Subgroup ($H$) information
| Description: | $C_3^5:D_6$ |
| Order: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(10,12)(11,14)(13,15)(16,18), (1,2,8)(3,7,6)(4,9,5)(10,11,13)(12,15,14) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^4:S_3^3$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{17}^2:(C_2\times C_{24})$, of order \(139968\)\(\medspace = 2^{6} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3\times (C_3\times \PSU(3,2)).S_3^3$ |
| $W$ | $C_3^2:S_3^3$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3^4:S_3^3$ |