Subgroup ($H$) information
| Description: | $C_3\wr A_4$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{2}, d, c^{2}, b^{6}, e, b^{9}cd^{2}e, c^{3}d^{2}e$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, and monomial (hence solvable).
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: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, 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 \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $S_3^4:S_3$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $W$ | $C_3^3:(S_3\times A_4)$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | $C_3$ | |||
| Normalizer: | $C_9:S_3\wr C_3$ | |||
| Minimal over-subgroups: | $(C_3^3\times C_9):A_4$ | $C_3^3:(S_3\times A_4)$ | ||
| Maximal under-subgroups: | $C_3\wr C_2^2$ | $C_3^3:A_4$ | $C_3^4:C_3$ | $C_3\times A_4$ |
Other information
| Möbius function | $3$ |
| Projective image | $C_9:S_3\wr C_3$ |