Subgroup ($H$) information
| Description: | $D_9\wr C_4.C_3$ |
| Order: | \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| Index: | \(2\) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$a^{3}b^{3}d^{24}e^{6}f^{3}, g^{3}, b^{3}d^{17}e^{2}f^{7}g^{4}, d^{12}g^{6}, b^{2}e^{6}f^{3}g^{8}, c^{2}d^{16}e^{4}fg^{6}, d^{18}e^{6}g^{4}, d^{24}e^{6}fg^{2}, a^{2}, e^{3}g^{6}, f^{3}g^{6}, d^{24}e^{3}f^{3}g^{7}, c^{3}d^{10}e^{2}f^{2}g^{8}, e^{4}g^{2}, d^{28}f^{3}g^{8}$
|
| Derived length: | $3$ |
The subgroup is normal, maximal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $D_9\wr C_4.C_6$ |
| Order: | \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| 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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9^4.C_4:D_4.C_6.C_2^2$, of order \(5038848\)\(\medspace = 2^{8} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $D_9\wr C_4.C_6$, of order \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \) |
| $W$ | $D_9\wr C_4.C_6$, of order \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $D_9\wr C_4.C_6$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $D_9\wr C_4.C_6$ |