Subgroup ($H$) information
| Description: | $C_9:S_3\wr C_3$ |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Index: | $1$ |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{3}, e, b^{8}, b^{6}, a^{2}, c^{3}d^{2}e, d, c^{2}, b^{9}$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, and monomial.
Ambient group ($G$) information
| Description: | $C_9:S_3\wr C_3$ |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_6^2.C_6.C_2$, of order \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_6^2.C_6.C_2$, of order \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| $W$ | $C_9:S_3\wr C_3$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | $C_1$ | ||||
| Normalizer: | $C_9:S_3\wr C_3$ | ||||
| Complements: | $C_1$ | ||||
| Maximal under-subgroups: | $(C_3^3\times C_9):A_4$ | $C_9:S_3^3$ | $C_3^3:(S_3\times A_4)$ | $(C_3^3\times C_9):C_6$ | $D_9:A_4$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_9:S_3\wr C_3$ |