Subgroup ($H$) information
| Description: | $C_3^5.D_6$ |
| Order: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Index: | \(2\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{3}d^{9}, c^{2}d^{2}, e, b^{3}, b^{2}c, d^{6}, c, a^{2}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^3.S_3^3$ |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.S_3^3$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_3^3.C_3^4.C_2^3$, of order \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| $W$ | $C_3^3.S_3^3$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3^3.S_3^3$ |