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^5:D_6$ |
| Order: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times (C_3\times \PSU(3,2)).S_3^3$ |
| $\operatorname{Aut}(H)$ | $\AGL(2,3)^2$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_3^3:S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3^3:S_3^2$ |