Subgroup ($H$) information
| Description: | $C_9:D_9^3$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Index: | \(2\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$abc^{4}d^{6}ef^{6}, d^{3}e^{6}f^{6}, de^{8}f^{5}, c^{12}e^{7}, e^{3}, c^{9}, c^{8}e^{2}, c^{6}e^{6}, f^{7}, f^{3}, b^{2}c^{4}e$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_9^4:(C_2\times D_4)$ |
| Order: | \(104976\)\(\medspace = 2^{4} \cdot 3^{8} \) |
| 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: | $C_2$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9^4.(C_6\times D_4).C_6.C_2^3$, of order \(15116544\)\(\medspace = 2^{8} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_9^4.C_6\wr S_4$, of order \(204073344\)\(\medspace = 2^{7} \cdot 3^{13} \) |
| $W$ | $C_9^4:(C_2\times D_4)$, of order \(104976\)\(\medspace = 2^{4} \cdot 3^{8} \) |
Related subgroups
| Centralizer: | $C_1$ | |||
| Normalizer: | $C_9^4:(C_2\times D_4)$ | |||
| Complements: | $C_2$ | |||
| Minimal over-subgroups: | $C_9^4:(C_2\times D_4)$ | |||
| Maximal under-subgroups: | $C_9^4.C_2^2$ | $C_9^4.C_2^2$ | $C_9\wr C_2^2$ | $C_3:D_9^3$ |
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 | $C_9^4:(C_2\times D_4)$ |