Subgroup ($H$) information
| Description: | $C_3^6.S_3^2:S_3^2$ |
| Order: | \(944784\)\(\medspace = 2^{4} \cdot 3^{10} \) |
| Index: | $1$ |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,24,21,18,14,12,31,4,27,34,9,28,2,22,19,17,15,10,32,6,25,35,7,30,3,23,20,16,13,11,33,5,26,36,8,29) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a semidirect factor, nonabelian, and a Hall subgroup. Whether it is a direct factor or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.S_3^2:S_3^2$ |
| Order: | \(944784\)\(\medspace = 2^{4} \cdot 3^{10} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_3^4.C_6^3.C_2^2$, of order \(51018336\)\(\medspace = 2^{5} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | $C_3^6.C_3^4.C_6^3.C_2^2$, of order \(51018336\)\(\medspace = 2^{5} \cdot 3^{13} \) |
| $W$ | $C_3^6.S_3^2:S_3^2$, of order \(944784\)\(\medspace = 2^{4} \cdot 3^{10} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^6.S_3^2:S_3^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.S_3^2:S_3^2$ |