Subgroup ($H$) information
| Description: | $C_9:D_9^3$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Index: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$d^{9}e^{11}f^{4}g^{6}, e^{6}g^{3}, e^{2}g, f^{2}g^{5}, f^{6}g^{6}, e^{9}f^{11}g^{7}, d^{14}e^{2}f^{2}g, d^{6}e^{6}f^{6}g^{3}, g^{4}, g^{3}, c^{3}d^{16}e^{14}f^{2}g^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^7.S_3\wr C_2^2$ |
| Order: | \(11337408\)\(\medspace = 2^{6} \cdot 3^{11} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_6\times S_3^2$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6^2:C_2^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.(C_3\times S_3\wr D_4)$, of order \(22674816\)\(\medspace = 2^{7} \cdot 3^{11} \) |
| $\operatorname{Aut}(H)$ | $C_9^4.C_6\wr S_4$, of order \(204073344\)\(\medspace = 2^{7} \cdot 3^{13} \) |
| $W$ | $C_3^7.S_3\wr C_2^2$, of order \(11337408\)\(\medspace = 2^{6} \cdot 3^{11} \) |
Related subgroups
| Centralizer: | $C_1$ |
| Normalizer: | $C_3^7.S_3\wr C_2^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7.S_3\wr C_2^2$ |